From 6cf80341770e0980e52481c8c5764e60f0ee971a Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Fri, 26 Jul 2024 19:57:46 +0200
Subject: [PATCH 01/54] push imu transformations alpha labs article for review

---
 alpha-lab/cards.json                   |  10 +
 alpha-lab/imu-transformations/index.md | 299 +++++++++++++++++++++++++
 2 files changed, 309 insertions(+)
 create mode 100644 alpha-lab/imu-transformations/index.md

diff --git a/alpha-lab/cards.json b/alpha-lab/cards.json
index 780507724..38b816834 100644
--- a/alpha-lab/cards.json
+++ b/alpha-lab/cards.json
@@ -148,5 +148,15 @@
     },
     "image": "/alpha-lab/tag-aligner.webp",
     "category": "Gaze Mapping"
+  },
+  {
+    "title": "IMU Coordinate System Snippets",
+    "details": "Use values from the IMU to express your data in a world reference frame.",
+    "link": {
+      "text": "View",
+      "href": "/alpha-lab/imu-transformations/"
+    },
+    "image": "/alpha-lab/imu-transformations.webp",
+    "category": "Coordinate Systems"
   }
 ]
diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
new file mode 100644
index 000000000..570a5bec6
--- /dev/null
+++ b/alpha-lab/imu-transformations/index.md
@@ -0,0 +1,299 @@
+---
+title: "IMU Coordinate System Snippets"
+description: "Use values from the IMU to express your data in a world reference frame."
+permalink: /alpha-lab/imu-transformations/
+meta:
+  - name: twitter:card
+    content: summary
+  - name: twitter:image
+    content: "https://i.ytimg.com/vi/nt_zNSBMJWI/maxresdefault.jpg"
+  - name: twitter:player
+    content: "https://www.youtube.com/embed/jag9EQB7840"
+  - name: twitter:width
+    content: "1280"
+  - name: twitter:height
+    content: "720"
+  - property: og:image
+    content: "https://i.ytimg.com/vi/nt_zNSBMJWI/maxresdefault.jpg"
+tags: [Neon, Cloud]
+---
+
+<script setup>
+import TagLinks from '@components/TagLinks.vue'
+</script>
+
+# IMU Coordinate System Snippets
+
+<TagLinks :tags="$frontmatter.tags" />
+
+<!-- <Youtube src="3N8jGLYCrNk"/> -->
+
+::: tip
+Compare IMU and gaze data in the same coordinate system to better understand how people coordinate head and eye movements.
+:::
+
+Here, we will provide some code snippets that can assist when analyzing the IMU data from Neon.
+
+We will use Python with the NumPy and SciPy packages for the code snippets below. Let’s start by loading those:
+
+```python
+import numpy as np
+from scipy.spatial.transform import Rotation as R
+```
+
+First, it is important to understand how the IMU data should be interpreted. The [orientation readings of the calibrated IMU](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data) are specified with respect to magnetic North and gravity:
+
+- Y axis = vector pointing towards magnetic North.
+  - Rotations about this axis are called “roll” and range from -180 to +180 degrees. Wearing Neon normally and sitting or standing upright roughly corresponds to a roll of 0 degrees. Rightward head tilt is positive roll; leftward head tilt is negative roll.
+- Z axis = vector pointing upwards, opposite to the direction of gravity.
+  - Rotations about this axis are called “yaw” and range from -180 to +180 degrees. When the IMU is calibrated, a Neon module oriented at magnetic North corresponds to a yaw of 0 degrees. Leftward head turn is positive yaw; rightward head turn is negative yaw.
+- X axis = cross-product of Y and Z, which is a vector pointing rightwards.
+  - Rotations about this axis are called “pitch” and range from -90 to +90 degrees. Wearing Neon normally and sitting or standing upright roughly corresponds to a pitch of 0 degrees. Backward head tilt is positive pitch; forward head tilt is negative pitch
+
+This coordinate system is known as the “global reference frame”. Here, we will refer to it as the world coordinate system. To be very exact, it is distinct from the IMU coordinate system. As the Neon module rotates, the IMU coordinate system rotates with it. The IMU actually measures the rotational difference between its coordinate system and the world coordinate system.
+
+The gyroscope values give the rotational velocity of roll, yaw, and pitch in degrees/s (i.e., they provide the change over time of each of these quantities).
+
+Note that Neon can sit differently on each wearer’s face, such that the headset is not necessarily in line with the naso-occiptal plane. For example, if the wearer is looking at magnetic North, the IMU might still report some deviation from neural orientation. 
+
+Now that we have laid out the relationship between the IMU and world coordinate systems, we can do some useful transformations.
+
+## Obtain IMU heading vectors
+
+An alternate representation of IMU data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to gravity.
+
+Note that a neutral IMU heading vector is unlikely to be perfectly aligned with the floor.
+
+:::: details Code
+```python
+def imu_heading_in_world(imu_quaternions):
+  """
+  Construct heading vectors from the IMU's quaternion values.
+        
+  Inputs:
+      - imu_quaternions (Nx4 np.array): A timeseries of quaternions
+      from Neon's IMU stream.
+        
+  Returns:
+      - headings_in_world (Nx3 np.array): A timeseries of IMU heading vectors
+      in the world coordinate system.
+  """
+
+  # We start by specifying the direction of a neutral heading vector
+  # in the IMU's coordinate system.
+  heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
+
+  # This array contains a timeseries of transformation matrices,
+  # as calculated from the IMU's timeseries of quaternions values.
+  # Each of these matrices sends points in the IMU coordinate system to their
+  # corresponding coordinates in the world coordinate system.
+  imu_to_world = R.from_quat(imu_quaternions).as_matrix()
+    
+  # We now apply each transformation matrix to the neutral IMU heading vector
+  # to obtain a timeseries of heading vectors in the world coordinate system.
+  headings_in_world = imu_to_world @ heading_neutral_in_imu_coords
+
+  return headings_in_world
+```
+::::
+
+## Transform IMU acceleration data to world coordinates
+
+We stated above that the Z axis of the world coordinate system is a vector pointing directly upwards, opposite gravity. However, the Z component of the IMU acceleration data points downwards, such that if you are standing on Earth, then positive acceleration values mean that you are accelerating towards the center of the Earth. The X and Y axes are the same and shared between the orientation and acceleration data provided by the IMU. The function below will transform the acceleration data to the world coordinate system described above.
+
+:::: details Code
+```python
+def imu_acceleration_in_world(accelerations_imu):
+  """
+  Transform the IMU's acceleration values to the world coordinate system.
+    
+  Inputs:
+    - accelerations_imu (Nx3 np.array): Timeseries of acceleration data
+    from the IMU, where columns are in the order: accel_x, accel_y, accel_z.
+    
+  Returns:
+    - accelerations_world (Nx3 np.array): Timeseries of acceleration data,
+    expressed in the world coordinate system.
+  """
+    
+  accelerations_world = accelerations_imu.copy()
+  accelerations_world[2] *= -1.0
+    
+  return accelerations_world
+```
+::::
+
+## Represent IMU and Gaze Data in the Same Coordinate System
+
+Neon simultaneously records gaze and IMU data, making it possible to study the relationship between head and eye movements.
+
+To facilitate comparison of gaze and head rotations, it can sometimes be easier to represent them in the same coordinate system. The coordinates of gaze are specified with respect to the scene camera coordinate system and the function below, `gaze_scene_to_world`, uses data from the IMU to transform gaze to the world coordinate system.
+
+::: tip
+💡 Reminder: The world coordinate system is defined by the direction away from the center of gravity, as well as the direction towards magnetic North.
+:::
+
+:::: details Code
+```python
+def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
+  """
+  Transform a 3D gaze ray to the world coordinate system.
+    
+  Note that the gaze data and the IMU quaternion should be sampled 
+  at the same timestamps. You can linearly interpolate the IMU data
+  to ensure this.
+    
+  The origin of the IMU coordinate system is the same as the
+  origin of the world coordinate system.
+    
+  This function makes use of the spherical_to_cartesian function,
+  defined below, that converts 3D gaze rays from spherical coordinates
+  to Cartesian coordinates.
+    
+  Inputs:
+    - gaze_elevations (Nx1 np.array): A timeseries of gaze elevations (degrees),
+    specified in the scene camera coordinate system.
+    - gaze_azimuths (Nx1 np.array): A timeseries of gaze azimuths (degrees),
+    specified in the scene camera coordinate system.
+    - imu_quaternions (Nx4 np.array): A timeseries of quaternion values
+    from Neon's IMU.
+    
+  Returns:
+    - gazes_in_world (Nx3 np.array): A timeseries of 3D Cartesian gaze
+    unit vectors, specified in the world coordinate system.
+  """
+    
+  # The IMU and scene camera coordinate systems have a fixed
+  # 102 degree rotation offset.
+  imu_scene_rotation_diff = np.deg2rad(-90 - 12)
+  
+  # This matrix takes points in the scene camera coordinate
+  # system and returns their corresponding coordinates in the
+  # IMU coordinate system.
+  scene_to_imu = np.array(
+      [
+          [1.0, 0.0, 0.0],
+          [
+              0.0,
+              np.cos(imu_scene_rotation_diff),
+              -np.sin(imu_scene_rotation_diff),
+          ],
+          [
+              0.0,
+              np.sin(imu_scene_rotation_diff),
+              np.cos(imu_scene_rotation_diff),
+          ],
+      ]
+  )
+  
+  # Neon provides 3D gaze in spherical coordinates by default, 
+  # so first transform the gaze data from spherical coordinates
+  # to Cartesian coordinates.
+  cart_gazes_in_scene = spherical_to_cartesian(gaze_elevations, gaze_azimuths)
+    
+  # Apply the transformation from the scene camera to the IMU coordinate system.
+  gazes_in_imu = scene_to_imu @ cart_gazes_in_scene
+  
+  # This array contains a timeseries of transformation matrices,
+  # as calculated from the IMU's timeseries of quaternions values.
+  # Each of these matrices sends points in the IMU coordinate system to their
+  # corresponding coordinates in the world coordinate system.
+  imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
+  
+  # Apply the transformations from the IMU to the world coordinate system.
+  gazes_in_world = [imu_to_world @ gaze for imu_to_world, gaze in zip(imu_to_world_matrices, gazes_in_imu.T)]
+  
+  return np.array(gazes_in_world)
+
+
+def spherical_to_cartesian(elevations, azimuths):
+  """
+  Convert Neon's spherical representation of 3D gaze to Cartesian coordinates.
+    
+  Inputs:
+    - elevations (Nx1 np.array): A timeseries of gaze elevations (degrees),
+    specified in the scene camera coordinate system.
+    - azimuths (Nx1 np.array): A timeseries of gaze azimuths (degrees),
+    specified in the scene camera coordinate system.
+    
+  Returns:
+    - cartesian_unit_vectors (Nx3 np.array): A timeseries of gaze unit
+    vectors, in Cartesian coordinates, specified in the scene camera
+    coordinate system.
+  """
+    
+  elevations_rad = np.deg2rad(elevations)
+  azimuths_rad = np.deg2rad(azimuths)
+
+  # Elevation of 0 in Neon system corresponds to Y = 0, but
+  # an elevation of 0 in traditional spherical coordinates would
+  # correspond to Y = 1, so first we convert elevation to the
+  # more traditional format.
+  elevations_rad = elevations_rad + np.pi / 2
+ 
+  # Azimuth of 0 in Neon system corresponds to X = 0, but
+  # an azimuth of 0 in traditional spherical coordinates would
+  # correspond to X = 1. Also, azimuth to the right in Neon is
+  # more positive, whereas it is more negative in traditional spherical coordiantes.
+  # So, first we convert azimuth to the more traditional format.
+  azimuths_rad = -azimuths_rad + np.pi / 2
+    
+  cartesian_unit_vectors = np.array([
+    np.sin(elevations_rad) * np.cos(azimuths_rad),
+    np.cos(elevations_rad),
+    np.sin(elevations_rad) * np.sin(azimuths_rad),
+  ])
+  
+  return cartesian_unit_vectors
+```
+::::
+
+## Analysis example
+
+Below is a brief example of running these commands on values from Pupil Cloud.
+
+:::: details Code
+```python
+import pandas as pd
+
+gaze = pd.read_csv("gaze.csv")
+imu = pd.read_csv("imu.csv")
+
+gaze_ts = gaze["timestamp [ns]"]
+imu_ts = imu["timestamp [ns]"]
+
+# We have more gaze datapoints (sampled at 200Hz) than
+# IMU datapoints (sampled at 110Hz), so linearly interpolate
+# the IMU datapoints to be congruent with gaze.
+quats_x_resampled = np.interp(gaze_ts, imu_ts, imu["quaternion x"])
+quats_y_resampled = np.interp(gaze_ts, imu_ts, imu["quaternion y"])
+quats_z_resampled = np.interp(gaze_ts, imu_ts, imu["quaternion z"])
+quats_w_resampled = np.interp(gaze_ts, imu_ts, imu["quaternion w"])
+
+quaternions_resampled = np.vstack(
+    [quats_x_resampled, quats_y_resampled, quats_z_resampled, quats_w_resampled]
+).T
+
+accels_x_resampled = np.interp(gaze_ts, imu_ts, imu["acceleration x [G]"])
+accels_y_resampled = np.interp(gaze_ts, imu_ts, imu["acceleration y [G]"])
+accels_z_resampled = np.interp(gaze_ts, imu_ts, imu["acceleration z [G]"])
+
+accelerations_resampled = np.vstack(
+    [accels_x_resampled, accels_y_resampled, accels_z_resampled]
+).T
+
+# Now, we can apply the functions.
+
+imu_headings = imu_heading_in_world(quaternions_resampled)
+world_gazes = gaze_scene_to_world(
+    gaze["elevation [deg]"],
+    gaze["azimuth [deg]"],
+    quaternions_resampled,
+)
+world_accelerations = imu_acceleration_in_world(accelerations_resampled)
+```
+::::
+
+::: tip
+Need assistance with aligning your AprilTags or applying the transformations to your Reference Image Mapper recordings? Or do you have something more custom in mind? Reach out to us via email at [info@pupil-labs.com](mailto:info@pupil-labs.com), on our [Discord server](https://pupil-labs.com/chat/), or visit our [Support Page](https://pupil-labs.com/products/support/) for dedicated support options.
+:::
\ No newline at end of file

From 2fa58eb604d2e158bf288cff042d21e93b200aea Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Fri, 26 Jul 2024 20:26:27 +0200
Subject: [PATCH 02/54] use terminology for matrix application that is in line
 with math literature

---
 alpha-lab/imu-transformations/index.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 570a5bec6..0c5c27ca5 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -167,8 +167,8 @@ def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
   # 102 degree rotation offset.
   imu_scene_rotation_diff = np.deg2rad(-90 - 12)
   
-  # This matrix takes points in the scene camera coordinate
-  # system and returns their corresponding coordinates in the
+  # This matrix send points in the scene camera coordinate
+  # system to their corresponding coordinates in the
   # IMU coordinate system.
   scene_to_imu = np.array(
       [

From 3e9d5bba85d148a13fb402a7a1775d8b59479815 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Fri, 26 Jul 2024 20:28:38 +0200
Subject: [PATCH 03/54] make it easier for users to learn why it is "-90 - 12"
 for scene->to->imu

---
 alpha-lab/imu-transformations/index.md | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 0c5c27ca5..7a52248ca 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -165,6 +165,8 @@ def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
     
   # The IMU and scene camera coordinate systems have a fixed
   # 102 degree rotation offset.
+  # See:
+  # https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data
   imu_scene_rotation_diff = np.deg2rad(-90 - 12)
   
   # This matrix send points in the scene camera coordinate

From 81efe892110b43bcfa7e8216d0c2d094c9a688c7 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Tue, 13 Aug 2024 00:32:57 +0200
Subject: [PATCH 04/54] incorporate latest changes, add supporting images

---
 .../imu-pitch-yaw-roll-black.png              | Bin 0 -> 183939 bytes
 alpha-lab/imu-transformations/index.md        | 155 ++++++++++--------
 alpha-lab/public/imu-transformations.webp     | Bin 0 -> 78386 bytes
 3 files changed, 91 insertions(+), 64 deletions(-)
 create mode 100644 alpha-lab/imu-transformations/imu-pitch-yaw-roll-black.png
 create mode 100644 alpha-lab/public/imu-transformations.webp

diff --git a/alpha-lab/imu-transformations/imu-pitch-yaw-roll-black.png b/alpha-lab/imu-transformations/imu-pitch-yaw-roll-black.png
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HcmV?d00001

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 7a52248ca..1e40f49a0 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -6,15 +6,15 @@ meta:
   - name: twitter:card
     content: summary
   - name: twitter:image
-    content: "https://i.ytimg.com/vi/nt_zNSBMJWI/maxresdefault.jpg"
+    content: "https://i.ytimg.com/vi/DMvmCpCL1ZQ/maxresdefault.jpg"
   - name: twitter:player
-    content: "https://www.youtube.com/embed/jag9EQB7840"
+    content: "https://www.youtube.com/embed/DMvmCpCL1ZQ"
   - name: twitter:width
     content: "1280"
   - name: twitter:height
     content: "720"
   - property: og:image
-    content: "https://i.ytimg.com/vi/nt_zNSBMJWI/maxresdefault.jpg"
+    content: "https://i.ytimg.com/vi/DMvmCpCL1ZQ/maxresdefault.jpg"
 tags: [Neon, Cloud]
 ---
 
@@ -32,7 +32,7 @@ import TagLinks from '@components/TagLinks.vue'
 Compare IMU and gaze data in the same coordinate system to better understand how people coordinate head and eye movements.
 :::
 
-Here, we will provide some code snippets that can assist when analyzing the IMU data from Neon.
+Neon comes equipped with an IMU that can be used to determine head pose. Here, we will provide some code snippets that can assist when analyzing the IMU data from Neon.
 
 We will use Python with the NumPy and SciPy packages for the code snippets below. Let’s start by loading those:
 
@@ -50,17 +50,29 @@ First, it is important to understand how the IMU data should be interpreted. The
 - X axis = cross-product of Y and Z, which is a vector pointing rightwards.
   - Rotations about this axis are called “pitch” and range from -90 to +90 degrees. Wearing Neon normally and sitting or standing upright roughly corresponds to a pitch of 0 degrees. Backward head tilt is positive pitch; forward head tilt is negative pitch
 
-This coordinate system is known as the “global reference frame”. Here, we will refer to it as the world coordinate system. To be very exact, it is distinct from the IMU coordinate system. As the Neon module rotates, the IMU coordinate system rotates with it. The IMU actually measures the rotational difference between its coordinate system and the world coordinate system.
+![Diagram of the three Euler orientation angles measured by Neon's IMU](./imu-pitch-yaw-roll-black.png)
+
+This coordinate system is known as the “global reference frame”. Here, we will refer to it as the world coordinate system. To be very exact, it is distinct from the IMU coordinate system - as the Neon module rotates, the IMU coordinate system rotates with it. The IMU actually measures the rotational difference between its coordinate system and the world coordinate system.
+
+::: tip
+💡 Note that Neon can sit differently on each wearer’s face, such that the headset is not necessarily in line with the naso-occipital plane. For example, if the wearer is looking at magnetic North, the IMU might still report some deviation from neural orientation.
+:::
 
 The gyroscope values give the rotational velocity of roll, yaw, and pitch in degrees/s (i.e., they provide the change over time of each of these quantities).
 
-Note that Neon can sit differently on each wearer’s face, such that the headset is not necessarily in line with the naso-occiptal plane. For example, if the wearer is looking at magnetic North, the IMU might still report some deviation from neural orientation. 
+The acceleration values measure translational (i.e., linear) acceleration, in terms of g-force (units of g), along the X, Y, and Z axes of the IMU’s coordinate system. That is, the reference frame for specifying acceleration values rotates with the IMU.
+
+::: tip
+💡 Note that [g-force](https://en.wikipedia.org/wiki/G-force) is not the same as free-fall acceleration due to gravity. For example, if you are standing on the surface of the Earth, then the g-force is equal and opposite to gravity, keeping you at rest. In that case, if you were wearing Neon normally and standing or sitting upright, then acceleration along the Z axis would be +1g, while it would be 0g along the X and Y axes.
+:::
+
+It can be helpful to also try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems.
 
 Now that we have laid out the relationship between the IMU and world coordinate systems, we can do some useful transformations.
 
 ## Obtain IMU heading vectors
 
-An alternate representation of IMU data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to gravity.
+An alternate representation of IMU data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
 
 Note that a neutral IMU heading vector is unlikely to be perfectly aligned with the floor.
 
@@ -71,12 +83,12 @@ def imu_heading_in_world(imu_quaternions):
   Construct heading vectors from the IMU's quaternion values.
         
   Inputs:
-      - imu_quaternions (Nx4 np.array): A timeseries of quaternions
-      from Neon's IMU stream.
+    - imu_quaternions (Nx4 np.array): A timeseries of quaternions
+    from Neon's IMU stream.
         
   Returns:
-      - headings_in_world (Nx3 np.array): A timeseries of IMU heading vectors
-      in the world coordinate system.
+    - headings_in_world (Nx3 np.array): A timeseries of IMU heading vectors
+    in the world coordinate system.
   """
 
   # We start by specifying the direction of a neutral heading vector
@@ -85,13 +97,14 @@ def imu_heading_in_world(imu_quaternions):
 
   # This array contains a timeseries of transformation matrices,
   # as calculated from the IMU's timeseries of quaternions values.
-  # Each of these matrices sends points in the IMU coordinate system to their
-  # corresponding coordinates in the world coordinate system.
-  imu_to_world = R.from_quat(imu_quaternions).as_matrix()
+  # Each of these matrices are used to transform points in the IMU
+  # coordinate system to their corresponding coordinates in the world
+  # coordinate system.
+  imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
     
   # We now apply each transformation matrix to the neutral IMU heading vector
   # to obtain a timeseries of heading vectors in the world coordinate system.
-  headings_in_world = imu_to_world @ heading_neutral_in_imu_coords
+  headings_in_world = imu_to_world_matrices @ heading_neutral_in_imu_coords
 
   return headings_in_world
 ```
@@ -99,27 +112,37 @@ def imu_heading_in_world(imu_quaternions):
 
 ## Transform IMU acceleration data to world coordinates
 
-We stated above that the Z axis of the world coordinate system is a vector pointing directly upwards, opposite gravity. However, the Z component of the IMU acceleration data points downwards, such that if you are standing on Earth, then positive acceleration values mean that you are accelerating towards the center of the Earth. The X and Y axes are the same and shared between the orientation and acceleration data provided by the IMU. The function below will transform the acceleration data to the world coordinate system described above.
+We mentioned above that the IMU’s acceleration data are specified with respect to the IMU’s coordinate system. Sometimes, it can be useful to have the acceleration data specified in the world coordinate system instead. The function below will perform this transformation.
+
+### 
 
 :::: details Code
 ```python
-def imu_acceleration_in_world(accelerations_imu):
+def imu_acceleration_in_world(imu_accelerations, imu_quaternions):
   """
   Transform the IMU's acceleration values to the world coordinate system.
     
   Inputs:
-    - accelerations_imu (Nx3 np.array): Timeseries of acceleration data
-    from the IMU, where columns are in the order: accel_x, accel_y, accel_z.
+    - accelerations_imu (Nx3 np.array): A timeseries of acceleration data
+    from the IMU, where columns are in the order: "acceleration x",
+    "acceleration y", "acceleration z".
+    - imu_quaternions (Nx4 np.array): A timeseries of quaternions
+    from Neon's IMU stream.
     
   Returns:
     - accelerations_world (Nx3 np.array): Timeseries of acceleration data,
     expressed in the world coordinate system.
   """
     
-  accelerations_world = accelerations_imu.copy()
-  accelerations_world[2] *= -1.0
+  # This array contains a timeseries of transformation matrices,
+  # as calculated from the IMU's timeseries of quaternions values.
+  # Each of these matrices sends points in the IMU coordinate system to their
+  # corresponding coordinates in the world coordinate system.
+  imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
     
-  return accelerations_world
+  accelerations_world = [imu_to_world @ imu_acceleration for imu_to_world, imu_acceleration in zip(imu_to_world_matrices, imu_accelerations)]
+    
+  return np.array(accelerations_world)
 ```
 ::::
 
@@ -145,6 +168,9 @@ def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
     
   The origin of the IMU coordinate system is the same as the
   origin of the world coordinate system.
+
+  The code in this function is adapted from the `plimu` visualization utility:
+  https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279
     
   This function makes use of the spherical_to_cartesian function,
   defined below, that converts 3D gaze rays from spherical coordinates
@@ -164,42 +190,41 @@ def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
   """
     
   # The IMU and scene camera coordinate systems have a fixed
-  # 102 degree rotation offset.
-  # See:
+  # 102 degree rotation offset. See:
   # https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data
   imu_scene_rotation_diff = np.deg2rad(-90 - 12)
   
-  # This matrix send points in the scene camera coordinate
-  # system to their corresponding coordinates in the
-  # IMU coordinate system.
+  # This matrix is used to transform points in the scene
+  # camera coordinate system to their corresponding coordinates
+  # in the IMU coordinate system.
   scene_to_imu = np.array(
       [
           [1.0, 0.0, 0.0],
           [
-              0.0,
-              np.cos(imu_scene_rotation_diff),
-              -np.sin(imu_scene_rotation_diff),
+            0.0,
+            np.cos(imu_scene_rotation_diff),
+            -np.sin(imu_scene_rotation_diff),
           ],
           [
-              0.0,
-              np.sin(imu_scene_rotation_diff),
-              np.cos(imu_scene_rotation_diff),
+            0.0,
+            np.sin(imu_scene_rotation_diff),
+            np.cos(imu_scene_rotation_diff),
           ],
       ]
   )
   
   # Neon provides 3D gaze in spherical coordinates by default, 
-  # so first transform the gaze data from spherical coordinates
+  # so we first transform the gaze data from spherical coordinates
   # to Cartesian coordinates.
   cart_gazes_in_scene = spherical_to_cartesian(gaze_elevations, gaze_azimuths)
     
   # Apply the transformation from the scene camera to the IMU coordinate system.
-  gazes_in_imu = scene_to_imu @ cart_gazes_in_scene
+  gazes_in_imu = scene_to_imu @ cart_gazes_in_scene.T
   
   # This array contains a timeseries of transformation matrices,
   # as calculated from the IMU's timeseries of quaternions values.
-  # Each of these matrices sends points in the IMU coordinate system to their
-  # corresponding coordinates in the world coordinate system.
+  # Each of these matrices are used to transform points in the IMU coordinate
+  # system to their corresponding coordinates in the world coordinate system.
   imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
   
   # Apply the transformations from the IMU to the world coordinate system.
@@ -223,7 +248,7 @@ def spherical_to_cartesian(elevations, azimuths):
     vectors, in Cartesian coordinates, specified in the scene camera
     coordinate system.
   """
-    
+
   elevations_rad = np.deg2rad(elevations)
   azimuths_rad = np.deg2rad(azimuths)
 
@@ -231,21 +256,22 @@ def spherical_to_cartesian(elevations, azimuths):
   # an elevation of 0 in traditional spherical coordinates would
   # correspond to Y = 1, so first we convert elevation to the
   # more traditional format.
-  elevations_rad = elevations_rad + np.pi / 2
+  elevations_rad += np.pi / 2
  
   # Azimuth of 0 in Neon system corresponds to X = 0, but
   # an azimuth of 0 in traditional spherical coordinates would
   # correspond to X = 1. Also, azimuth to the right in Neon is
   # more positive, whereas it is more negative in traditional spherical coordiantes.
   # So, first we convert azimuth to the more traditional format.
-  azimuths_rad = -azimuths_rad + np.pi / 2
-    
+  azimuths_rad *= -1.0
+  azimuths_rad += np.pi / 2
+
   cartesian_unit_vectors = np.array([
     np.sin(elevations_rad) * np.cos(azimuths_rad),
     np.cos(elevations_rad),
     np.sin(elevations_rad) * np.sin(azimuths_rad),
-  ])
-  
+  ]).T
+
   return cartesian_unit_vectors
 ```
 ::::
@@ -267,35 +293,36 @@ imu_ts = imu["timestamp [ns]"]
 # We have more gaze datapoints (sampled at 200Hz) than
 # IMU datapoints (sampled at 110Hz), so linearly interpolate
 # the IMU datapoints to be congruent with gaze.
-quats_x_resampled = np.interp(gaze_ts, imu_ts, imu["quaternion x"])
-quats_y_resampled = np.interp(gaze_ts, imu_ts, imu["quaternion y"])
-quats_z_resampled = np.interp(gaze_ts, imu_ts, imu["quaternion z"])
-quats_w_resampled = np.interp(gaze_ts, imu_ts, imu["quaternion w"])
-
-quaternions_resampled = np.vstack(
-    [quats_x_resampled, quats_y_resampled, quats_z_resampled, quats_w_resampled]
-).T
-
-accels_x_resampled = np.interp(gaze_ts, imu_ts, imu["acceleration x [G]"])
-accels_y_resampled = np.interp(gaze_ts, imu_ts, imu["acceleration y [G]"])
-accels_z_resampled = np.interp(gaze_ts, imu_ts, imu["acceleration z [G]"])
-
-accelerations_resampled = np.vstack(
-    [accels_x_resampled, accels_y_resampled, accels_z_resampled]
-).T
+quaternions_resampled = np.array([
+  np.interp(gaze_ts, imu_ts, imu["quaternion x"]),
+  np.interp(gaze_ts, imu_ts, imu["quaternion y"]),
+  np.interp(gaze_ts, imu_ts, imu["quaternion z"]),
+  np.interp(gaze_ts, imu_ts, imu["quaternion w"]),
+]).T
+
+accelerations_resampled = np.array([
+  np.interp(gaze_ts, imu_ts, imu["acceleration x [g]"]),
+  np.interp(gaze_ts, imu_ts, imu["acceleration y [g]"]),
+  np.interp(gaze_ts, imu_ts, imu["acceleration z [g]"]),
+]).T
 
 # Now, we can apply the functions.
 
 imu_headings = imu_heading_in_world(quaternions_resampled)
+
 world_gazes = gaze_scene_to_world(
-    gaze["elevation [deg]"],
-    gaze["azimuth [deg]"],
-    quaternions_resampled,
+  gaze["elevation [deg]"],
+  gaze["azimuth [deg]"],
+  quaternions_resampled,
+)
+
+world_accelerations = imu_acceleration_in_world(
+  accelerations_resampled,
+  quaternions_resampled,
 )
-world_accelerations = imu_acceleration_in_world(accelerations_resampled)
 ```
 ::::
 
 ::: tip
-Need assistance with aligning your AprilTags or applying the transformations to your Reference Image Mapper recordings? Or do you have something more custom in mind? Reach out to us via email at [info@pupil-labs.com](mailto:info@pupil-labs.com), on our [Discord server](https://pupil-labs.com/chat/), or visit our [Support Page](https://pupil-labs.com/products/support/) for dedicated support options.
+Need assistance with the IMU code in this article? Or do you have something more custom in mind? Reach out to us via email at [info@pupil-labs.com](mailto:info@pupil-labs.com), on our [Discord server](https://pupil-labs.com/chat/), or visit our [Support Page](https://pupil-labs.com/products/support/) for dedicated support options.
 :::
\ No newline at end of file
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From 50f2b25efdfabad8993a873de403e1912b48d95e Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Tue, 13 Aug 2024 01:16:26 +0200
Subject: [PATCH 05/54] add eyestate_to_world function

---
 alpha-lab/imu-transformations/index.md | 117 +++++++++++++++++++++++++
 1 file changed, 117 insertions(+)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 1e40f49a0..f02b622f4 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -276,6 +276,103 @@ def spherical_to_cartesian(elevations, azimuths):
 ```
 ::::
 
+## Represent IMU and 3D Eyestate data in the same coordinate system
+
+The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) provided by Neon are aligned with the scene camera coordinate system. This means we can reuse elements of the `gaze_scene_to_world` function to reconstruct the pose of the eyes as a wearer rotates their head. Essentially, you would be representing eyestate in the world coordinate system of the IMU.
+
+The function below performs that transformation for you, returning eyeball centers and optical axes in the world coordinate system.
+
+:::: details Code
+```python
+def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
+  """
+  Transforms 3D eyestate data to the world coordinate system of the IMU.
+  
+  Note that the 3D eyestate data and the IMU quaternions should be sampled 
+  at the same timestamps. You can linearly interpolate the IMU data
+  to ensure this.
+    
+  The origin of the IMU coordinate system is the same as the
+  origin of the world coordinate system.
+    
+  The code in this function is adapted from the `plimu` visualization utility:
+  https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279
+    
+  Inputs:
+    - eyeball_centers (Nx3 np.array): A timeseries of eyeball center
+    estimates (mm) for one eye, as provided by Neon's 3D eyestate data stream.
+    - optical_axes (Nx3 np.array): A timeseries of optical axis estimates
+    (3D unit vectors) for the same eye, as provided by Neon's 3D eyestate data stream.
+    - imu_quaternions (Nx4 np.array): A timeseries of quaternion values from
+    Neon's IMU.
+    
+  Returns:
+    - eyeball_centers_in_world (Nx3 np.array): A timeseries of eyeball center
+    estimates (mm) for one eye, transformed to the world coordinate system of the IMU.
+    - optical_axes_in_world (Nx3 np.array): A timeseries of optical axis estimates
+    (3D unit vectors) for the same eye, transformed to the world coordinate system of the IMU.
+  """
+  
+  # The eyeball centers are specified in terms of their distance from
+  # the center of the scene camera, so to accurately convert them to
+  # world coordinates, we need the position of the scene camera in
+  # the IMU coordinate system. Here, we express that position
+  # in millimeters.
+  scene_camera_position_in_imu = np.array([[0.0, -3.996, -11.172]])
+
+  # The coordinate system of 3D eyestate is aligned with
+  # the coordinate system of the scene camera, so we can
+  # make use of the scene->to->IMU transformation procedure.
+  #
+  # The IMU and scene camera coordinate systems have a fixed
+  # 102 degree rotation offset. See here:
+  # https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data
+  imu_scene_rotation_diff = np.deg2rad(-90 - 12)
+  
+  # This matrix is used to transform points in the scene
+  # camera coordinate system to their corresponding coordinates
+  # in the IMU coordinate system.
+  scene_to_imu = np.array(
+      [
+        [1.0, 0.0, 0.0],
+        [
+          0.0,
+          np.cos(imu_scene_rotation_diff),
+          -np.sin(imu_scene_rotation_diff),
+        ],
+        [
+          0.0,
+          np.sin(imu_scene_rotation_diff),
+          np.cos(imu_scene_rotation_diff),
+        ],
+      ]
+  )
+  
+  # Convert the coordinates of the eyeball center and optical axes
+  # to the IMU coordinate system.
+  eyeball_centers_in_imu = scene_to_imu @ eyeball_centers.T
+  optical_axes_in_imu = scene_to_imu @ optical_axes.T
+    
+  # Take into account that the scene camera is offset from the IMU.
+  eyeball_centers_in_imu[0, :] += scene_cam_in_imu[0]
+  eyeball_centers_in_imu[1, :] += scene_cam_in_imu[1]
+  eyeball_centers_in_imu[2, :] += scene_cam_in_imu[2]
+  
+  # This array contains a timeseries of transformation matrices,
+  # as calculated from the IMU's timeseries of quaternions values.
+  # Each of these matrices are used to transform points in the IMU
+  # coordinate system to their corresponding coordinates in the world
+  # coordinate system.
+  imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
+  
+  # Apply the transformations from the IMU to the world coordinate system.
+  eyeball_centers_in_world = [imu_to_world @ eye_center for imu_to_world, eye_center in zip(imu_to_world_matrices, eyeball_centers_in_imu.T)]
+  optical_axes_in_world = [imu_to_world @ optical_axis for imu_to_world, optical_axis in zip(imu_to_world_matrices, optical_axes_in_imu.T)]
+    
+  return eyeball_centers_in_world, optical_axes_in_world
+```
+::::
+
 ## Analysis example
 
 Below is a brief example of running these commands on values from Pupil Cloud.
@@ -285,9 +382,11 @@ Below is a brief example of running these commands on values from Pupil Cloud.
 import pandas as pd
 
 gaze = pd.read_csv("gaze.csv")
+eye3d = pd.read_csv("3d_eye_states.csv")
 imu = pd.read_csv("imu.csv")
 
 gaze_ts = gaze["timestamp [ns]"]
+eye3d_ts = eye3d["timestamp [ns]"]
 imu_ts = imu["timestamp [ns]"]
 
 # We have more gaze datapoints (sampled at 200Hz) than
@@ -306,6 +405,18 @@ accelerations_resampled = np.array([
   np.interp(gaze_ts, imu_ts, imu["acceleration z [g]"]),
 ]).T
 
+optical_axes_left = np.array([
+  eye3d["optical axis left x"],
+  eye3d["optical axis left y"],
+  eye3d["optical axis left z"],
+]).T
+
+eyeball_centers_left = np.array([
+  eye3d["eyeball center left x [mm]"],
+  eye3d["eyeball center left y [mm]"],
+  eye3d["eyeball center left z [mm]"],
+]).T
+
 # Now, we can apply the functions.
 
 imu_headings = imu_heading_in_world(quaternions_resampled)
@@ -316,6 +427,12 @@ world_gazes = gaze_scene_to_world(
   quaternions_resampled,
 )
 
+eye_centers_left_world, optical_axes_left_world = eyestate_to_world(
+  eyeball_centers_left,
+  optical_axes_left,
+  quaternions_resampled,
+)
+
 world_accelerations = imu_acceleration_in_world(
   accelerations_resampled,
   quaternions_resampled,

From 8ea73ad44090302def074ad975f332029e67ea66 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Thu, 15 Aug 2024 11:31:06 +0200
Subject: [PATCH 06/54] add cartesian_to_spherical_world function and some
 explanation; small cleanup based on test/validations

---
 alpha-lab/imu-transformations/index.md | 68 +++++++++++++++++++++++---
 1 file changed, 61 insertions(+), 7 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index f02b622f4..668ba5db0 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -216,7 +216,7 @@ def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
   # Neon provides 3D gaze in spherical coordinates by default, 
   # so we first transform the gaze data from spherical coordinates
   # to Cartesian coordinates.
-  cart_gazes_in_scene = spherical_to_cartesian(gaze_elevations, gaze_azimuths)
+  cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
     
   # Apply the transformation from the scene camera to the IMU coordinate system.
   gazes_in_imu = scene_to_imu @ cart_gazes_in_scene.T
@@ -233,7 +233,7 @@ def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
   return np.array(gazes_in_world)
 
 
-def spherical_to_cartesian(elevations, azimuths):
+def spherical_to_cartesian_scene(elevations, azimuths):
   """
   Convert Neon's spherical representation of 3D gaze to Cartesian coordinates.
     
@@ -318,7 +318,7 @@ def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
   # world coordinates, we need the position of the scene camera in
   # the IMU coordinate system. Here, we express that position
   # in millimeters.
-  scene_camera_position_in_imu = np.array([[0.0, -3.996, -11.172]])
+  scene_camera_position_in_imu = np.array([0.0, -3.996, -11.172])
 
   # The coordinate system of 3D eyestate is aligned with
   # the coordinate system of the scene camera, so we can
@@ -354,9 +354,9 @@ def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
   optical_axes_in_imu = scene_to_imu @ optical_axes.T
     
   # Take into account that the scene camera is offset from the IMU.
-  eyeball_centers_in_imu[0, :] += scene_cam_in_imu[0]
-  eyeball_centers_in_imu[1, :] += scene_cam_in_imu[1]
-  eyeball_centers_in_imu[2, :] += scene_cam_in_imu[2]
+  eyeball_centers_in_imu[0, :] += scene_camera_position_in_imu[0]
+  eyeball_centers_in_imu[1, :] += scene_camera_position_in_imu[1]
+  eyeball_centers_in_imu[2, :] += scene_camera_position_in_imu[2]
   
   # This array contains a timeseries of transformation matrices,
   # as calculated from the IMU's timeseries of quaternions values.
@@ -369,7 +369,59 @@ def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
   eyeball_centers_in_world = [imu_to_world @ eye_center for imu_to_world, eye_center in zip(imu_to_world_matrices, eyeball_centers_in_imu.T)]
   optical_axes_in_world = [imu_to_world @ optical_axis for imu_to_world, optical_axis in zip(imu_to_world_matrices, optical_axes_in_imu.T)]
     
-  return eyeball_centers_in_world, optical_axes_in_world
+  return np.array(eyeball_centers_in_world), np.array(optical_axes_in_world)
+```
+::::
+
+## Convert (Cartesian) world points to spherical coordinates
+
+When studying head orientation and gaze orientation as observers navigate a 3D environment, it can be useful to know how much these quantities deviate from pointing at a given landmark. For instance, you might want to know when someone’s gaze or heading deviates from pointing at the horizon. This can be simplified by representing data in spherical coordinates. The orientation values from the IMU are already in such a format. For the values returned by  gaze_scene_to_world, the function below will do the necessary transformation. When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds to keeping your head neutral while gazing at the horizon.
+
+:::: details Code
+```python
+def cartesian_to_spherical_world(world_points_3d):
+  """
+  Convert points 3D Cartesian world coordinates to spherical coordinates.
+
+  The origin of world coordinate system is the same as the origin of the
+  IMU coordinate system.
+
+  When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds to keeping your head neutral while gazing at the horizon.
+
+  Inputs:
+    - world_points_3d (Nx3 np.array): A collection of 3D points in 
+    Cartesian world coordinates.
+        
+  Returns:
+    - elevation (Nx1 np.array): The corresponding elevation value, in radians,
+    for each 3D world point (i.e., vertical rotation away from neutral orientation).
+    Note: neutral orientation = 0 elevation; orientation upwards is positive;
+    orientation downwards is negative.
+    - azimuth (Nx1 np.array): The corresponding azimuth value, in radians, for
+    each 3D world point (i.e., horizontal rotation away from neutral orientation).
+    Note: neutral orientation = 0 azimuth, orientation rightwards is negative;
+    orientation leftwards is positive.
+  """
+
+  x = world_points_3d[:, 0]
+  y = world_points_3d[:, 1]
+  z = world_points_3d[:, 2]
+
+  radii = np.sqrt(x**2 + y**2 + z**2)
+
+  elevation = -(np.arccos(z / radii) - np.pi / 2)
+  azimuth = np.arctan2(y, x) - np.pi/2
+  
+  # Keep all azimuth values in the range of [-180, 180] to remain consistent
+  # with the yaw orientation values provided by the IMU
+  azimuth[azimuth < -np.pi] += 2 * np.pi
+  azimuth[azimuth > np.pi] -= 2 * np.pi
+
+  # Convert from radians to degrees
+  elevation = np.rad2deg(elevation)
+  azimuth = np.rad2deg(azimuth)
+
+  return elevation, azimuth
 ```
 ::::
 
@@ -437,6 +489,8 @@ world_accelerations = imu_acceleration_in_world(
   accelerations_resampled,
   quaternions_resampled,
 )
+
+gaze_elevations_world, gaze_azimuths_world = cartesian_to_spherical_world(world_gazes)
 ```
 ::::
 

From 43a8e7dbc93fd82bedb1ab044e15eab9130818fa Mon Sep 17 00:00:00 2001
From: N-M-T <neil87@mail.com>
Date: Fri, 16 Aug 2024 12:11:05 +0800
Subject: [PATCH 07/54] @n-m-t 's edits

---
 alpha-lab/.vitepress/config.mts        |  6 ++
 alpha-lab/cards.json                   |  4 +-
 alpha-lab/imu-transformations/index.md | 97 +++++++++++++++-----------
 3 files changed, 64 insertions(+), 43 deletions(-)

diff --git a/alpha-lab/.vitepress/config.mts b/alpha-lab/.vitepress/config.mts
index e98080c8a..7d4e663fc 100644
--- a/alpha-lab/.vitepress/config.mts
+++ b/alpha-lab/.vitepress/config.mts
@@ -6,6 +6,12 @@ import { config as default_config, theme_config as default_theme_config } from "
 let theme_config_additions = {
   sidebar: [
     { text: "Welcome", link: "/" },
+    {
+      text: "Coordinate Systems",
+      items: [
+        { text: "IMU Transformations", link: "/imu-transformations/" },
+      ],
+    },
     {
       text: "Gaze Mapping",
       items: [
diff --git a/alpha-lab/cards.json b/alpha-lab/cards.json
index 38b816834..ed602bfed 100644
--- a/alpha-lab/cards.json
+++ b/alpha-lab/cards.json
@@ -150,8 +150,8 @@
     "category": "Gaze Mapping"
   },
   {
-    "title": "IMU Coordinate System Snippets",
-    "details": "Use values from the IMU to express your data in a world reference frame.",
+    "title": "IMU Transformations",
+    "details": "Transform IMU data into different representations and coordinate systems with these code snippets.",
     "link": {
       "text": "View",
       "href": "/alpha-lab/imu-transformations/"
diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 668ba5db0..aa3fba931 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -1,6 +1,6 @@
 ---
-title: "IMU Coordinate System Snippets"
-description: "Use values from the IMU to express your data in a world reference frame."
+title: "Transform IMU Data"
+description: "Transform IMU data into different representations and coordinate systems."
 permalink: /alpha-lab/imu-transformations/
 meta:
   - name: twitter:card
@@ -22,59 +22,73 @@ tags: [Neon, Cloud]
 import TagLinks from '@components/TagLinks.vue'
 </script>
 
-# IMU Coordinate System Snippets
+# IMU Transformations
 
 <TagLinks :tags="$frontmatter.tags" />
 
 <!-- <Youtube src="3N8jGLYCrNk"/> -->
 
 ::: tip
-Compare IMU and gaze data in the same coordinate system to better understand how people coordinate head and eye movements.
+Want to compare IMU and gaze data in the same coordinate system to better understand how people coordinate 
+head and eye movements? The transformation functions in this tutorial will show you how!
 :::
 
-Neon comes equipped with an IMU that can be used to determine head pose. Here, we will provide some code snippets that can assist when analyzing the IMU data from Neon.
+This guide contains various transformation functions that can assist when analysing Neon's IMU data.
 
-We will use Python with the NumPy and SciPy packages for the code snippets below. Let’s start by loading those:
+First, it's important to understand how the IMU data should be interpreted. 
+The [orientation readings of the calibrated IMU](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data) 
+are specified with respect to magnetic North and gravity. We will refer to this as the 'world coordinate system'.
 
-```python
-import numpy as np
-from scipy.spatial.transform import Rotation as R
-```
-
-First, it is important to understand how the IMU data should be interpreted. The [orientation readings of the calibrated IMU](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data) are specified with respect to magnetic North and gravity:
+To be exact, the 'world coordinate system' is distinct from the 'IMU coordinate system' - as the Neon module rotates, 
+the IMU coordinate system rotates with it. The IMU actually measures the rotational difference between its coordinate 
+system and the world coordinate system.
 
 - Y axis = vector pointing towards magnetic North.
-  - Rotations about this axis are called “roll” and range from -180 to +180 degrees. Wearing Neon normally and sitting or standing upright roughly corresponds to a roll of 0 degrees. Rightward head tilt is positive roll; leftward head tilt is negative roll.
+  - Rotations about this axis are called “roll” and range from -180 to +180 degrees. Wearing Neon normally in an upright position roughly corresponds to a roll of 0 degrees. Rightward head tilt is positive roll; leftward head tilt is negative roll.
 - Z axis = vector pointing upwards, opposite to the direction of gravity.
   - Rotations about this axis are called “yaw” and range from -180 to +180 degrees. When the IMU is calibrated, a Neon module oriented at magnetic North corresponds to a yaw of 0 degrees. Leftward head turn is positive yaw; rightward head turn is negative yaw.
 - X axis = cross-product of Y and Z, which is a vector pointing rightwards.
-  - Rotations about this axis are called “pitch” and range from -90 to +90 degrees. Wearing Neon normally and sitting or standing upright roughly corresponds to a pitch of 0 degrees. Backward head tilt is positive pitch; forward head tilt is negative pitch
+  - Rotations about this axis are called “pitch” and range from -90 to +90 degrees. Wearing Neon normally in an upright position roughly corresponds to a pitch of 0 degrees. Backward head tilt is positive pitch; forward head tilt is negative pitch
 
 ![Diagram of the three Euler orientation angles measured by Neon's IMU](./imu-pitch-yaw-roll-black.png)
 
-This coordinate system is known as the “global reference frame”. Here, we will refer to it as the world coordinate system. To be very exact, it is distinct from the IMU coordinate system - as the Neon module rotates, the IMU coordinate system rotates with it. The IMU actually measures the rotational difference between its coordinate system and the world coordinate system.
-
 ::: tip
-💡 Note that Neon can sit differently on each wearer’s face, such that the headset is not necessarily in line with the naso-occipital plane. For example, if the wearer is looking at magnetic North, the IMU might still report some deviation from neural orientation.
+💡 Note that Neon can sit differently on each wearer’s face, such that the headset is not necessarily in 
+line with a given plane. For example, if the wearer is facing magnetic North, the IMU might 
+still report some deviation from neural orientation.
 :::
 
-The gyroscope values give the rotational velocity of roll, yaw, and pitch in degrees/s (i.e., they provide the change over time of each of these quantities).
+The gyroscope values give the rotational velocity of roll, yaw, and pitch in degrees/s (i.e., they provide the change 
+over time of each of these quantities).
+
+The acceleration values measure translational (i.e., linear) acceleration, in terms of g-force (units of g), 
+along the X, Y, and Z axes of the IMU’s coordinate system. The reference frame for specifying acceleration values 
+rotates with the IMU.
+
 
-The acceleration values measure translational (i.e., linear) acceleration, in terms of g-force (units of g), along the X, Y, and Z axes of the IMU’s coordinate system. That is, the reference frame for specifying acceleration values rotates with the IMU.
 
 ::: tip
-💡 Note that [g-force](https://en.wikipedia.org/wiki/G-force) is not the same as free-fall acceleration due to gravity. For example, if you are standing on the surface of the Earth, then the g-force is equal and opposite to gravity, keeping you at rest. In that case, if you were wearing Neon normally and standing or sitting upright, then acceleration along the Z axis would be +1g, while it would be 0g along the X and Y axes.
+💡 [g-force](https://en.wikipedia.org/wiki/G-force) is not the same as free-fall acceleration due 
+to gravity. For example, if you are standing on the surface of the Earth, then the g-force is equal and 
+opposite to gravity, keeping you at rest. In that case, if you were wearing Neon normally in an upright position, then 
+acceleration along the Z axis would be +1g, while it would be 0g along the X and Y axes.
 :::
 
 It can be helpful to also try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems.
 
-Now that we have laid out the relationship between the IMU and world coordinate systems, we can do some useful transformations.
+Now that we have laid out the relationship between the IMU and world coordinate systems, and how to interpret the
+readings, we can do some useful transformations.
 
-## Obtain IMU heading vectors
+We will use Python with the NumPy and SciPy packages for the code snippets below. Let’s start by importing those:
 
-An alternate representation of IMU data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
+```python
+import numpy as np
+from scipy.spatial.transform import Rotation as R
+```
 
-Note that a neutral IMU heading vector is unlikely to be perfectly aligned with the floor.
+## Obtain IMU Heading Vectors
+
+An alternate representation of IMU data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
 
 :::: details Code
 ```python
@@ -110,7 +124,7 @@ def imu_heading_in_world(imu_quaternions):
 ```
 ::::
 
-## Transform IMU acceleration data to world coordinates
+## Transform IMU Acceleration Data to World Coordinates
 
 We mentioned above that the IMU’s acceleration data are specified with respect to the IMU’s coordinate system. Sometimes, it can be useful to have the acceleration data specified in the world coordinate system instead. The function below will perform this transformation.
 
@@ -148,13 +162,10 @@ def imu_acceleration_in_world(imu_accelerations, imu_quaternions):
 
 ## Represent IMU and Gaze Data in the Same Coordinate System
 
-Neon simultaneously records gaze and IMU data, making it possible to study the relationship between head and eye movements.
+Neon simultaneously records gaze and IMU data, making it possible to study the relationship between head and 
+eye movements.
 
-To facilitate comparison of gaze and head rotations, it can sometimes be easier to represent them in the same coordinate system. The coordinates of gaze are specified with respect to the scene camera coordinate system and the function below, `gaze_scene_to_world`, uses data from the IMU to transform gaze to the world coordinate system.
-
-::: tip
-💡 Reminder: The world coordinate system is defined by the direction away from the center of gravity, as well as the direction towards magnetic North.
-:::
+To facilitate the comparison, it can be useful to represent them in the same coordinate system. The coordinates of gaze are specified with respect to the scene camera coordinate system and the function below, `gaze_scene_to_world`, uses data from the IMU to transform gaze to the world coordinate system.
 
 :::: details Code
 ```python
@@ -276,17 +287,17 @@ def spherical_to_cartesian_scene(elevations, azimuths):
 ```
 ::::
 
-## Represent IMU and 3D Eyestate data in the same coordinate system
+## Represent IMU and 3D Eyestate in the Same Coordinate System
 
-The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) provided by Neon are aligned with the scene camera coordinate system. This means we can reuse elements of the `gaze_scene_to_world` function to reconstruct the pose of the eyes as a wearer rotates their head. Essentially, you would be representing eyestate in the world coordinate system of the IMU.
-
-The function below performs that transformation for you, returning eyeball centers and optical axes in the world coordinate system.
+The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) 
+provided by Neon are aligned with the scene camera coordinate system. This means we can reuse elements of 
+the `gaze_scene_to_world` function to reconstruct the pose of the eyes in the world coordinate system.
 
 :::: details Code
 ```python
 def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
   """
-  Transforms 3D eyestate data to the world coordinate system of the IMU.
+  Transforms 3D eyestate data to the world coordinate system.
   
   Note that the 3D eyestate data and the IMU quaternions should be sampled 
   at the same timestamps. You can linearly interpolate the IMU data
@@ -373,9 +384,15 @@ def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
 ```
 ::::
 
-## Convert (Cartesian) world points to spherical coordinates
+## Convert (Cartesian) World Points to Spherical Coordinates
 
-When studying head orientation and gaze orientation as observers navigate a 3D environment, it can be useful to know how much these quantities deviate from pointing at a given landmark. For instance, you might want to know when someone’s gaze or heading deviates from pointing at the horizon. This can be simplified by representing data in spherical coordinates. The orientation values from the IMU are already in such a format. For the values returned by  gaze_scene_to_world, the function below will do the necessary transformation. When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds to keeping your head neutral while gazing at the horizon.
+When studying head orientation and gaze orientation as observers navigate a 3D environment, it can be useful 
+to know how much these quantities deviate from pointing at a given landmark. For instance, you might want to 
+know when someone’s gaze or heading deviates from pointing at the horizon. This can be simplified by 
+representing data in spherical coordinates. The orientation values from the IMU are already in such a format. 
+For the values returned by  `gaze_scene_to_world`, the function below will do the necessary transformation. 
+When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds to keeping your head 
+neutral while gazing at the horizon.
 
 :::: details Code
 ```python
@@ -427,12 +444,10 @@ def cartesian_to_spherical_world(world_points_3d):
 
 ## Analysis example
 
-Below is a brief example of running these commands on values from Pupil Cloud.
+Below is a brief example of how to run the functions on this page using IMU data downloaded from Pupil Cloud.
 
 :::: details Code
 ```python
-import pandas as pd
-
 gaze = pd.read_csv("gaze.csv")
 eye3d = pd.read_csv("3d_eye_states.csv")
 imu = pd.read_csv("imu.csv")

From 6c5841dc839b45ec7d41621ed1b7e8d36af987d1 Mon Sep 17 00:00:00 2001
From: "Neil M. Thomas" <neil87@mail.com>
Date: Fri, 16 Aug 2024 12:19:06 +0800
Subject: [PATCH 08/54] Update index.md

---
 alpha-lab/imu-transformations/index.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index aa3fba931..1e407db5f 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -55,7 +55,7 @@ system and the world coordinate system.
 ::: tip
 💡 Note that Neon can sit differently on each wearer’s face, such that the headset is not necessarily in 
 line with a given plane. For example, if the wearer is facing magnetic North, the IMU might 
-still report some deviation from neural orientation.
+still report some deviation from a neutral orientation.
 :::
 
 The gyroscope values give the rotational velocity of roll, yaw, and pitch in degrees/s (i.e., they provide the change 
@@ -511,4 +511,4 @@ gaze_elevations_world, gaze_azimuths_world = cartesian_to_spherical_world(world_
 
 ::: tip
 Need assistance with the IMU code in this article? Or do you have something more custom in mind? Reach out to us via email at [info@pupil-labs.com](mailto:info@pupil-labs.com), on our [Discord server](https://pupil-labs.com/chat/), or visit our [Support Page](https://pupil-labs.com/products/support/) for dedicated support options.
-:::
\ No newline at end of file
+:::

From 19a6d1fc764c2d79eee51319f9e20a75b78b112b Mon Sep 17 00:00:00 2001
From: "Neil M. Thomas" <neil87@mail.com>
Date: Fri, 16 Aug 2024 13:09:22 +0800
Subject: [PATCH 09/54] Update index.md

---
 alpha-lab/imu-transformations/index.md | 10 ++++------
 1 file changed, 4 insertions(+), 6 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 1e407db5f..e9cc8b5af 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -40,7 +40,7 @@ The [orientation readings of the calibrated IMU](https://docs.pupil-labs.com/neo
 are specified with respect to magnetic North and gravity. We will refer to this as the 'world coordinate system'.
 
 To be exact, the 'world coordinate system' is distinct from the 'IMU coordinate system' - as the Neon module rotates, 
-the IMU coordinate system rotates with it. The IMU actually measures the rotational difference between its coordinate 
+the IMU coordinate system rotates with it. The IMU's orientation readings actually specify the rotational difference between its coordinate 
 system and the world coordinate system.
 
 - Y axis = vector pointing towards magnetic North.
@@ -58,15 +58,13 @@ line with a given plane. For example, if the wearer is facing magnetic North, th
 still report some deviation from a neutral orientation.
 :::
 
-The gyroscope values give the rotational velocity of roll, yaw, and pitch in degrees/s (i.e., they provide the change 
-over time of each of these quantities).
+The raw gyroscope values give the rotational velocity about the X, Y, and Z axes of the IMU’s coordinate system. Their units are degrees/s (i.e., they provide the change 
+over time of each of these quantities). The reference frame for specifying rotational velocities rotates with the IMU.
 
-The acceleration values measure translational (i.e., linear) acceleration, in terms of g-force (units of g), 
+The raw acceleration values measure translational (i.e., linear) acceleration, in terms of g-force (units of g), 
 along the X, Y, and Z axes of the IMU’s coordinate system. The reference frame for specifying acceleration values 
 rotates with the IMU.
 
-
-
 ::: tip
 💡 [g-force](https://en.wikipedia.org/wiki/G-force) is not the same as free-fall acceleration due 
 to gravity. For example, if you are standing on the surface of the Earth, then the g-force is equal and 

From a860f452ce1f20e9df8eea95ee6e0a3d3fe97450 Mon Sep 17 00:00:00 2001
From: "Neil M. Thomas" <neil87@mail.com>
Date: Fri, 16 Aug 2024 18:06:17 +0800
Subject: [PATCH 10/54] Add YT link

---
 alpha-lab/imu-transformations/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index e9cc8b5af..54a42042d 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -26,7 +26,7 @@ import TagLinks from '@components/TagLinks.vue'
 
 <TagLinks :tags="$frontmatter.tags" />
 
-<!-- <Youtube src="3N8jGLYCrNk"/> -->
+<Youtube src="expAuDaIllU"/>
 
 ::: tip
 Want to compare IMU and gaze data in the same coordinate system to better understand how people coordinate 

From b499ddd7defd2893ee78cd40be14a1b28e355e54 Mon Sep 17 00:00:00 2001
From: "Neil M. Thomas" <neil87@mail.com>
Date: Fri, 16 Aug 2024 18:21:08 +0800
Subject: [PATCH 11/54] Remove YT link

---
 alpha-lab/imu-transformations/index.md | 2 --
 1 file changed, 2 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 54a42042d..7647ffc19 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -26,8 +26,6 @@ import TagLinks from '@components/TagLinks.vue'
 
 <TagLinks :tags="$frontmatter.tags" />
 
-<Youtube src="expAuDaIllU"/>
-
 ::: tip
 Want to compare IMU and gaze data in the same coordinate system to better understand how people coordinate 
 head and eye movements? The transformation functions in this tutorial will show you how!

From 3ba910fd8035ab969fb9733999d27c259e6498c1 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Mon, 19 Aug 2024 10:46:53 +0200
Subject: [PATCH 12/54] fix some typo's, clarify meaning of spherical
 coordinates in world system a bit better

---
 alpha-lab/imu-transformations/index.md | 47 ++++++++++++++------------
 1 file changed, 26 insertions(+), 21 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 7647ffc19..bfb8a421d 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -97,8 +97,8 @@ def imu_heading_in_world(imu_quaternions):
     from Neon's IMU stream.
         
   Returns:
-    - headings_in_world (Nx3 np.array): A timeseries of IMU heading vectors
-    in the world coordinate system.
+    - headings_in_world (Nx3 np.array): The corresponding timeseries of
+    IMU heading vectors in the world coordinate system.
   """
 
   # We start by specifying the direction of a neutral heading vector
@@ -140,8 +140,8 @@ def imu_acceleration_in_world(imu_accelerations, imu_quaternions):
     from Neon's IMU stream.
     
   Returns:
-    - accelerations_world (Nx3 np.array): Timeseries of acceleration data,
-    expressed in the world coordinate system.
+    - accelerations_world (Nx3 np.array): The corresponding timeseries of
+    acceleration data, expressed in the world coordinate system.
   """
     
   # This array contains a timeseries of transformation matrices,
@@ -179,7 +179,7 @@ def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
   The code in this function is adapted from the `plimu` visualization utility:
   https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279
     
-  This function makes use of the spherical_to_cartesian function,
+  This function makes use of the spherical_to_cartesian_scene function,
   defined below, that converts 3D gaze rays from spherical coordinates
   to Cartesian coordinates.
     
@@ -192,8 +192,8 @@ def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
     from Neon's IMU.
     
   Returns:
-    - gazes_in_world (Nx3 np.array): A timeseries of 3D Cartesian gaze
-    unit vectors, specified in the world coordinate system.
+    - gazes_in_world (Nx3 np.array): The corresponding timeseries of
+    3D Cartesian gaze unit vectors, specified in the world coordinate system.
   """
     
   # The IMU and scene camera coordinate systems have a fixed
@@ -314,10 +314,12 @@ def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
     Neon's IMU.
     
   Returns:
-    - eyeball_centers_in_world (Nx3 np.array): A timeseries of eyeball center
-    estimates (mm) for one eye, transformed to the world coordinate system of the IMU.
-    - optical_axes_in_world (Nx3 np.array): A timeseries of optical axis estimates
-    (3D unit vectors) for the same eye, transformed to the world coordinate system of the IMU.
+    - eyeball_centers_in_world (Nx3 np.array): The corresponding timeseries of
+    eyeball center estimates (mm) for one eye, transformed to the world coordinate
+    system of the IMU.
+    - optical_axes_in_world (Nx3 np.array): The corresponding timeseries of optical
+    axis estimates (3D unit vectors) for the same eye, transformed to the world
+    coordinate system of the IMU.
   """
   
   # The eyeball centers are specified in terms of their distance from
@@ -325,7 +327,7 @@ def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
   # world coordinates, we need the position of the scene camera in
   # the IMU coordinate system. Here, we express that position
   # in millimeters.
-  scene_camera_position_in_imu = np.array([0.0, -3.996, -11.172])
+  scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
 
   # The coordinate system of 3D eyestate is aligned with
   # the coordinate system of the scene camera, so we can
@@ -385,10 +387,10 @@ def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
 When studying head orientation and gaze orientation as observers navigate a 3D environment, it can be useful 
 to know how much these quantities deviate from pointing at a given landmark. For instance, you might want to 
 know when someone’s gaze or heading deviates from pointing at the horizon. This can be simplified by 
-representing data in spherical coordinates. The orientation values from the IMU are already in such a format. 
+converting world points from Cartesian to spherical coordinates. The orientation values from the IMU are already in such a format. 
 For the values returned by  `gaze_scene_to_world`, the function below will do the necessary transformation. 
-When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds to keeping your head 
-neutral while gazing at the horizon.
+When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds to a neutral orientation:
+i.e., aimed at magnetic North and parallel to the horizon.
 
 :::: details Code
 ```python
@@ -396,24 +398,25 @@ def cartesian_to_spherical_world(world_points_3d):
   """
   Convert points 3D Cartesian world coordinates to spherical coordinates.
 
-  The origin of world coordinate system is the same as the origin of the
+  The origin of the world coordinate system is the same as the origin of the
   IMU coordinate system.
 
-  When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds to keeping your head neutral while gazing at the horizon.
+  When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds
+  to a neutral orientation: i.e., aimed at magnetic North and parallel to the horizon.
 
   Inputs:
     - world_points_3d (Nx3 np.array): A collection of 3D points in 
     Cartesian world coordinates.
         
   Returns:
-    - elevation (Nx1 np.array): The corresponding elevation value, in radians,
+    - elevation (Nx1 np.array): The corresponding elevation value, in degrees,
     for each 3D world point (i.e., vertical rotation away from neutral orientation).
     Note: neutral orientation = 0 elevation; orientation upwards is positive;
     orientation downwards is negative.
-    - azimuth (Nx1 np.array): The corresponding azimuth value, in radians, for
+    - azimuth (Nx1 np.array): The corresponding azimuth value, in degrees, for
     each 3D world point (i.e., horizontal rotation away from neutral orientation).
-    Note: neutral orientation = 0 azimuth, orientation rightwards is negative;
-    orientation leftwards is positive.
+    Note: neutral orientation = 0 azimuth, orientation leftwards is positive;
+    orientation rightwards is negative.
   """
 
   x = world_points_3d[:, 0]
@@ -468,6 +471,8 @@ accelerations_resampled = np.array([
   np.interp(gaze_ts, imu_ts, imu["acceleration z [g]"]),
 ]).T
 
+# Put the eyestate data in a format that is compatible with
+# the transformation functions from this article.
 optical_axes_left = np.array([
   eye3d["optical axis left x"],
   eye3d["optical axis left y"],

From ca541bac10425104e70c51a3dbc589849d122c96 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Wed, 21 Aug 2024 08:48:27 +0200
Subject: [PATCH 13/54] try out code snippets vitepress extension

---
 alpha-lab/imu-transformations/index.md        | 48 +++++--------------
 .../pl_imu_transformations.py                 | 40 ++++++++++++++++
 2 files changed, 51 insertions(+), 37 deletions(-)
 create mode 100644 alpha-lab/imu-transformations/pl_imu_transformations.py

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index bfb8a421d..21e94ac8d 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -75,50 +75,24 @@ It can be helpful to also try out our IMU visualization utility, [plimu](https:/
 Now that we have laid out the relationship between the IMU and world coordinate systems, and how to interpret the
 readings, we can do some useful transformations.
 
-We will use Python with the NumPy and SciPy packages for the code snippets below. Let’s start by importing those:
-
-```python
-import numpy as np
-from scipy.spatial.transform import Rotation as R
-```
-
 ## Obtain IMU Heading Vectors
 
 An alternate representation of IMU data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
 
-:::: details Code
-```python
-def imu_heading_in_world(imu_quaternions):
-  """
-  Construct heading vectors from the IMU's quaternion values.
-        
-  Inputs:
-    - imu_quaternions (Nx4 np.array): A timeseries of quaternions
-    from Neon's IMU stream.
-        
-  Returns:
-    - headings_in_world (Nx3 np.array): The corresponding timeseries of
-    IMU heading vectors in the world coordinate system.
-  """
+We start by specifying a neutral heading vector:
 
-  # We start by specifying the direction of a neutral heading vector
-  # in the IMU's coordinate system.
-  heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
+<<< @/imu-transformations/pl_imu_transformations.py#neutral_heading
 
-  # This array contains a timeseries of transformation matrices,
-  # as calculated from the IMU's timeseries of quaternions values.
-  # Each of these matrices are used to transform points in the IMU
-  # coordinate system to their corresponding coordinates in the world
-  # coordinate system.
-  imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
-    
-  # We now apply each transformation matrix to the neutral IMU heading vector
-  # to obtain a timeseries of heading vectors in the world coordinate system.
-  headings_in_world = imu_to_world_matrices @ heading_neutral_in_imu_coords
+Then, we can obtain a timeseries of transformation matrices, using the timeseries of quaternion values
+obtained from the IMU. These matrices can be used to transform points the local IMU coordinate system
+to the world coordinate system:
 
-  return headings_in_world
-```
-::::
+<<< @/imu-transformations/pl_imu_transformations.py#imu_world_matrices
+
+Finally, we obtain a timeseries of heading vectors by applying each transformation matrix, in turn,
+to the neutral heading vector:
+
+<<< @/imu-transformations/pl_imu_transformations.py#heading_in_world
 
 ## Transform IMU Acceleration Data to World Coordinates
 
diff --git a/alpha-lab/imu-transformations/pl_imu_transformations.py b/alpha-lab/imu-transformations/pl_imu_transformations.py
new file mode 100644
index 000000000..9e3bf0994
--- /dev/null
+++ b/alpha-lab/imu-transformations/pl_imu_transformations.py
@@ -0,0 +1,40 @@
+import numpy as np
+from scipy.spatial.transform import Rotation as R
+
+def imu_heading_in_world(imu_quaternions):
+    """
+    Construct heading vectors from the IMU's quaternion values.
+
+    Inputs:
+      - imu_quaternions (Nx4 np.array): A timeseries of quaternions
+      from Neon's IMU stream.
+
+    Returns:
+      - headings_in_world (Nx3 np.array): The corresponding timeseries of
+      IMU heading vectors in the world coordinate system.
+    """
+
+    # We start by specifying the direction of a neutral heading vector
+    # in the IMU's coordinate system.
+    # region neutral_heading
+    heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
+    # endregion neutral_heading
+
+    # This array contains a timeseries of transformation matrices,
+    # as calculated from the IMU's timeseries of quaternions values.
+    # Each of these matrices are used to transform points in the IMU
+    # coordinate system to their corresponding coordinates in the world
+    # coordinate system.
+    # region imu_world_matrices
+    imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
+    # endregion imu_world_matrices
+
+    # We now apply each transformation matrix to the neutral IMU heading vector
+    # to obtain a timeseries of heading vectors in the world coordinate system.
+    # region heading_in_world
+    headings_in_world = imu_to_world_matrices @ heading_neutral_in_imu_coords
+    # endregion heading_in_world
+    
+    return headings_in_world
+
+

From ff08b4b8d6c7cde33c45c42792837de5de2bd04f Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Wed, 21 Aug 2024 10:03:01 +0200
Subject: [PATCH 14/54] test submodule + line specific code snippet import

---
 .gitmodules                                | 3 +++
 alpha-lab/47303f69efcff77bd634c82c86612b84 | 1 +
 alpha-lab/imu-transformations/index.md     | 2 +-
 3 files changed, 5 insertions(+), 1 deletion(-)
 create mode 100644 .gitmodules
 create mode 160000 alpha-lab/47303f69efcff77bd634c82c86612b84

diff --git a/.gitmodules b/.gitmodules
new file mode 100644
index 000000000..3f6be0d5a
--- /dev/null
+++ b/.gitmodules
@@ -0,0 +1,3 @@
+[submodule "alpha-lab/47303f69efcff77bd634c82c86612b84"]
+	path = alpha-lab/47303f69efcff77bd634c82c86612b84
+	url = https://gist.github.com/47303f69efcff77bd634c82c86612b84.git
diff --git a/alpha-lab/47303f69efcff77bd634c82c86612b84 b/alpha-lab/47303f69efcff77bd634c82c86612b84
new file mode 160000
index 000000000..542043d8a
--- /dev/null
+++ b/alpha-lab/47303f69efcff77bd634c82c86612b84
@@ -0,0 +1 @@
+Subproject commit 542043d8a4500ae908af6abe9846266a76bd1eeb
diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 21e94ac8d..314c98498 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -81,7 +81,7 @@ An alternate representation of IMU data is a heading vector that points outwards
 
 We start by specifying a neutral heading vector:
 
-<<< @/imu-transformations/pl_imu_transformations.py#neutral_heading
+<<< @/47303f69efcff77bd634c82c86612b84/pl_imu_transformations.py#L20
 
 Then, we can obtain a timeseries of transformation matrices, using the timeseries of quaternion values
 obtained from the IMU. These matrices can be used to transform points the local IMU coordinate system

From ec8dcb97cee3796c89c0842042ec393bdf3308cc Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Wed, 21 Aug 2024 11:45:07 +0200
Subject: [PATCH 15/54] test variants for "inlining" the code in the docs

---
 alpha-lab/imu-transformations/index.md        | 192 +++++++-----------
 .../pl_imu_transformations.py                 |  12 +-
 2 files changed, 72 insertions(+), 132 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 314c98498..82fa7f4b6 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -77,22 +77,41 @@ readings, we can do some useful transformations.
 
 ## Obtain IMU Heading Vectors
 
-An alternate representation of IMU data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
+An alternate representation of IMU data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity:
 
-We start by specifying a neutral heading vector:
+- **Rob Variant 1:** Write out scipy module reference in full:
 
-<<< @/47303f69efcff77bd634c82c86612b84/pl_imu_transformations.py#L20
+```python
+imu_neutral_heading = np.array([0.0, 1.0, 0.0])
+world_rotation_matrices = scipy.spatial.transform.Rotation.from_quat(imu_quaternions).as_matrix()
+headings_in_world = world_rotation_matrices @ imu_neutral_heading
+```
 
-Then, we can obtain a timeseries of transformation matrices, using the timeseries of quaternion values
-obtained from the IMU. These matrices can be used to transform points the local IMU coordinate system
-to the world coordinate system:
+- **Rob Variant 2:** Show conventional way to import Rotation sub-module of scipy:
 
-<<< @/imu-transformations/pl_imu_transformations.py#imu_world_matrices
+```python
+from scipy.spatial.transform import Rotation as R
+
+imu_neutral_heading = np.array([0.0, 1.0, 0.0])
+world_rotation_matrices = R.from_quat(imu_quaternions).as_matrix()
+headings_in_world = world_rotation_matrices @ imu_neutral_heading
+```
+
+- **Rob Variant 3:** Just write the conventional way without showing the import:
+
+```python
+imu_neutral_heading = np.array([0.0, 1.0, 0.0])
+world_rotation_matrices = R.from_quat(imu_quaternions).as_matrix()
+headings_in_world = world_rotation_matrices @ imu_neutral_heading
+```
 
-Finally, we obtain a timeseries of heading vectors by applying each transformation matrix, in turn,
-to the neutral heading vector:
+- **Rob Variant 4:** Do it as Python-esque pseudo-code:
 
-<<< @/imu-transformations/pl_imu_transformations.py#heading_in_world
+```python
+imu_neutral_heading = np.array([0.0, 1.0, 0.0])
+world_rotation_matrices = rotation_matrices_from_quaternions(imu_quaternions)
+headings_in_world = world_rotation_matrices @ imu_neutral_heading
+```
 
 ## Transform IMU Acceleration Data to World Coordinates
 
@@ -137,125 +156,52 @@ eye movements.
 
 To facilitate the comparison, it can be useful to represent them in the same coordinate system. The coordinates of gaze are specified with respect to the scene camera coordinate system and the function below, `gaze_scene_to_world`, uses data from the IMU to transform gaze to the world coordinate system.
 
-:::: details Code
-```python
-def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
-  """
-  Transform a 3D gaze ray to the world coordinate system.
-    
-  Note that the gaze data and the IMU quaternion should be sampled 
-  at the same timestamps. You can linearly interpolate the IMU data
-  to ensure this.
-    
-  The origin of the IMU coordinate system is the same as the
-  origin of the world coordinate system.
+The IMU and scene camera coordinate systems have [a fixed 102 degree rotation offset](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data). Knowing this, we can build a matrix to transform points in the scene camera coordinate system to their corresponding coordinates in the IMU coordinate system:
 
-  The code in this function is adapted from the `plimu` visualization utility:
-  https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279
-    
-  This function makes use of the spherical_to_cartesian_scene function,
-  defined below, that converts 3D gaze rays from spherical coordinates
-  to Cartesian coordinates.
-    
-  Inputs:
-    - gaze_elevations (Nx1 np.array): A timeseries of gaze elevations (degrees),
-    specified in the scene camera coordinate system.
-    - gaze_azimuths (Nx1 np.array): A timeseries of gaze azimuths (degrees),
-    specified in the scene camera coordinate system.
-    - imu_quaternions (Nx4 np.array): A timeseries of quaternion values
-    from Neon's IMU.
-    
-  Returns:
-    - gazes_in_world (Nx3 np.array): The corresponding timeseries of
-    3D Cartesian gaze unit vectors, specified in the world coordinate system.
-  """
-    
-  # The IMU and scene camera coordinate systems have a fixed
-  # 102 degree rotation offset. See:
-  # https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data
-  imu_scene_rotation_diff = np.deg2rad(-90 - 12)
-  
-  # This matrix is used to transform points in the scene
-  # camera coordinate system to their corresponding coordinates
-  # in the IMU coordinate system.
-  scene_to_imu = np.array(
-      [
-          [1.0, 0.0, 0.0],
-          [
-            0.0,
-            np.cos(imu_scene_rotation_diff),
-            -np.sin(imu_scene_rotation_diff),
-          ],
-          [
-            0.0,
-            np.sin(imu_scene_rotation_diff),
-            np.cos(imu_scene_rotation_diff),
-          ],
-      ]
-  )
-  
-  # Neon provides 3D gaze in spherical coordinates by default, 
-  # so we first transform the gaze data from spherical coordinates
-  # to Cartesian coordinates.
-  cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
-    
-  # Apply the transformation from the scene camera to the IMU coordinate system.
-  gazes_in_imu = scene_to_imu @ cart_gazes_in_scene.T
-  
-  # This array contains a timeseries of transformation matrices,
-  # as calculated from the IMU's timeseries of quaternions values.
-  # Each of these matrices are used to transform points in the IMU coordinate
-  # system to their corresponding coordinates in the world coordinate system.
-  imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
-  
-  # Apply the transformations from the IMU to the world coordinate system.
-  gazes_in_world = [imu_to_world @ gaze for imu_to_world, gaze in zip(imu_to_world_matrices, gazes_in_imu.T)]
+```
+imu_scene_rotation_diff = deg2rad(-90 - 12)
+
+scene_to_imu = array(
+  [
+    [1.0, 0.0, 0.0],
+    [
+      0.0,
+      cos(imu_scene_rotation_diff),
+      -sin(imu_scene_rotation_diff),
+    ],
+    [
+      0.0,
+      sin(imu_scene_rotation_diff),
+      cos(imu_scene_rotation_diff),
+    ],
+  ]
+)
+```
   
-  return np.array(gazes_in_world)
+Neon provides 3D gaze in spherical coordinates by default, so we first need to transform the gaze data from spherical coordinates to Cartesian coordinates.
+
+```
+cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
+```
 
+Now, we can apply the transformation from the scene camera to the IMU coordinate system:
 
-def spherical_to_cartesian_scene(elevations, azimuths):
-  """
-  Convert Neon's spherical representation of 3D gaze to Cartesian coordinates.
-    
-  Inputs:
-    - elevations (Nx1 np.array): A timeseries of gaze elevations (degrees),
-    specified in the scene camera coordinate system.
-    - azimuths (Nx1 np.array): A timeseries of gaze azimuths (degrees),
-    specified in the scene camera coordinate system.
-    
-  Returns:
-    - cartesian_unit_vectors (Nx3 np.array): A timeseries of gaze unit
-    vectors, in Cartesian coordinates, specified in the scene camera
-    coordinate system.
-  """
+```
+gazes_in_imu = scene_to_imu @ cart_gazes_in_scene.T
+```
+
+Using the timeseries of quaternion values from the IMU, we can construct a timeseries of transformation matrices.
+Each of these matrices are used to transform points in the IMU coordinate system to their corresponding coordinates in the world coordinate system:
 
-  elevations_rad = np.deg2rad(elevations)
-  azimuths_rad = np.deg2rad(azimuths)
-
-  # Elevation of 0 in Neon system corresponds to Y = 0, but
-  # an elevation of 0 in traditional spherical coordinates would
-  # correspond to Y = 1, so first we convert elevation to the
-  # more traditional format.
-  elevations_rad += np.pi / 2
- 
-  # Azimuth of 0 in Neon system corresponds to X = 0, but
-  # an azimuth of 0 in traditional spherical coordinates would
-  # correspond to X = 1. Also, azimuth to the right in Neon is
-  # more positive, whereas it is more negative in traditional spherical coordiantes.
-  # So, first we convert azimuth to the more traditional format.
-  azimuths_rad *= -1.0
-  azimuths_rad += np.pi / 2
-
-  cartesian_unit_vectors = np.array([
-    np.sin(elevations_rad) * np.cos(azimuths_rad),
-    np.cos(elevations_rad),
-    np.sin(elevations_rad) * np.sin(azimuths_rad),
-  ]).T
-
-  return cartesian_unit_vectors
 ```
-::::
+imu_to_world_matrices = rotation_matrices_from_quaternions(imu_quaternions)
+```
+  
+Finally, we can apply the transformations from the IMU to the world coordinate system:
+
+```
+gazes_in_world = [imu_to_world @ gaze for imu_to_world, gaze in zip(imu_to_world_matrices, gazes_in_imu.T)]
+```
 
 ## Represent IMU and 3D Eyestate in the Same Coordinate System
 
diff --git a/alpha-lab/imu-transformations/pl_imu_transformations.py b/alpha-lab/imu-transformations/pl_imu_transformations.py
index 9e3bf0994..81987b13e 100644
--- a/alpha-lab/imu-transformations/pl_imu_transformations.py
+++ b/alpha-lab/imu-transformations/pl_imu_transformations.py
@@ -16,24 +16,18 @@ def imu_heading_in_world(imu_quaternions):
 
     # We start by specifying the direction of a neutral heading vector
     # in the IMU's coordinate system.
-    # region neutral_heading
-    heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
-    # endregion neutral_heading
+    imu_neutral_heading = np.array([0.0, 1.0, 0.0])
 
     # This array contains a timeseries of transformation matrices,
     # as calculated from the IMU's timeseries of quaternions values.
     # Each of these matrices are used to transform points in the IMU
     # coordinate system to their corresponding coordinates in the world
     # coordinate system.
-    # region imu_world_matrices
-    imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
-    # endregion imu_world_matrices
+    world_rotation_matrices = R.from_quat(imu_quaternions).as_matrix()
 
     # We now apply each transformation matrix to the neutral IMU heading vector
     # to obtain a timeseries of heading vectors in the world coordinate system.
-    # region heading_in_world
-    headings_in_world = imu_to_world_matrices @ heading_neutral_in_imu_coords
-    # endregion heading_in_world
+    headings_in_world = world_rotation_matrices @ imu_neutral_heading
     
     return headings_in_world
 

From 2e39a75dd4868b630a04fd0fb8f18edddb0c0d06 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Wed, 21 Aug 2024 11:55:00 +0200
Subject: [PATCH 16/54] make scene_to_imu pseudo-code more compact

---
 alpha-lab/imu-transformations/index.md | 23 ++++-------------------
 1 file changed, 4 insertions(+), 19 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 82fa7f4b6..b7de7b3c2 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -160,25 +160,10 @@ The IMU and scene camera coordinate systems have [a fixed 102 degree rotation of
 
 ```
 imu_scene_rotation_diff = deg2rad(-90 - 12)
-
-scene_to_imu = array(
-  [
-    [1.0, 0.0, 0.0],
-    [
-      0.0,
-      cos(imu_scene_rotation_diff),
-      -sin(imu_scene_rotation_diff),
-    ],
-    [
-      0.0,
-      sin(imu_scene_rotation_diff),
-      cos(imu_scene_rotation_diff),
-    ],
-  ]
-)
+scene_to_imu = yz_rotation_matrix_from_angle(imu_scene_rotation_diff)
 ```
   
-Neon provides 3D gaze in spherical coordinates by default, so we first need to transform the gaze data from spherical coordinates to Cartesian coordinates.
+Neon provides 3D gaze in spherical coordinates by default, so next, we need to transform the gaze data from spherical coordinates to Cartesian coordinates.
 
 ```
 cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
@@ -187,7 +172,7 @@ cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuth
 Now, we can apply the transformation from the scene camera to the IMU coordinate system:
 
 ```
-gazes_in_imu = scene_to_imu @ cart_gazes_in_scene.T
+gazes_in_imu = scene_to_imu @ cart_gazes_in_scene
 ```
 
 Using the timeseries of quaternion values from the IMU, we can construct a timeseries of transformation matrices.
@@ -200,7 +185,7 @@ imu_to_world_matrices = rotation_matrices_from_quaternions(imu_quaternions)
 Finally, we can apply the transformations from the IMU to the world coordinate system:
 
 ```
-gazes_in_world = [imu_to_world @ gaze for imu_to_world, gaze in zip(imu_to_world_matrices, gazes_in_imu.T)]
+gazes_in_world = [imu_to_world @ gaze for imu_to_world, gaze in zip(imu_to_world_matrices, gazes_in_imu)]
 ```
 
 ## Represent IMU and 3D Eyestate in the Same Coordinate System

From 156298fc580c877d0eef22e81e9f9cc3aecc38fa Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Wed, 21 Aug 2024 12:17:48 +0200
Subject: [PATCH 17/54] implement @mgg's codegroup and include approach

---
 alpha-lab/imu-transformations/index.md        | 36 ++++---------------
 .../pl_imu_transformations_no_comments.py     |  8 +++++
 2 files changed, 15 insertions(+), 29 deletions(-)
 create mode 100644 alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index b7de7b3c2..3988a07e5 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -79,39 +79,17 @@ readings, we can do some useful transformations.
 
 An alternate representation of IMU data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity:
 
-- **Rob Variant 1:** Write out scipy module reference in full:
+::: code-group
 
-```python
-imu_neutral_heading = np.array([0.0, 1.0, 0.0])
-world_rotation_matrices = scipy.spatial.transform.Rotation.from_quat(imu_quaternions).as_matrix()
-headings_in_world = world_rotation_matrices @ imu_neutral_heading
+```python:line-numbers {16,23,27} [imu_heading_in_world()]
+<!--@include: ./pl_imu_transformations_no_comments.py{2,8}-->
 ```
 
-- **Rob Variant 2:** Show conventional way to import Rotation sub-module of scipy:
-
-```python
-from scipy.spatial.transform import Rotation as R
-
-imu_neutral_heading = np.array([0.0, 1.0, 0.0])
-world_rotation_matrices = R.from_quat(imu_quaternions).as_matrix()
-headings_in_world = world_rotation_matrices @ imu_neutral_heading
-```
-
-- **Rob Variant 3:** Just write the conventional way without showing the import:
-
-```python
-imu_neutral_heading = np.array([0.0, 1.0, 0.0])
-world_rotation_matrices = R.from_quat(imu_quaternions).as_matrix()
-headings_in_world = world_rotation_matrices @ imu_neutral_heading
-```
-
-- **Rob Variant 4:** Do it as Python-esque pseudo-code:
-
-```python
-imu_neutral_heading = np.array([0.0, 1.0, 0.0])
-world_rotation_matrices = rotation_matrices_from_quaternions(imu_quaternions)
-headings_in_world = world_rotation_matrices @ imu_neutral_heading
+```python:line-numbers {18,25,29} [(with comments)]
+<!--@include: ./pl_imu_transformations.py{2,32}-->
 ```
+:::
+ 
 
 ## Transform IMU Acceleration Data to World Coordinates
 
diff --git a/alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py b/alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py
new file mode 100644
index 000000000..86bcd9236
--- /dev/null
+++ b/alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py
@@ -0,0 +1,8 @@
+import numpy as np
+from scipy.spatial.transform import Rotation as R
+
+def imu_heading_in_world(imu_quaternions):
+    imu_neutral_heading = np.array([0.0, 1.0, 0.0])
+    world_rotation_matrices = R.from_quat(imu_quaternions).as_matrix()
+    headings_in_world = world_rotation_matrices @ imu_neutral_heading
+    return headings_in_world

From 9d960348ef6ddbbe95205a6f07b15650ae872678 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Wed, 21 Aug 2024 12:30:41 +0200
Subject: [PATCH 18/54] add gaze_scene_to_world to codegroup test

---
 alpha-lab/imu-transformations/index.md        |  35 ++----
 .../pl_imu_transformations.py                 | 119 ++++++++++++++++++
 .../pl_imu_transformations_no_comments.py     |  49 ++++++++
 3 files changed, 175 insertions(+), 28 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 3988a07e5..e98849031 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -81,7 +81,7 @@ An alternate representation of IMU data is a heading vector that points outwards
 
 ::: code-group
 
-```python:line-numbers {16,23,27} [imu_heading_in_world()]
+```python [imu_heading_in_world()]
 <!--@include: ./pl_imu_transformations_no_comments.py{2,8}-->
 ```
 
@@ -134,37 +134,16 @@ eye movements.
 
 To facilitate the comparison, it can be useful to represent them in the same coordinate system. The coordinates of gaze are specified with respect to the scene camera coordinate system and the function below, `gaze_scene_to_world`, uses data from the IMU to transform gaze to the world coordinate system.
 
-The IMU and scene camera coordinate systems have [a fixed 102 degree rotation offset](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data). Knowing this, we can build a matrix to transform points in the scene camera coordinate system to their corresponding coordinates in the IMU coordinate system:
-
-```
-imu_scene_rotation_diff = deg2rad(-90 - 12)
-scene_to_imu = yz_rotation_matrix_from_angle(imu_scene_rotation_diff)
-```
-  
-Neon provides 3D gaze in spherical coordinates by default, so next, we need to transform the gaze data from spherical coordinates to Cartesian coordinates.
-
-```
-cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
-```
-
-Now, we can apply the transformation from the scene camera to the IMU coordinate system:
-
-```
-gazes_in_imu = scene_to_imu @ cart_gazes_in_scene
-```
-
-Using the timeseries of quaternion values from the IMU, we can construct a timeseries of transformation matrices.
-Each of these matrices are used to transform points in the IMU coordinate system to their corresponding coordinates in the world coordinate system:
+::: code-group
 
+```python [gaze_scene_to_world() & spherical_to_cartesian_scene()]
+<!--@include: ./pl_imu_transformations_no_comments.py{11,57}-->
 ```
-imu_to_world_matrices = rotation_matrices_from_quaternions(imu_quaternions)
-```
-  
-Finally, we can apply the transformations from the IMU to the world coordinate system:
 
+```python:line-numbers {34,39-53,58,61,67,70-75,94,95,101,108-119} [(with comments)]
+<!--@include: ./pl_imu_transformations.py{35,153}-->
 ```
-gazes_in_world = [imu_to_world @ gaze for imu_to_world, gaze in zip(imu_to_world_matrices, gazes_in_imu)]
-```
+:::
 
 ## Represent IMU and 3D Eyestate in the Same Coordinate System
 
diff --git a/alpha-lab/imu-transformations/pl_imu_transformations.py b/alpha-lab/imu-transformations/pl_imu_transformations.py
index 81987b13e..e3362e5a1 100644
--- a/alpha-lab/imu-transformations/pl_imu_transformations.py
+++ b/alpha-lab/imu-transformations/pl_imu_transformations.py
@@ -32,3 +32,122 @@ def imu_heading_in_world(imu_quaternions):
     return headings_in_world
 
 
+def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
+    """
+    Transform a 3D gaze ray to the world coordinate system.
+
+    Note that the gaze data and the IMU quaternion should be sampled
+    at the same timestamps. You can linearly interpolate the IMU data
+    to ensure this.
+
+    The origin of the IMU coordinate system is the same as the
+    origin of the world coordinate system.
+    The code in this function is adapted from the `plimu` visualization utility:
+    https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279
+
+    This function makes use of the spherical_to_cartesian_scene function,
+    defined below, that converts 3D gaze rays from spherical coordinates
+    to Cartesian coordinates.
+
+    Inputs:
+      - gaze_elevations (Nx1 np.array): A timeseries of gaze elevations (degrees),
+      specified in the scene camera coordinate system.
+      - gaze_azimuths (Nx1 np.array): A timeseries of gaze azimuths (degrees),
+      specified in the scene camera coordinate system.
+      - imu_quaternions (Nx4 np.array): A timeseries of quaternion values
+      from Neon's IMU.
+
+    Returns:
+      - gazes_in_world (Nx3 np.array): The corresponding timeseries of
+      3D Cartesian gaze unit vectors, specified in the world coordinate system.
+    """
+
+    # The IMU and scene camera coordinate systems have a fixed
+    # 102 degree rotation offset. See:
+    # https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data
+    imu_scene_rotation_diff = np.deg2rad(-90 - 12)
+
+    # This matrix is used to transform points in the scene
+    # camera coordinate system to their corresponding coordinates
+    # in the IMU coordinate system.
+    scene_to_imu = np.array(
+        [
+            [1.0, 0.0, 0.0],
+            [
+                0.0,
+                np.cos(imu_scene_rotation_diff),
+                -np.sin(imu_scene_rotation_diff),
+            ],
+            [
+                0.0,
+                np.sin(imu_scene_rotation_diff),
+                np.cos(imu_scene_rotation_diff),
+            ],
+        ]
+    )
+
+    # Neon provides 3D gaze in spherical coordinates by default,
+    # so we first transform the gaze data from spherical coordinates
+    # to Cartesian coordinates.
+    cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
+
+    # Apply the transformation from the scene camera to the IMU coordinate system.
+    gazes_in_imu = scene_to_imu @ cart_gazes_in_scene.T
+
+    # This array contains a timeseries of transformation matrices,
+    # as calculated from the IMU's timeseries of quaternions values.
+    # Each of these matrices are used to transform points in the IMU coordinate
+    # system to their corresponding coordinates in the world coordinate system.
+    imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
+
+    # Apply the transformations from the IMU to the world coordinate system.
+    gazes_in_world = [
+        imu_to_world @ gaze
+        for imu_to_world, gaze in zip(imu_to_world_matrices, gazes_in_imu.T)
+    ]
+
+    return np.array(gazes_in_world)
+
+
+def spherical_to_cartesian_scene(elevations, azimuths):
+    """
+    Convert Neon's spherical representation of 3D gaze to Cartesian coordinates.
+
+    Inputs:
+      - elevations (Nx1 np.array): A timeseries of gaze elevations (degrees),
+      specified in the scene camera coordinate system.
+      - azimuths (Nx1 np.array): A timeseries of gaze azimuths (degrees),
+      specified in the scene camera coordinate system.
+
+    Returns:
+      - cartesian_unit_vectors (Nx3 np.array): A timeseries of gaze unit
+      vectors, in Cartesian coordinates, specified in the scene camera
+      coordinate system.
+    """
+
+    elevations_rad = np.deg2rad(elevations)
+    azimuths_rad = np.deg2rad(azimuths)
+
+    # Elevation of 0 in Neon system corresponds to Y = 0, but
+    # an elevation of 0 in traditional spherical coordinates would
+    # correspond to Y = 1, so first we convert elevation to the
+    # more traditional format.
+    elevations_rad += np.pi / 2
+
+    # Azimuth of 0 in Neon system corresponds to X = 0, but
+    # an azimuth of 0 in traditional spherical coordinates would
+    # correspond to X = 1. Also, azimuth to the right in Neon is
+    # more positive, whereas it is more negative in traditional spherical coordiantes.
+    # So, first we convert azimuth to the more traditional format.
+    azimuths_rad *= -1.0
+    azimuths_rad += np.pi / 2
+
+    cartesian_unit_vectors = np.array(
+        [
+            np.sin(elevations_rad) * np.cos(azimuths_rad),
+            np.cos(elevations_rad),
+            np.sin(elevations_rad) * np.sin(azimuths_rad),
+        ]
+    ).T
+
+    return cartesian_unit_vectors
diff --git a/alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py b/alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py
index 86bcd9236..2af0fb828 100644
--- a/alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py
+++ b/alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py
@@ -6,3 +6,52 @@ def imu_heading_in_world(imu_quaternions):
     world_rotation_matrices = R.from_quat(imu_quaternions).as_matrix()
     headings_in_world = world_rotation_matrices @ imu_neutral_heading
     return headings_in_world
+
+
+def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
+    imu_scene_rotation_diff = np.deg2rad(-90 - 12)
+    scene_to_imu = np.array(
+        [
+            [1.0, 0.0, 0.0],
+            [
+                0.0,
+                np.cos(imu_scene_rotation_diff),
+                -np.sin(imu_scene_rotation_diff),
+            ],
+            [
+                0.0,
+                np.sin(imu_scene_rotation_diff),
+                np.cos(imu_scene_rotation_diff),
+            ],
+        ]
+    )
+
+    cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
+    gazes_in_imu = scene_to_imu @ cart_gazes_in_scene.T
+    imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
+    gazes_in_world = [
+        imu_to_world @ gaze
+        for imu_to_world, gaze in zip(imu_to_world_matrices, gazes_in_imu.T)
+    ]
+
+    return np.array(gazes_in_world)
+
+
+def spherical_to_cartesian_scene(elevations, azimuths):
+    elevations_rad = np.deg2rad(elevations)
+    azimuths_rad = np.deg2rad(azimuths)
+
+    elevations_rad += np.pi / 2
+
+    azimuths_rad *= -1.0
+    azimuths_rad += np.pi / 2
+
+    cartesian_unit_vectors = np.array(
+        [
+            np.sin(elevations_rad) * np.cos(azimuths_rad),
+            np.cos(elevations_rad),
+            np.sin(elevations_rad) * np.sin(azimuths_rad),
+        ]
+    ).T
+
+    return cartesian_unit_vectors

From b3ad60f49efb6bcccc332424d7423f68acd83e19 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Wed, 21 Aug 2024 13:56:37 +0200
Subject: [PATCH 19/54] cleanup submodule & code snippet imports, make imu
 alpha lab opening text more concise

---
 .gitmodules                                   |   4 +-
 alpha-lab/47303f69efcff77bd634c82c86612b84    |   1 -
 alpha-lab/imu-transformations/index.md        | 245 +-----------------
 .../pl-imu-transformations                    |   1 +
 .../pl_imu_transformations.py                 | 153 -----------
 .../pl_imu_transformations_no_comments.py     |  57 ----
 6 files changed, 16 insertions(+), 445 deletions(-)
 delete mode 160000 alpha-lab/47303f69efcff77bd634c82c86612b84
 create mode 160000 alpha-lab/imu-transformations/pl-imu-transformations
 delete mode 100644 alpha-lab/imu-transformations/pl_imu_transformations.py
 delete mode 100644 alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py

diff --git a/.gitmodules b/.gitmodules
index 3f6be0d5a..32e785748 100644
--- a/.gitmodules
+++ b/.gitmodules
@@ -1,3 +1,3 @@
-[submodule "alpha-lab/47303f69efcff77bd634c82c86612b84"]
-	path = alpha-lab/47303f69efcff77bd634c82c86612b84
+[submodule "alpha-lab/imu-transformations/pl-imu-transformations"]
+	path = alpha-lab/imu-transformations/pl-imu-transformations
 	url = https://gist.github.com/47303f69efcff77bd634c82c86612b84.git
diff --git a/alpha-lab/47303f69efcff77bd634c82c86612b84 b/alpha-lab/47303f69efcff77bd634c82c86612b84
deleted file mode 160000
index 542043d8a..000000000
--- a/alpha-lab/47303f69efcff77bd634c82c86612b84
+++ /dev/null
@@ -1 +0,0 @@
-Subproject commit 542043d8a4500ae908af6abe9846266a76bd1eeb
diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index e98849031..e449d179d 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -33,216 +33,43 @@ head and eye movements? The transformation functions in this tutorial will show
 
 This guide contains various transformation functions that can assist when analysing Neon's IMU data.
 
-First, it's important to understand how the IMU data should be interpreted. 
-The [orientation readings of the calibrated IMU](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data) 
-are specified with respect to magnetic North and gravity. We will refer to this as the 'world coordinate system'.
-
-To be exact, the 'world coordinate system' is distinct from the 'IMU coordinate system' - as the Neon module rotates, 
-the IMU coordinate system rotates with it. The IMU's orientation readings actually specify the rotational difference between its coordinate 
-system and the world coordinate system.
-
-- Y axis = vector pointing towards magnetic North.
-  - Rotations about this axis are called “roll” and range from -180 to +180 degrees. Wearing Neon normally in an upright position roughly corresponds to a roll of 0 degrees. Rightward head tilt is positive roll; leftward head tilt is negative roll.
-- Z axis = vector pointing upwards, opposite to the direction of gravity.
-  - Rotations about this axis are called “yaw” and range from -180 to +180 degrees. When the IMU is calibrated, a Neon module oriented at magnetic North corresponds to a yaw of 0 degrees. Leftward head turn is positive yaw; rightward head turn is negative yaw.
-- X axis = cross-product of Y and Z, which is a vector pointing rightwards.
-  - Rotations about this axis are called “pitch” and range from -90 to +90 degrees. Wearing Neon normally in an upright position roughly corresponds to a pitch of 0 degrees. Backward head tilt is positive pitch; forward head tilt is negative pitch
-
-![Diagram of the three Euler orientation angles measured by Neon's IMU](./imu-pitch-yaw-roll-black.png)
-
-::: tip
-💡 Note that Neon can sit differently on each wearer’s face, such that the headset is not necessarily in 
-line with a given plane. For example, if the wearer is facing magnetic North, the IMU might 
-still report some deviation from a neutral orientation.
-:::
-
-The raw gyroscope values give the rotational velocity about the X, Y, and Z axes of the IMU’s coordinate system. Their units are degrees/s (i.e., they provide the change 
-over time of each of these quantities). The reference frame for specifying rotational velocities rotates with the IMU.
-
-The raw acceleration values measure translational (i.e., linear) acceleration, in terms of g-force (units of g), 
-along the X, Y, and Z axes of the IMU’s coordinate system. The reference frame for specifying acceleration values 
-rotates with the IMU.
-
-::: tip
-💡 [g-force](https://en.wikipedia.org/wiki/G-force) is not the same as free-fall acceleration due 
-to gravity. For example, if you are standing on the surface of the Earth, then the g-force is equal and 
-opposite to gravity, keeping you at rest. In that case, if you were wearing Neon normally in an upright position, then 
-acceleration along the Z axis would be +1g, while it would be 0g along the X and Y axes.
-:::
-
-It can be helpful to also try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems.
-
-Now that we have laid out the relationship between the IMU and world coordinate systems, and how to interpret the
-readings, we can do some useful transformations.
+While working through this guide, it can be helpful to try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems.
 
 ## Obtain IMU Heading Vectors
 
 An alternate representation of IMU data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity:
 
-::: code-group
-
-```python [imu_heading_in_world()]
-<!--@include: ./pl_imu_transformations_no_comments.py{2,8}-->
-```
-
-```python:line-numbers {18,25,29} [(with comments)]
-<!--@include: ./pl_imu_transformations.py{2,32}-->
+```python
+<!--@include: ./pl-imu-transformations/pl_imu_transformations_no_comments.py{2,8}-->
 ```
-:::
  
-
 ## Transform IMU Acceleration Data to World Coordinates
 
-We mentioned above that the IMU’s acceleration data are specified with respect to the IMU’s coordinate system. Sometimes, it can be useful to have the acceleration data specified in the world coordinate system instead. The function below will perform this transformation.
-
-### 
+We mentioned above that the IMU’s acceleration data are specified with respect to the IMU’s coordinate system. Sometimes, it can be useful to have the acceleration data specified in the world coordinate system instead. The function below will perform this transformation:
 
-:::: details Code
 ```python
-def imu_acceleration_in_world(imu_accelerations, imu_quaternions):
-  """
-  Transform the IMU's acceleration values to the world coordinate system.
-    
-  Inputs:
-    - accelerations_imu (Nx3 np.array): A timeseries of acceleration data
-    from the IMU, where columns are in the order: "acceleration x",
-    "acceleration y", "acceleration z".
-    - imu_quaternions (Nx4 np.array): A timeseries of quaternions
-    from Neon's IMU stream.
-    
-  Returns:
-    - accelerations_world (Nx3 np.array): The corresponding timeseries of
-    acceleration data, expressed in the world coordinate system.
-  """
-    
-  # This array contains a timeseries of transformation matrices,
-  # as calculated from the IMU's timeseries of quaternions values.
-  # Each of these matrices sends points in the IMU coordinate system to their
-  # corresponding coordinates in the world coordinate system.
-  imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
-    
-  accelerations_world = [imu_to_world @ imu_acceleration for imu_to_world, imu_acceleration in zip(imu_to_world_matrices, imu_accelerations)]
-    
-  return np.array(accelerations_world)
+<!--@include: ./pl-imu-transformations/pl_imu_transformations_no_comments.py{11,21}-->
 ```
-::::
 
 ## Represent IMU and Gaze Data in the Same Coordinate System
 
-Neon simultaneously records gaze and IMU data, making it possible to study the relationship between head and 
-eye movements.
+Neon simultaneously records gaze and IMU data, making it possible to study the relationship between head and eye movements.
 
-To facilitate the comparison, it can be useful to represent them in the same coordinate system. The coordinates of gaze are specified with respect to the scene camera coordinate system and the function below, `gaze_scene_to_world`, uses data from the IMU to transform gaze to the world coordinate system.
+To facilitate the comparison, it can be useful to represent them in the same coordinate system. The coordinates of gaze are specified with respect to the scene camera coordinate system and the function below, `gaze_scene_to_world`, uses data from the IMU to transform gaze to the world coordinate system:
 
-::: code-group
-
-```python [gaze_scene_to_world() & spherical_to_cartesian_scene()]
-<!--@include: ./pl_imu_transformations_no_comments.py{11,57}-->
-```
-
-```python:line-numbers {34,39-53,58,61,67,70-75,94,95,101,108-119} [(with comments)]
-<!--@include: ./pl_imu_transformations.py{35,153}-->
+```python
+<!--@include: ./pl-imu-transformations/pl_imu_transformations_no_comments.py{24,73}-->
 ```
-:::
 
 ## Represent IMU and 3D Eyestate in the Same Coordinate System
 
 The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) 
 provided by Neon are aligned with the scene camera coordinate system. This means we can reuse elements of 
-the `gaze_scene_to_world` function to reconstruct the pose of the eyes in the world coordinate system.
+the `gaze_scene_to_world` function to reconstruct the pose of the eyes in the world coordinate system:
 
-:::: details Code
 ```python
-def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
-  """
-  Transforms 3D eyestate data to the world coordinate system.
-  
-  Note that the 3D eyestate data and the IMU quaternions should be sampled 
-  at the same timestamps. You can linearly interpolate the IMU data
-  to ensure this.
-    
-  The origin of the IMU coordinate system is the same as the
-  origin of the world coordinate system.
-    
-  The code in this function is adapted from the `plimu` visualization utility:
-  https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279
-    
-  Inputs:
-    - eyeball_centers (Nx3 np.array): A timeseries of eyeball center
-    estimates (mm) for one eye, as provided by Neon's 3D eyestate data stream.
-    - optical_axes (Nx3 np.array): A timeseries of optical axis estimates
-    (3D unit vectors) for the same eye, as provided by Neon's 3D eyestate data stream.
-    - imu_quaternions (Nx4 np.array): A timeseries of quaternion values from
-    Neon's IMU.
-    
-  Returns:
-    - eyeball_centers_in_world (Nx3 np.array): The corresponding timeseries of
-    eyeball center estimates (mm) for one eye, transformed to the world coordinate
-    system of the IMU.
-    - optical_axes_in_world (Nx3 np.array): The corresponding timeseries of optical
-    axis estimates (3D unit vectors) for the same eye, transformed to the world
-    coordinate system of the IMU.
-  """
-  
-  # The eyeball centers are specified in terms of their distance from
-  # the center of the scene camera, so to accurately convert them to
-  # world coordinates, we need the position of the scene camera in
-  # the IMU coordinate system. Here, we express that position
-  # in millimeters.
-  scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
-
-  # The coordinate system of 3D eyestate is aligned with
-  # the coordinate system of the scene camera, so we can
-  # make use of the scene->to->IMU transformation procedure.
-  #
-  # The IMU and scene camera coordinate systems have a fixed
-  # 102 degree rotation offset. See here:
-  # https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data
-  imu_scene_rotation_diff = np.deg2rad(-90 - 12)
-  
-  # This matrix is used to transform points in the scene
-  # camera coordinate system to their corresponding coordinates
-  # in the IMU coordinate system.
-  scene_to_imu = np.array(
-      [
-        [1.0, 0.0, 0.0],
-        [
-          0.0,
-          np.cos(imu_scene_rotation_diff),
-          -np.sin(imu_scene_rotation_diff),
-        ],
-        [
-          0.0,
-          np.sin(imu_scene_rotation_diff),
-          np.cos(imu_scene_rotation_diff),
-        ],
-      ]
-  )
-  
-  # Convert the coordinates of the eyeball center and optical axes
-  # to the IMU coordinate system.
-  eyeball_centers_in_imu = scene_to_imu @ eyeball_centers.T
-  optical_axes_in_imu = scene_to_imu @ optical_axes.T
-    
-  # Take into account that the scene camera is offset from the IMU.
-  eyeball_centers_in_imu[0, :] += scene_camera_position_in_imu[0]
-  eyeball_centers_in_imu[1, :] += scene_camera_position_in_imu[1]
-  eyeball_centers_in_imu[2, :] += scene_camera_position_in_imu[2]
-  
-  # This array contains a timeseries of transformation matrices,
-  # as calculated from the IMU's timeseries of quaternions values.
-  # Each of these matrices are used to transform points in the IMU
-  # coordinate system to their corresponding coordinates in the world
-  # coordinate system.
-  imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
-  
-  # Apply the transformations from the IMU to the world coordinate system.
-  eyeball_centers_in_world = [imu_to_world @ eye_center for imu_to_world, eye_center in zip(imu_to_world_matrices, eyeball_centers_in_imu.T)]
-  optical_axes_in_world = [imu_to_world @ optical_axis for imu_to_world, optical_axis in zip(imu_to_world_matrices, optical_axes_in_imu.T)]
-    
-  return np.array(eyeball_centers_in_world), np.array(optical_axes_in_world)
+<!--@include: ./pl-imu-transformations/pl_imu_transformations_no_comments.py{76,119}-->
 ```
-::::
 
 ## Convert (Cartesian) World Points to Spherical Coordinates
 
@@ -254,60 +81,15 @@ For the values returned by  `gaze_scene_to_world`, the function below will do th
 When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds to a neutral orientation:
 i.e., aimed at magnetic North and parallel to the horizon.
 
-:::: details Code
+
 ```python
-def cartesian_to_spherical_world(world_points_3d):
-  """
-  Convert points 3D Cartesian world coordinates to spherical coordinates.
-
-  The origin of the world coordinate system is the same as the origin of the
-  IMU coordinate system.
-
-  When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds
-  to a neutral orientation: i.e., aimed at magnetic North and parallel to the horizon.
-
-  Inputs:
-    - world_points_3d (Nx3 np.array): A collection of 3D points in 
-    Cartesian world coordinates.
-        
-  Returns:
-    - elevation (Nx1 np.array): The corresponding elevation value, in degrees,
-    for each 3D world point (i.e., vertical rotation away from neutral orientation).
-    Note: neutral orientation = 0 elevation; orientation upwards is positive;
-    orientation downwards is negative.
-    - azimuth (Nx1 np.array): The corresponding azimuth value, in degrees, for
-    each 3D world point (i.e., horizontal rotation away from neutral orientation).
-    Note: neutral orientation = 0 azimuth, orientation leftwards is positive;
-    orientation rightwards is negative.
-  """
-
-  x = world_points_3d[:, 0]
-  y = world_points_3d[:, 1]
-  z = world_points_3d[:, 2]
-
-  radii = np.sqrt(x**2 + y**2 + z**2)
-
-  elevation = -(np.arccos(z / radii) - np.pi / 2)
-  azimuth = np.arctan2(y, x) - np.pi/2
-  
-  # Keep all azimuth values in the range of [-180, 180] to remain consistent
-  # with the yaw orientation values provided by the IMU
-  azimuth[azimuth < -np.pi] += 2 * np.pi
-  azimuth[azimuth > np.pi] -= 2 * np.pi
-
-  # Convert from radians to degrees
-  elevation = np.rad2deg(elevation)
-  azimuth = np.rad2deg(azimuth)
-
-  return elevation, azimuth
+<!--@include: ./pl-imu-transformations/pl_imu_transformations_no_comments.py{122,138}-->
 ```
-::::
 
 ## Analysis example
 
 Below is a brief example of how to run the functions on this page using IMU data downloaded from Pupil Cloud.
 
-:::: details Code
 ```python
 gaze = pd.read_csv("gaze.csv")
 eye3d = pd.read_csv("3d_eye_states.csv")
@@ -370,7 +152,6 @@ world_accelerations = imu_acceleration_in_world(
 
 gaze_elevations_world, gaze_azimuths_world = cartesian_to_spherical_world(world_gazes)
 ```
-::::
 
 ::: tip
 Need assistance with the IMU code in this article? Or do you have something more custom in mind? Reach out to us via email at [info@pupil-labs.com](mailto:info@pupil-labs.com), on our [Discord server](https://pupil-labs.com/chat/), or visit our [Support Page](https://pupil-labs.com/products/support/) for dedicated support options.
diff --git a/alpha-lab/imu-transformations/pl-imu-transformations b/alpha-lab/imu-transformations/pl-imu-transformations
new file mode 160000
index 000000000..0b702c0a7
--- /dev/null
+++ b/alpha-lab/imu-transformations/pl-imu-transformations
@@ -0,0 +1 @@
+Subproject commit 0b702c0a75efcb1b82321042e17a025cf9a639c9
diff --git a/alpha-lab/imu-transformations/pl_imu_transformations.py b/alpha-lab/imu-transformations/pl_imu_transformations.py
deleted file mode 100644
index e3362e5a1..000000000
--- a/alpha-lab/imu-transformations/pl_imu_transformations.py
+++ /dev/null
@@ -1,153 +0,0 @@
-import numpy as np
-from scipy.spatial.transform import Rotation as R
-
-def imu_heading_in_world(imu_quaternions):
-    """
-    Construct heading vectors from the IMU's quaternion values.
-
-    Inputs:
-      - imu_quaternions (Nx4 np.array): A timeseries of quaternions
-      from Neon's IMU stream.
-
-    Returns:
-      - headings_in_world (Nx3 np.array): The corresponding timeseries of
-      IMU heading vectors in the world coordinate system.
-    """
-
-    # We start by specifying the direction of a neutral heading vector
-    # in the IMU's coordinate system.
-    imu_neutral_heading = np.array([0.0, 1.0, 0.0])
-
-    # This array contains a timeseries of transformation matrices,
-    # as calculated from the IMU's timeseries of quaternions values.
-    # Each of these matrices are used to transform points in the IMU
-    # coordinate system to their corresponding coordinates in the world
-    # coordinate system.
-    world_rotation_matrices = R.from_quat(imu_quaternions).as_matrix()
-
-    # We now apply each transformation matrix to the neutral IMU heading vector
-    # to obtain a timeseries of heading vectors in the world coordinate system.
-    headings_in_world = world_rotation_matrices @ imu_neutral_heading
-    
-    return headings_in_world
-
-
-def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
-    """
-    Transform a 3D gaze ray to the world coordinate system.
-
-    Note that the gaze data and the IMU quaternion should be sampled
-    at the same timestamps. You can linearly interpolate the IMU data
-    to ensure this.
-
-    The origin of the IMU coordinate system is the same as the
-    origin of the world coordinate system.
-    The code in this function is adapted from the `plimu` visualization utility:
-    https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279
-
-    This function makes use of the spherical_to_cartesian_scene function,
-    defined below, that converts 3D gaze rays from spherical coordinates
-    to Cartesian coordinates.
-
-    Inputs:
-      - gaze_elevations (Nx1 np.array): A timeseries of gaze elevations (degrees),
-      specified in the scene camera coordinate system.
-      - gaze_azimuths (Nx1 np.array): A timeseries of gaze azimuths (degrees),
-      specified in the scene camera coordinate system.
-      - imu_quaternions (Nx4 np.array): A timeseries of quaternion values
-      from Neon's IMU.
-
-    Returns:
-      - gazes_in_world (Nx3 np.array): The corresponding timeseries of
-      3D Cartesian gaze unit vectors, specified in the world coordinate system.
-    """
-
-    # The IMU and scene camera coordinate systems have a fixed
-    # 102 degree rotation offset. See:
-    # https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data
-    imu_scene_rotation_diff = np.deg2rad(-90 - 12)
-
-    # This matrix is used to transform points in the scene
-    # camera coordinate system to their corresponding coordinates
-    # in the IMU coordinate system.
-    scene_to_imu = np.array(
-        [
-            [1.0, 0.0, 0.0],
-            [
-                0.0,
-                np.cos(imu_scene_rotation_diff),
-                -np.sin(imu_scene_rotation_diff),
-            ],
-            [
-                0.0,
-                np.sin(imu_scene_rotation_diff),
-                np.cos(imu_scene_rotation_diff),
-            ],
-        ]
-    )
-
-    # Neon provides 3D gaze in spherical coordinates by default,
-    # so we first transform the gaze data from spherical coordinates
-    # to Cartesian coordinates.
-    cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
-
-    # Apply the transformation from the scene camera to the IMU coordinate system.
-    gazes_in_imu = scene_to_imu @ cart_gazes_in_scene.T
-
-    # This array contains a timeseries of transformation matrices,
-    # as calculated from the IMU's timeseries of quaternions values.
-    # Each of these matrices are used to transform points in the IMU coordinate
-    # system to their corresponding coordinates in the world coordinate system.
-    imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
-
-    # Apply the transformations from the IMU to the world coordinate system.
-    gazes_in_world = [
-        imu_to_world @ gaze
-        for imu_to_world, gaze in zip(imu_to_world_matrices, gazes_in_imu.T)
-    ]
-
-    return np.array(gazes_in_world)
-
-
-def spherical_to_cartesian_scene(elevations, azimuths):
-    """
-    Convert Neon's spherical representation of 3D gaze to Cartesian coordinates.
-
-    Inputs:
-      - elevations (Nx1 np.array): A timeseries of gaze elevations (degrees),
-      specified in the scene camera coordinate system.
-      - azimuths (Nx1 np.array): A timeseries of gaze azimuths (degrees),
-      specified in the scene camera coordinate system.
-
-    Returns:
-      - cartesian_unit_vectors (Nx3 np.array): A timeseries of gaze unit
-      vectors, in Cartesian coordinates, specified in the scene camera
-      coordinate system.
-    """
-
-    elevations_rad = np.deg2rad(elevations)
-    azimuths_rad = np.deg2rad(azimuths)
-
-    # Elevation of 0 in Neon system corresponds to Y = 0, but
-    # an elevation of 0 in traditional spherical coordinates would
-    # correspond to Y = 1, so first we convert elevation to the
-    # more traditional format.
-    elevations_rad += np.pi / 2
-
-    # Azimuth of 0 in Neon system corresponds to X = 0, but
-    # an azimuth of 0 in traditional spherical coordinates would
-    # correspond to X = 1. Also, azimuth to the right in Neon is
-    # more positive, whereas it is more negative in traditional spherical coordiantes.
-    # So, first we convert azimuth to the more traditional format.
-    azimuths_rad *= -1.0
-    azimuths_rad += np.pi / 2
-
-    cartesian_unit_vectors = np.array(
-        [
-            np.sin(elevations_rad) * np.cos(azimuths_rad),
-            np.cos(elevations_rad),
-            np.sin(elevations_rad) * np.sin(azimuths_rad),
-        ]
-    ).T
-
-    return cartesian_unit_vectors
diff --git a/alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py b/alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py
deleted file mode 100644
index 2af0fb828..000000000
--- a/alpha-lab/imu-transformations/pl_imu_transformations_no_comments.py
+++ /dev/null
@@ -1,57 +0,0 @@
-import numpy as np
-from scipy.spatial.transform import Rotation as R
-
-def imu_heading_in_world(imu_quaternions):
-    imu_neutral_heading = np.array([0.0, 1.0, 0.0])
-    world_rotation_matrices = R.from_quat(imu_quaternions).as_matrix()
-    headings_in_world = world_rotation_matrices @ imu_neutral_heading
-    return headings_in_world
-
-
-def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
-    imu_scene_rotation_diff = np.deg2rad(-90 - 12)
-    scene_to_imu = np.array(
-        [
-            [1.0, 0.0, 0.0],
-            [
-                0.0,
-                np.cos(imu_scene_rotation_diff),
-                -np.sin(imu_scene_rotation_diff),
-            ],
-            [
-                0.0,
-                np.sin(imu_scene_rotation_diff),
-                np.cos(imu_scene_rotation_diff),
-            ],
-        ]
-    )
-
-    cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
-    gazes_in_imu = scene_to_imu @ cart_gazes_in_scene.T
-    imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
-    gazes_in_world = [
-        imu_to_world @ gaze
-        for imu_to_world, gaze in zip(imu_to_world_matrices, gazes_in_imu.T)
-    ]
-
-    return np.array(gazes_in_world)
-
-
-def spherical_to_cartesian_scene(elevations, azimuths):
-    elevations_rad = np.deg2rad(elevations)
-    azimuths_rad = np.deg2rad(azimuths)
-
-    elevations_rad += np.pi / 2
-
-    azimuths_rad *= -1.0
-    azimuths_rad += np.pi / 2
-
-    cartesian_unit_vectors = np.array(
-        [
-            np.sin(elevations_rad) * np.cos(azimuths_rad),
-            np.cos(elevations_rad),
-            np.sin(elevations_rad) * np.sin(azimuths_rad),
-        ]
-    ).T
-
-    return cartesian_unit_vectors

From cc8a873914078ca107392e5a97dc5003dd3298f4 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Wed, 21 Aug 2024 14:12:33 +0200
Subject: [PATCH 20/54] small cleanup

---
 alpha-lab/imu-transformations/index.md | 13 ++++++-------
 1 file changed, 6 insertions(+), 7 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index e449d179d..2aa7c1648 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -75,7 +75,7 @@ the `gaze_scene_to_world` function to reconstruct the pose of the eyes in the wo
 
 When studying head orientation and gaze orientation as observers navigate a 3D environment, it can be useful 
 to know how much these quantities deviate from pointing at a given landmark. For instance, you might want to 
-know when someone’s gaze or heading deviates from pointing at the horizon. This can be simplified by 
+know when someone’s gaze or heading deviates from parallel with the horizon. This can be simplified by 
 converting world points from Cartesian to spherical coordinates. The orientation values from the IMU are already in such a format. 
 For the values returned by  `gaze_scene_to_world`, the function below will do the necessary transformation. 
 When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds to a neutral orientation:
@@ -96,7 +96,6 @@ eye3d = pd.read_csv("3d_eye_states.csv")
 imu = pd.read_csv("imu.csv")
 
 gaze_ts = gaze["timestamp [ns]"]
-eye3d_ts = eye3d["timestamp [ns]"]
 imu_ts = imu["timestamp [ns]"]
 
 # We have more gaze datapoints (sampled at 200Hz) than
@@ -133,6 +132,11 @@ eyeball_centers_left = np.array([
 
 imu_headings = imu_heading_in_world(quaternions_resampled)
 
+world_accelerations = imu_acceleration_in_world(
+  accelerations_resampled,
+  quaternions_resampled,
+)
+
 world_gazes = gaze_scene_to_world(
   gaze["elevation [deg]"],
   gaze["azimuth [deg]"],
@@ -145,11 +149,6 @@ eye_centers_left_world, optical_axes_left_world = eyestate_to_world(
   quaternions_resampled,
 )
 
-world_accelerations = imu_acceleration_in_world(
-  accelerations_resampled,
-  quaternions_resampled,
-)
-
 gaze_elevations_world, gaze_azimuths_world = cartesian_to_spherical_world(world_gazes)
 ```
 

From 623274be1fb16ba148bbe7cdfad4b683c2106096 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Wed, 21 Aug 2024 15:02:31 +0200
Subject: [PATCH 21/54] update github workflows to accomodate importing code
 snippets from a gist (as submodule) - relevant for IMU alpha lab and later
 articles

---
 .github/workflows/PR-checks.yml               | 3 ++-
 .github/workflows/build-deploy-production.yml | 3 ++-
 .github/workflows/build-deploy-staging.yml    | 3 ++-
 .github/workflows/build-only.yml              | 3 ++-
 4 files changed, 8 insertions(+), 4 deletions(-)

diff --git a/.github/workflows/PR-checks.yml b/.github/workflows/PR-checks.yml
index ec237fd3e..bab62093d 100644
--- a/.github/workflows/PR-checks.yml
+++ b/.github/workflows/PR-checks.yml
@@ -27,9 +27,10 @@ jobs:
         working-directory: ./${{ matrix.component }}
         run: npm install
 
-      - name: Copy node_modules
+      - name: Copy node_modules & init submodules
         shell: bash
         run: |
+          git submodule update --init --recursive
           cp -r ${{ matrix.component }}/node_modules .
           ls -la
 
diff --git a/.github/workflows/build-deploy-production.yml b/.github/workflows/build-deploy-production.yml
index 682612ab0..17aa6638a 100644
--- a/.github/workflows/build-deploy-production.yml
+++ b/.github/workflows/build-deploy-production.yml
@@ -122,9 +122,10 @@ jobs:
         run: npm install
       
       # Copy node modules to provide dependencies for shared components in root folder
-      - name: Copy node_modules
+      - name: Copy node_modules & init submodules
         shell: bash
         run: |
+          git submodule update --init --recursive
           cp -r alpha-lab/node_modules .
           ls -la
 
diff --git a/.github/workflows/build-deploy-staging.yml b/.github/workflows/build-deploy-staging.yml
index 92b61a2bf..4d425cee1 100644
--- a/.github/workflows/build-deploy-staging.yml
+++ b/.github/workflows/build-deploy-staging.yml
@@ -120,9 +120,10 @@ jobs:
         run: npm install
       
       # Copy node modules to provide dependencies for shared components in root folder
-      - name: Copy node_modules
+      - name: Copy node_modules & init submodules
         shell: bash
         run: |
+          git submodule update --init --recursive
           cp -r alpha-lab/node_modules .
           ls -la
 
diff --git a/.github/workflows/build-only.yml b/.github/workflows/build-only.yml
index cab837c5e..449e4244e 100644
--- a/.github/workflows/build-only.yml
+++ b/.github/workflows/build-only.yml
@@ -120,9 +120,10 @@ jobs:
         run: npm install
       
       # Copy node modules to provide dependencies for shared components in root folder
-      - name: Copy node_modules
+      - name: Copy node_modules & init submodules
         shell: bash
         run: |
+          git submodule update --init --recursive
           cp -r alpha-lab/node_modules .
           ls -la
 

From b96ddb2a2d1063e5e7f3268ac888180690b660fe Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Thu, 22 Aug 2024 11:55:29 +0200
Subject: [PATCH 22/54] transition away from gist approach,
 restructure/compactify imu code, change structure of article accordingly

---
 .gitmodules                                   |   3 -
 alpha-lab/imu-transformations/index.md        | 204 ++++++++++++++++--
 .../pl-imu-transformations                    |   1 -
 3 files changed, 183 insertions(+), 25 deletions(-)
 delete mode 100644 .gitmodules
 delete mode 160000 alpha-lab/imu-transformations/pl-imu-transformations

diff --git a/.gitmodules b/.gitmodules
deleted file mode 100644
index 32e785748..000000000
--- a/.gitmodules
+++ /dev/null
@@ -1,3 +0,0 @@
-[submodule "alpha-lab/imu-transformations/pl-imu-transformations"]
-	path = alpha-lab/imu-transformations/pl-imu-transformations
-	url = https://gist.github.com/47303f69efcff77bd634c82c86612b84.git
diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 2aa7c1648..76f54de73 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -35,40 +35,173 @@ This guide contains various transformation functions that can assist when analys
 
 While working through this guide, it can be helpful to try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems.
 
+## Transforming to World Coordinates
+
+One of the key steps when dealing with the IMU is the transformation that take coordinates in the local IMU coordinate system to their corresponding coordinates in the world coordinate system. The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function, defined below, applies the transformation and will be used throughout this article.
+
+```python
+def transform_imu_to_world(imu_coordinates, imu_quaternions):
+    # This array contains a timeseries of transformation matrices,
+    # as calculated from the IMU's timeseries of quaternions values.
+    imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
+    
+    if len(imu_coordinates) == 1:
+        return imu_to_world_matrices @ imu_coordinates
+    else:
+        return np.array([
+            imu_to_world @ imu_coord
+            for imu_to_world, imu_coord in zip(
+                imu_to_world_matrices, imu_coordinates
+            )
+        ])
+```
+
+Now that we have the `transform_imu_to_world` function, let's use it!
+
 ## Obtain IMU Heading Vectors
 
-An alternate representation of IMU data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity:
+One representation of IMU orientation data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity:
 
 ```python
-<!--@include: ./pl-imu-transformations/pl_imu_transformations_no_comments.py{2,8}-->
+heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
+headings_in_world = transform_imu_to_world(heading_neutral_in_imu_coords, imu_quaternions)
 ```
- 
-## Transform IMU Acceleration Data to World Coordinates
 
-We mentioned above that the IMU’s acceleration data are specified with respect to the IMU’s coordinate system. Sometimes, it can be useful to have the acceleration data specified in the world coordinate system instead. The function below will perform this transformation:
+## Transform Acceleration Data to World Coordinates
+
+The IMU’s acceleration data are specified in IMU's local coordinate system. Sometimes, it can be useful to have the acceleration data specified in the world coordinate system instead:
 
 ```python
-<!--@include: ./pl-imu-transformations/pl_imu_transformations_no_comments.py{11,21}-->
+accelerations_in_world = transform_imu_to_world(imu_accelerations, imu_quaternions)
 ```
 
-## Represent IMU and Gaze Data in the Same Coordinate System
+## Transform Gaze Data to World Coordinates
 
 Neon simultaneously records gaze and IMU data, making it possible to study the relationship between head and eye movements.
 
-To facilitate the comparison, it can be useful to represent them in the same coordinate system. The coordinates of gaze are specified with respect to the scene camera coordinate system and the function below, `gaze_scene_to_world`, uses data from the IMU to transform gaze to the world coordinate system:
+To facilitate the comparison, it can be useful to represent them in the same coordinate system. The coordinates of gaze are specified with respect to the scene camera coordinate system and the function below, `gaze_scene_to_world`, uses data from the IMU to transform gaze to the world coordinate system. The `gaze_scene_to_world` function depends on the `transform_scene_to_imu` and `spherical_to_cartesian_scene`, which we define first.
 
 ```python
-<!--@include: ./pl-imu-transformations/pl_imu_transformations_no_comments.py{24,73}-->
+def transform_scene_to_imu(coords_in_scene, translation = None):
+    # The IMU and scene camera coordinate systems have a fixed
+    # 102 degree rotation offset. See:
+    # https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data
+    imu_scene_rotation_diff = np.deg2rad(-102)
+    scene_to_imu = np.array(
+        [
+            [1.0, 0.0, 0.0],
+            [
+                0.0,
+                np.cos(imu_scene_rotation_diff),
+                -np.sin(imu_scene_rotation_diff),
+            ],
+            [
+                0.0,
+                np.sin(imu_scene_rotation_diff),
+                np.cos(imu_scene_rotation_diff),
+            ],
+        ]
+    )
+
+    if translation:
+        scene_to_imu_homogeneous = np.zeros((4, 4))
+        scene_to_imu_homogeneous[:3, :3] = scene_to_imu
+        scene_to_imu_homogeneous[:3, 3] = translation
+
+        coords_in_scene_homogeneous = cv2.convertPointsToHomogeneous(coords_in_scene)
+        return cv2.convertPointsFromHomogeneous(scene_to_imu_homogeneous @ coords_in_scene_homogeneous)
+    else:
+        return scene_to_imu @ coords_in_scene
+
+
+def spherical_to_cartesian_scene(elevations, azimuths):
+    """
+        Convert Neon's spherical representation of 3D gaze to Cartesian coordinates.
+    """
+
+    elevations_rad = np.deg2rad(elevations)
+    azimuths_rad = np.deg2rad(azimuths)
+
+    # Elevation of 0 in Neon system corresponds to Y = 0, but
+    # an elevation of 0 in traditional spherical coordinates would
+    # correspond to Y = 1, so we convert elevation to the
+    # more traditional format.
+    elevations_rad += np.pi / 2
+
+    # Azimuth of 0 in Neon system corresponds to X = 0, but
+    # an azimuth of 0 in traditional spherical coordinates would
+    # correspond to X = 1. Also, azimuth to the right in Neon is
+    # more positive, whereas it is more negative in traditional spherical coordiantes.
+    # So, we convert azimuth to the more traditional format.
+    azimuths_rad *= -1.0
+    azimuths_rad += np.pi / 2
+
+    return np.array(
+        [
+            np.sin(elevations_rad) * np.cos(azimuths_rad),
+            np.cos(elevations_rad),
+            np.sin(elevations_rad) * np.sin(azimuths_rad),
+        ]
+    ).T
 ```
 
-## Represent IMU and 3D Eyestate in the Same Coordinate System
+Now, we have the tools to build the `gaze_scene_to_world` function. Note that the origin of the IMU coordinate system is the same as the origin of the world coordinate system.:
 
-The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) 
-provided by Neon are aligned with the scene camera coordinate system. This means we can reuse elements of 
-the `gaze_scene_to_world` function to reconstruct the pose of the eyes in the world coordinate system:
+```python
+def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
+    """
+        Note that the gaze data and the IMU quaternions should be sampled
+        at the same timestamps. You can linearly interpolate the IMU data
+        to ensure this.
+
+        The code in this function is adapted from the `plimu` visualization utility:
+        https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279
+
+        This function makes use of the spherical_to_cartesian_scene function,
+        defined below, that converts Neon's 3D gaze rays from spherical coordinates
+        to Cartesian coordinates.
+    """
+
+    cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
+    gazes_in_imu = transform_scene_to_imu(cart_gazes_in_scene.T)
+    return imu_to_world(gazes_in_imu.T, imu_quaternions)
+```
+
+## Transform 3D Eyestate to World Coordinates
+
+The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) provided by Neon are aligned with the scene camera coordinate system. This means we can reuse elements of the `gaze_scene_to_world` function to reconstruct the pose of the eyes in the world coordinate system. Note that the origin of the IMU coordinate system is the same as the origin of the world coordinate system.
 
 ```python
-<!--@include: ./pl-imu-transformations/pl_imu_transformations_no_comments.py{76,119}-->
+def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
+    """
+        Note that the 3D eyestate data and the IMU quaternions should be sampled
+        at the same timestamps. You can linearly interpolate the IMU data
+        to ensure this.
+
+        The code in this function is adapted from the `plimu` visualization utility:
+        https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279
+    """
+
+    # The eyeball centers are specified in terms of their distance from
+    # the center of the scene camera, so to accurately convert them to
+    # world coordinates, we need the position of the scene camera in
+    # the IMU coordinate system. Here, we express that position
+    # in millimeters.
+    scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
+
+    # The coordinate system of 3D eyestate is aligned with
+    # the coordinate system of the scene camera, so we can
+    # make use of the scene->to->IMU transformation procedure.
+    #
+    # We just need to take into account that the scene camera is
+    # offset from the IMU.
+    eyeball_centers_in_imu = transform_scene_to_imu(eyeball_centers.T, translation=scene_camera_position_in_imu)
+    optical_axes_in_imu = transform_scene_to_imu(optical_axes.T, translation=scene_camera_position_in_imu)
+
+    eyeball_centers_in_world = imu_to_world(eyeball_centers_in_imu.T, imu_quaternions)
+    optical_axes_in_world = imu_to_world(optical_axes_in_imu.T, imu_quaternions)
+
+    return eyeball_centers_in_world, optical_axes_in_world
 ```
 
 ## Convert (Cartesian) World Points to Spherical Coordinates
@@ -81,9 +214,40 @@ For the values returned by  `gaze_scene_to_world`, the function below will do th
 When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds to a neutral orientation:
 i.e., aimed at magnetic North and parallel to the horizon.
 
-
 ```python
-<!--@include: ./pl-imu-transformations/pl_imu_transformations_no_comments.py{122,138}-->
+def cartesian_to_spherical_world(world_points_3d):
+    """
+        Convert points in 3D Cartesian world coordinates to spherical coordinates.
+    
+        When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds
+        to a neutral orientation: i.e., aimed at magnetic North and parallel to the horizon.
+    
+        Note:
+            - For elevation: neutral orientation = 0 elevation; orientation upwards is positive;
+            orientation downwards is negative.
+            - For azimuth: neutral orientation = 0 azimuth, orientation leftwards is positive;
+            orientation rightwards is negative.
+    """
+
+    x = world_points_3d[:, 0]
+    y = world_points_3d[:, 1]
+    z = world_points_3d[:, 2]
+
+    radii = np.sqrt(x**2 + y**2 + z**2)
+
+    elevation = -(np.arccos(z / radii) - np.pi / 2)
+    azimuth = np.arctan2(y, x) - np.pi / 2
+
+    # Keep all azimuth values in the range of [-180, 180] to remain consistent
+    # with the yaw orientation values provided by the IMU
+    azimuth[azimuth < -np.pi] += 2 * np.pi
+    azimuth[azimuth > np.pi] -= 2 * np.pi
+
+    # Convert from radians to degrees
+    elevation = np.rad2deg(elevation)
+    azimuth = np.rad2deg(azimuth)
+
+    return elevation, azimuth
 ```
 
 ## Analysis example
@@ -130,12 +294,10 @@ eyeball_centers_left = np.array([
 
 # Now, we can apply the functions.
 
-imu_headings = imu_heading_in_world(quaternions_resampled)
+heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
+headings_in_world = transform_imu_to_world(heading_neutral_in_imu_coords, quaternions_resampled)
 
-world_accelerations = imu_acceleration_in_world(
-  accelerations_resampled,
-  quaternions_resampled,
-)
+accelerations_in_world = transform_imu_to_world(accelerations_resampled, quaternions_resampled)
 
 world_gazes = gaze_scene_to_world(
   gaze["elevation [deg]"],
diff --git a/alpha-lab/imu-transformations/pl-imu-transformations b/alpha-lab/imu-transformations/pl-imu-transformations
deleted file mode 160000
index 0b702c0a7..000000000
--- a/alpha-lab/imu-transformations/pl-imu-transformations
+++ /dev/null
@@ -1 +0,0 @@
-Subproject commit 0b702c0a75efcb1b82321042e17a025cf9a639c9

From 9c9c72ba083a709cd48aa6fbc2cf54faa92e8730 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Thu, 22 Aug 2024 12:39:52 +0200
Subject: [PATCH 23/54] adapt text + code better to new format

---
 alpha-lab/imu-transformations/index.md | 230 +++++++++----------------
 1 file changed, 80 insertions(+), 150 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 76f54de73..6365d2d80 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -33,11 +33,13 @@ head and eye movements? The transformation functions in this tutorial will show
 
 This guide contains various transformation functions that can assist when analysing Neon's IMU data.
 
-While working through this guide, it can be helpful to try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems.
+While working through this guide, it can be helpful to try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems. For example, some of the code in this article is [adapted from it](https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279).
 
-## Transforming to World Coordinates
+## Transform to World Coordinates
 
-One of the key steps when dealing with the IMU is the transformation that take coordinates in the local IMU coordinate system to their corresponding coordinates in the world coordinate system. The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function, defined below, applies the transformation and will be used throughout this article.
+One of the key steps when dealing with the IMU is the transformation that takes coordinates in the local IMU coordinate system to their corresponding coordinates in the world coordinate system. The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function is defined below and will be used throughout this article.
+
+Note that the origin of the IMU coordinate system is the same as the origin of the world coordinate system.
 
 ```python
 def transform_imu_to_world(imu_coordinates, imu_quaternions):
@@ -60,33 +62,40 @@ Now that we have the `transform_imu_to_world` function, let's use it!
 
 ## Obtain IMU Heading Vectors
 
-One representation of IMU orientation data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity:
+An alternative representation of IMU orientation data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity:
 
 ```python
 heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
-headings_in_world = transform_imu_to_world(heading_neutral_in_imu_coords, imu_quaternions)
+headings_in_world = transform_imu_to_world(
+    heading_neutral_in_imu_coords,
+    imu_quaternions,
+)
 ```
 
-## Transform Acceleration Data to World Coordinates
+## Acceleration in World
 
-The IMU’s acceleration data are specified in IMU's local coordinate system. Sometimes, it can be useful to have the acceleration data specified in the world coordinate system instead:
+The IMU’s acceleration data are specified in its local coordinate system. Sometimes, it can be useful to have the acceleration data specified in the world coordinate system instead:
 
 ```python
-accelerations_in_world = transform_imu_to_world(imu_accelerations, imu_quaternions)
+accelerations_in_world = transform_imu_to_world(
+    imu_accelerations,
+    imu_quaternions,
+)
 ```
 
-## Transform Gaze Data to World Coordinates
+## Gaze to World Coordinates
 
 Neon simultaneously records gaze and IMU data, making it possible to study the relationship between head and eye movements.
 
-To facilitate the comparison, it can be useful to represent them in the same coordinate system. The coordinates of gaze are specified with respect to the scene camera coordinate system and the function below, `gaze_scene_to_world`, uses data from the IMU to transform gaze to the world coordinate system. The `gaze_scene_to_world` function depends on the `transform_scene_to_imu` and `spherical_to_cartesian_scene`, which we define first.
+To facilitate the comparison, it can be useful to represent these data streams in the same coordinate system. An important step is accounting for [the fixed 102 degree rotation offset between the scene camera and IMU coordinate systems](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data), as depicted below.
+
+![Diagrams showing the 102 degree rotation offset between the IMU and scene camera coordinate systems.](./imu-scene_camera_offset-black.jpg)
+
+We can use data from the IMU to transform gaze from scene camera coordinates to world coordinates. This is facilitated by the `transform_scene_to_imu` and `spherical_to_cartesian_scene` functions:
 
 ```python
 def transform_scene_to_imu(coords_in_scene, translation = None):
-    # The IMU and scene camera coordinate systems have a fixed
-    # 102 degree rotation offset. See:
-    # https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data
-    imu_scene_rotation_diff = np.deg2rad(-102)
+    imu_scene_rotation_diff = np.deg2rad(-90 - 12)
     scene_to_imu = np.array(
         [
             [1.0, 0.0, 0.0],
@@ -109,14 +118,16 @@ def transform_scene_to_imu(coords_in_scene, translation = None):
         scene_to_imu_homogeneous[:3, 3] = translation
 
         coords_in_scene_homogeneous = cv2.convertPointsToHomogeneous(coords_in_scene)
-        return cv2.convertPointsFromHomogeneous(scene_to_imu_homogeneous @ coords_in_scene_homogeneous)
+        return cv2.convertPointsFromHomogeneous(
+            scene_to_imu_homogeneous @ coords_in_scene_homogeneous
+        )
     else:
         return scene_to_imu @ coords_in_scene
 
 
 def spherical_to_cartesian_scene(elevations, azimuths):
     """
-        Convert Neon's spherical representation of 3D gaze to Cartesian coordinates.
+    Convert Neon's spherical representation of 3D gaze to Cartesian coordinates.
     """
 
     elevations_rad = np.deg2rad(elevations)
@@ -131,8 +142,9 @@ def spherical_to_cartesian_scene(elevations, azimuths):
     # Azimuth of 0 in Neon system corresponds to X = 0, but
     # an azimuth of 0 in traditional spherical coordinates would
     # correspond to X = 1. Also, azimuth to the right in Neon is
-    # more positive, whereas it is more negative in traditional spherical coordiantes.
-    # So, we convert azimuth to the more traditional format.
+    # more positive, whereas it is more negative in traditional
+    # spherical coordiantes. So, we convert azimuth to the more
+    # traditional format.
     azimuths_rad *= -1.0
     azimuths_rad += np.pi / 2
 
@@ -145,88 +157,71 @@ def spherical_to_cartesian_scene(elevations, azimuths):
     ).T
 ```
 
-Now, we have the tools to build the `gaze_scene_to_world` function. Note that the origin of the IMU coordinate system is the same as the origin of the world coordinate system.:
+Now we have the tools to convert gaze data to world coordinates:
 
 ```python
-def gaze_scene_to_world(gaze_elevations, gaze_azimuths, imu_quaternions):
-    """
-        Note that the gaze data and the IMU quaternions should be sampled
-        at the same timestamps. You can linearly interpolate the IMU data
-        to ensure this.
+# The gaze data and the IMU quaternions should be sampled at the
+# same timestamps. You can linearly interpolate the IMU data to
+# ensure this.
 
-        The code in this function is adapted from the `plimu` visualization utility:
-        https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279
-
-        This function makes use of the spherical_to_cartesian_scene function,
-        defined below, that converts Neon's 3D gaze rays from spherical coordinates
-        to Cartesian coordinates.
-    """
-
-    cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
-    gazes_in_imu = transform_scene_to_imu(cart_gazes_in_scene.T)
-    return imu_to_world(gazes_in_imu.T, imu_quaternions)
+cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
+gazes_in_imu = transform_scene_to_imu(cart_gazes_in_scene.T)
+gazes_in_world = transform_imu_to_world(gazes_in_imu.T, imu_quaternions)
 ```
 
-## Transform 3D Eyestate to World Coordinates
+## Eyestate to World Coordinates
 
-The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) provided by Neon are aligned with the scene camera coordinate system. This means we can reuse elements of the `gaze_scene_to_world` function to reconstruct the pose of the eyes in the world coordinate system. Note that the origin of the IMU coordinate system is the same as the origin of the world coordinate system.
+The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) provided by Neon are aligned with the scene camera coordinate system. To reconstruct the pose of the eyes in the world coordinate system, we need to also consider that the scene camera coordinate system is positioned downwards and a bit back from the IMU:
 
 ```python
-def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
-    """
-        Note that the 3D eyestate data and the IMU quaternions should be sampled
-        at the same timestamps. You can linearly interpolate the IMU data
-        to ensure this.
-
-        The code in this function is adapted from the `plimu` visualization utility:
-        https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279
-    """
+# The 3D eyestate data and the IMU quaternions should be sampled
+# at the same timestamps. You can linearly interpolate the IMU data
+# to ensure this.
+
+# The eyeball centers are specified in terms of their distance from
+# the center of the scene camera, so to accurately convert them to
+# world coordinates, we need to account for the position of the
+# scene camera in the IMU coordinate system. Here, we express that
+# position in millimeters.
+scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
+
+eyeball_centers_in_imu = transform_scene_to_imu(
+    eyeball_centers.T,
+    translation=scene_camera_position_in_imu,
+)
+eyeball_centers_in_world = transform_imu_to_world(
+    eyeball_centers_in_imu.T,
+    imu_quaternions,
+)
 
-    # The eyeball centers are specified in terms of their distance from
-    # the center of the scene camera, so to accurately convert them to
-    # world coordinates, we need the position of the scene camera in
-    # the IMU coordinate system. Here, we express that position
-    # in millimeters.
-    scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
-
-    # The coordinate system of 3D eyestate is aligned with
-    # the coordinate system of the scene camera, so we can
-    # make use of the scene->to->IMU transformation procedure.
-    #
-    # We just need to take into account that the scene camera is
-    # offset from the IMU.
-    eyeball_centers_in_imu = transform_scene_to_imu(eyeball_centers.T, translation=scene_camera_position_in_imu)
-    optical_axes_in_imu = transform_scene_to_imu(optical_axes.T, translation=scene_camera_position_in_imu)
-
-    eyeball_centers_in_world = imu_to_world(eyeball_centers_in_imu.T, imu_quaternions)
-    optical_axes_in_world = imu_to_world(optical_axes_in_imu.T, imu_quaternions)
-
-    return eyeball_centers_in_world, optical_axes_in_world
+optical_axes_in_imu = transform_scene_to_imu(
+    optical_axes.T,
+    translation=scene_camera_position_in_imu,
+)
+optical_axes_in_world = transform_imu_to_world(
+    optical_axes_in_imu.T,
+    imu_quaternions
+)
 ```
 
-## Convert (Cartesian) World Points to Spherical Coordinates
+## World Spherical Coordinates
 
-When studying head orientation and gaze orientation as observers navigate a 3D environment, it can be useful 
-to know how much these quantities deviate from pointing at a given landmark. For instance, you might want to 
-know when someone’s gaze or heading deviates from parallel with the horizon. This can be simplified by 
-converting world points from Cartesian to spherical coordinates. The orientation values from the IMU are already in such a format. 
-For the values returned by  `gaze_scene_to_world`, the function below will do the necessary transformation. 
-When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds to a neutral orientation:
-i.e., aimed at magnetic North and parallel to the horizon.
+When studying head orientation and gaze orientation as observers navigate a 3D environment, it can be useful to know how much these quantities deviate from pointing at a given landmark or direction. For instance, you might want to know when someone’s gaze or heading deviates from parallel with the horizon. This can be simplified by converting world points from Cartesian to spherical coordinates. The orientation values from the IMU are already in such a format. For gaze data in world coordinates, the function below will do the necessary transformation. When wearing Neon normally, an elevation and azimuth of 0 degrees corresponds to a neutral orientation: i.e., aimed at magnetic North and parallel to the horizon.
 
 ```python
 def cartesian_to_spherical_world(world_points_3d):
     """
-        Convert points in 3D Cartesian world coordinates to spherical coordinates.
-    
-        When wearing Neon normally, then an elevation and azimuth of 0 degrees corresponds
-        to a neutral orientation: i.e., aimed at magnetic North and parallel to the horizon.
+    Convert points in 3D Cartesian world coordinates to spherical coordinates.
     
-        Note:
-            - For elevation: neutral orientation = 0 elevation; orientation upwards is positive;
-            orientation downwards is negative.
-            - For azimuth: neutral orientation = 0 azimuth, orientation leftwards is positive;
-            orientation rightwards is negative.
+    For elevation:
+      - Neutral orientation = 0 (i.e., parallel with horizon)
+      - Upwards is positive
+      - Downwards is negative
+
+    For azimuth:
+      - Neutral orientation = 0 (i.e., aligned with magnetic North)
+      - Leftwards is positive
+      - Rightwards is negative
     """
 
     x = world_points_3d[:, 0]
@@ -238,82 +233,17 @@ def cartesian_to_spherical_world(world_points_3d):
     elevation = -(np.arccos(z / radii) - np.pi / 2)
     azimuth = np.arctan2(y, x) - np.pi / 2
 
-    # Keep all azimuth values in the range of [-180, 180] to remain consistent
-    # with the yaw orientation values provided by the IMU
+    # Keep all azimuth values in the range of [-180, 180] to remain
+    # consistent with the yaw orientation values provided by the IMU.
     azimuth[azimuth < -np.pi] += 2 * np.pi
     azimuth[azimuth > np.pi] -= 2 * np.pi
 
-    # Convert from radians to degrees
     elevation = np.rad2deg(elevation)
     azimuth = np.rad2deg(azimuth)
 
     return elevation, azimuth
 ```
 
-## Analysis example
-
-Below is a brief example of how to run the functions on this page using IMU data downloaded from Pupil Cloud.
-
-```python
-gaze = pd.read_csv("gaze.csv")
-eye3d = pd.read_csv("3d_eye_states.csv")
-imu = pd.read_csv("imu.csv")
-
-gaze_ts = gaze["timestamp [ns]"]
-imu_ts = imu["timestamp [ns]"]
-
-# We have more gaze datapoints (sampled at 200Hz) than
-# IMU datapoints (sampled at 110Hz), so linearly interpolate
-# the IMU datapoints to be congruent with gaze.
-quaternions_resampled = np.array([
-  np.interp(gaze_ts, imu_ts, imu["quaternion x"]),
-  np.interp(gaze_ts, imu_ts, imu["quaternion y"]),
-  np.interp(gaze_ts, imu_ts, imu["quaternion z"]),
-  np.interp(gaze_ts, imu_ts, imu["quaternion w"]),
-]).T
-
-accelerations_resampled = np.array([
-  np.interp(gaze_ts, imu_ts, imu["acceleration x [g]"]),
-  np.interp(gaze_ts, imu_ts, imu["acceleration y [g]"]),
-  np.interp(gaze_ts, imu_ts, imu["acceleration z [g]"]),
-]).T
-
-# Put the eyestate data in a format that is compatible with
-# the transformation functions from this article.
-optical_axes_left = np.array([
-  eye3d["optical axis left x"],
-  eye3d["optical axis left y"],
-  eye3d["optical axis left z"],
-]).T
-
-eyeball_centers_left = np.array([
-  eye3d["eyeball center left x [mm]"],
-  eye3d["eyeball center left y [mm]"],
-  eye3d["eyeball center left z [mm]"],
-]).T
-
-# Now, we can apply the functions.
-
-heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
-headings_in_world = transform_imu_to_world(heading_neutral_in_imu_coords, quaternions_resampled)
-
-accelerations_in_world = transform_imu_to_world(accelerations_resampled, quaternions_resampled)
-
-world_gazes = gaze_scene_to_world(
-  gaze["elevation [deg]"],
-  gaze["azimuth [deg]"],
-  quaternions_resampled,
-)
-
-eye_centers_left_world, optical_axes_left_world = eyestate_to_world(
-  eyeball_centers_left,
-  optical_axes_left,
-  quaternions_resampled,
-)
-
-gaze_elevations_world, gaze_azimuths_world = cartesian_to_spherical_world(world_gazes)
-```
-
 ::: tip
 Need assistance with the IMU code in this article? Or do you have something more custom in mind? Reach out to us via email at [info@pupil-labs.com](mailto:info@pupil-labs.com), on our [Discord server](https://pupil-labs.com/chat/), or visit our [Support Page](https://pupil-labs.com/products/support/) for dedicated support options.
 :::

From d458968b7979541ae919c8db234df8b19ecc2a05 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Thu, 22 Aug 2024 12:48:29 +0200
Subject: [PATCH 24/54] undo github actions submodule changes; add imu+scene
 diagram as png

---
 .github/workflows/PR-checks.yml                |   3 +--
 .github/workflows/build-deploy-production.yml  |   3 +--
 .github/workflows/build-deploy-staging.yml     |   3 +--
 .github/workflows/build-only.yml               |   3 +--
 .../imu-scene_camera_offset-black.png          | Bin 0 -> 145548 bytes
 alpha-lab/imu-transformations/index.md         |   2 +-
 6 files changed, 5 insertions(+), 9 deletions(-)
 create mode 100644 alpha-lab/imu-transformations/imu-scene_camera_offset-black.png

diff --git a/.github/workflows/PR-checks.yml b/.github/workflows/PR-checks.yml
index bab62093d..ec237fd3e 100644
--- a/.github/workflows/PR-checks.yml
+++ b/.github/workflows/PR-checks.yml
@@ -27,10 +27,9 @@ jobs:
         working-directory: ./${{ matrix.component }}
         run: npm install
 
-      - name: Copy node_modules & init submodules
+      - name: Copy node_modules
         shell: bash
         run: |
-          git submodule update --init --recursive
           cp -r ${{ matrix.component }}/node_modules .
           ls -la
 
diff --git a/.github/workflows/build-deploy-production.yml b/.github/workflows/build-deploy-production.yml
index 17aa6638a..682612ab0 100644
--- a/.github/workflows/build-deploy-production.yml
+++ b/.github/workflows/build-deploy-production.yml
@@ -122,10 +122,9 @@ jobs:
         run: npm install
       
       # Copy node modules to provide dependencies for shared components in root folder
-      - name: Copy node_modules & init submodules
+      - name: Copy node_modules
         shell: bash
         run: |
-          git submodule update --init --recursive
           cp -r alpha-lab/node_modules .
           ls -la
 
diff --git a/.github/workflows/build-deploy-staging.yml b/.github/workflows/build-deploy-staging.yml
index 4d425cee1..92b61a2bf 100644
--- a/.github/workflows/build-deploy-staging.yml
+++ b/.github/workflows/build-deploy-staging.yml
@@ -120,10 +120,9 @@ jobs:
         run: npm install
       
       # Copy node modules to provide dependencies for shared components in root folder
-      - name: Copy node_modules & init submodules
+      - name: Copy node_modules
         shell: bash
         run: |
-          git submodule update --init --recursive
           cp -r alpha-lab/node_modules .
           ls -la
 
diff --git a/.github/workflows/build-only.yml b/.github/workflows/build-only.yml
index 449e4244e..cab837c5e 100644
--- a/.github/workflows/build-only.yml
+++ b/.github/workflows/build-only.yml
@@ -120,10 +120,9 @@ jobs:
         run: npm install
       
       # Copy node modules to provide dependencies for shared components in root folder
-      - name: Copy node_modules & init submodules
+      - name: Copy node_modules
         shell: bash
         run: |
-          git submodule update --init --recursive
           cp -r alpha-lab/node_modules .
           ls -la
 
diff --git a/alpha-lab/imu-transformations/imu-scene_camera_offset-black.png b/alpha-lab/imu-transformations/imu-scene_camera_offset-black.png
new file mode 100644
index 0000000000000000000000000000000000000000..808f96257312e2e3f806fc00e9676dfb7592586a
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diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 6365d2d80..d9f0de144 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -89,7 +89,7 @@ Neon simultaneously records gaze and IMU data, making it possible to study the r
 
 To facilitate the comparison, it can be useful to represent these data streams in the same coordinate system. An important step is accounting for [the fixed 102 degree rotation offset between the scene camera and IMU coordinate systems](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data), as depicted below.
 
-![Diagrams showing the 102 degree rotation offset between the IMU and scene camera coordinate systems.](./imu-scene_camera_offset-black.jpg)
+![Diagrams showing the 102 degree rotation offset between the IMU and scene camera coordinate systems.](./imu-scene_camera_offset-black.png)
 
 We can use data from the IMU to transform gaze from scene camera coordinates to world coordinates. This is facilitated by the `transform_scene_to_imu` and `spherical_to_cartesian_scene` functions:
 

From cf2a6fea5451e4e554328c8a257662bca95477bf Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Thu, 22 Aug 2024 13:00:12 +0200
Subject: [PATCH 25/54] tiny cleanup

---
 alpha-lab/imu-transformations/index.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index d9f0de144..a143043eb 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -37,7 +37,7 @@ While working through this guide, it can be helpful to try out our IMU visualiza
 
 ## Transform to World Coordinates
 
-One of the key steps when dealing with the IMU is the transformation that takes coordinates in the local IMU coordinate system to their corresponding coordinates in the world coordinate system. The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function is defined below and will be used throughout this article.
+One of the key steps when dealing with the IMU is the transformation that takes coordinates in the local IMU coordinate system to their corresponding coordinates in the world coordinate system. The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function, defined below, will be used throughout this article.
 
 Note that the origin of the IMU coordinate system is the same as the origin of the world coordinate system.
 
@@ -206,7 +206,7 @@ optical_axes_in_world = transform_imu_to_world(
 
 ## World Spherical Coordinates
 
-When studying head orientation and gaze orientation as observers navigate a 3D environment, it can be useful to know how much these quantities deviate from pointing at a given landmark or direction. For instance, you might want to know when someone’s gaze or heading deviates from parallel with the horizon. This can be simplified by converting world points from Cartesian to spherical coordinates. The orientation values from the IMU are already in such a format. For gaze data in world coordinates, the function below will do the necessary transformation. When wearing Neon normally, an elevation and azimuth of 0 degrees corresponds to a neutral orientation: i.e., aimed at magnetic North and parallel to the horizon.
+When studying head orientation and gaze orientation as observers navigate a 3D environment, it can be useful to know how much these quantities deviate from pointing at a given landmark or direction. For instance, you might want to know when someone’s gaze or heading deviates from parallel with the horizon. This can be simplified by converting world points from Cartesian to spherical coordinates. The [Euler angles from the IMU](https://docs.pupil-labs.com/neon/data-collection/data-streams/#euler-angles) are already in a compatible format. For gaze data in world coordinates, the `cartesian_to_spherical_world` function below will do the necessary transformation. When wearing Neon normally, an elevation and azimuth of 0 degrees corresponds to a neutral orientation: i.e., aimed at magnetic North and parallel to the horizon.
 
 ```python
 def cartesian_to_spherical_world(world_points_3d):

From 01f3160e7488ef253ee253c05716fc993bd4c6d1 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Thu, 22 Aug 2024 13:39:00 +0200
Subject: [PATCH 26/54] make R import from scipy clearer; fix small code typo

---
 alpha-lab/imu-transformations/index.md | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index a143043eb..bf2a1662d 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -42,12 +42,14 @@ One of the key steps when dealing with the IMU is the transformation that takes
 Note that the origin of the IMU coordinate system is the same as the origin of the world coordinate system.
 
 ```python
+from scipy.spatial.transform import Rotation as R
+
 def transform_imu_to_world(imu_coordinates, imu_quaternions):
     # This array contains a timeseries of transformation matrices,
     # as calculated from the IMU's timeseries of quaternions values.
     imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()
     
-    if len(imu_coordinates) == 1:
+    if np.ndim(imu_coordinates) == 1:
         return imu_to_world_matrices @ imu_coordinates
     else:
         return np.array([

From eb76bb4a89556badab01d1f6be8934eb0990ed4e Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Thu, 22 Aug 2024 13:54:31 +0200
Subject: [PATCH 27/54] tidy formatting a bit

---
 alpha-lab/imu-transformations/index.md | 18 ++++++------------
 1 file changed, 6 insertions(+), 12 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index bf2a1662d..1d7df53a5 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -69,8 +69,7 @@ An alternative representation of IMU orientation data is a heading vector that p
 ```python
 heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
 headings_in_world = transform_imu_to_world(
-    heading_neutral_in_imu_coords,
-    imu_quaternions,
+    heading_neutral_in_imu_coords, imu_quaternions,
 )
 ```
 
@@ -80,8 +79,7 @@ The IMU’s acceleration data are specified in its local coordinate system. Some
 
 ```python
 accelerations_in_world = transform_imu_to_world(
-    imu_accelerations,
-    imu_quaternions,
+    imu_accelerations, imu_quaternions,
 )
 ```
 
@@ -188,21 +186,17 @@ The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/dat
 scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
 
 eyeball_centers_in_imu = transform_scene_to_imu(
-    eyeball_centers.T,
-    translation=scene_camera_position_in_imu,
+    eyeball_centers.T, translation=scene_camera_position_in_imu,
 )
 eyeball_centers_in_world = transform_imu_to_world(
-    eyeball_centers_in_imu.T,
-    imu_quaternions,
+    eyeball_centers_in_imu.T, imu_quaternions,
 )
 
 optical_axes_in_imu = transform_scene_to_imu(
-    optical_axes.T,
-    translation=scene_camera_position_in_imu,
+    optical_axes.T, translation=scene_camera_position_in_imu,
 )
 optical_axes_in_world = transform_imu_to_world(
-    optical_axes_in_imu.T,
-    imu_quaternions
+    optical_axes_in_imu.T, imu_quaternions,
 )
 ```
 

From 98342426deec756d5944d2c6c8291da8f2478300 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Thu, 22 Aug 2024 15:11:36 +0200
Subject: [PATCH 28/54] fix homogeneous part of transform_scene_to_imu based on
 results of my test code

---
 alpha-lab/imu-transformations/index.md | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 1d7df53a5..685e753ba 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -117,10 +117,10 @@ def transform_scene_to_imu(coords_in_scene, translation = None):
         scene_to_imu_homogeneous[:3, :3] = scene_to_imu
         scene_to_imu_homogeneous[:3, 3] = translation
 
-        coords_in_scene_homogeneous = cv2.convertPointsToHomogeneous(coords_in_scene)
-        return cv2.convertPointsFromHomogeneous(
-            scene_to_imu_homogeneous @ coords_in_scene_homogeneous
-        )
+        coords_in_scene_homogeneous = cv2.convertPointsToHomogeneous(coords_in_scene.T)
+        coords_in_imu_homogeneous = scene_to_imu_homogeneous @ coords_in_scene_homogeneous.reshape(-1, 4).T
+        coords_in_imu = cv2.convertPointsFromHomogeneous(coords_in_imu_homogeneous.T)
+        return coords_in_imu.reshape(-1, 3).T
     else:
         return scene_to_imu @ coords_in_scene
 
@@ -193,7 +193,7 @@ eyeball_centers_in_world = transform_imu_to_world(
 )
 
 optical_axes_in_imu = transform_scene_to_imu(
-    optical_axes.T, translation=scene_camera_position_in_imu,
+    optical_axes.T,
 )
 optical_axes_in_world = transform_imu_to_world(
     optical_axes_in_imu.T, imu_quaternions,

From c1b6182315337f41b038d077bafab3b7b0c7d211 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Thu, 22 Aug 2024 16:56:38 +0200
Subject: [PATCH 29/54] further generalize + compress + simplify; re-tested new
 functions and they needed a tweak to work correctly again; they are back in a
 ready state

---
 alpha-lab/imu-transformations/index.md | 43 ++++++++++++++------------
 1 file changed, 23 insertions(+), 20 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 685e753ba..a1301ade6 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -94,7 +94,7 @@ To facilitate the comparison, it can be useful to represent these data streams i
 We can use data from the IMU to transform gaze from scene camera coordinates to world coordinates. This is facilitated by the `transform_scene_to_imu` and `spherical_to_cartesian_scene` functions:
 
 ```python
-def transform_scene_to_imu(coords_in_scene, translation = None):
+def transform_scene_to_imu(coords_in_scene, translation=None):
     imu_scene_rotation_diff = np.deg2rad(-90 - 12)
     scene_to_imu = np.array(
         [
@@ -112,17 +112,18 @@ def transform_scene_to_imu(coords_in_scene, translation = None):
         ]
     )
 
-    if translation:
+    if translation is not None:
         scene_to_imu_homogeneous = np.zeros((4, 4))
         scene_to_imu_homogeneous[:3, :3] = scene_to_imu
         scene_to_imu_homogeneous[:3, 3] = translation
 
-        coords_in_scene_homogeneous = cv2.convertPointsToHomogeneous(coords_in_scene.T)
+        coords_in_scene_homogeneous = cv2.convertPointsToHomogeneous(coords_in_scene)
         coords_in_imu_homogeneous = scene_to_imu_homogeneous @ coords_in_scene_homogeneous.reshape(-1, 4).T
-        coords_in_imu = cv2.convertPointsFromHomogeneous(coords_in_imu_homogeneous.T)
-        return coords_in_imu.reshape(-1, 3).T
+        coords_in_imu = cv2.convertPointsFromHomogeneous(coords_in_imu_homogeneous.T).reshape(-1, 3)
     else:
-        return scene_to_imu @ coords_in_scene
+        coords_in_imu = (scene_to_imu @ coords_in_scene.T).T
+
+    return coords_in_imu
 
 
 def spherical_to_cartesian_scene(elevations, azimuths):
@@ -157,7 +158,15 @@ def spherical_to_cartesian_scene(elevations, azimuths):
     ).T
 ```
 
-Now we have the tools to convert gaze data to world coordinates:
+Putting these together, we can build a general `transform_scene_to_world` function:
+
+```python
+def transform_scene_to_world(coords_in_scene, imu_quaternions, translation=None):
+    coords_in_imu = transform_scene_to_imu(coords_in_scene, translation)
+    return transform_imu_to_world(coords_in_imu, imu_quaternions)
+```
+
+Now we have the tools to convert 3D gaze data to world coordinates:
 
 ```python
 # The gaze data and the IMU quaternions should be sampled at the
@@ -165,8 +174,7 @@ Now we have the tools to convert gaze data to world coordinates:
 # ensure this.
 
 cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
-gazes_in_imu = transform_scene_to_imu(cart_gazes_in_scene.T)
-gazes_in_world = transform_imu_to_world(gazes_in_imu.T, imu_quaternions)
+cart_gazes_in_world = transform_scene_to_world(cart_gazes_in_scene, imu_quaternions)
 ```
 
 ## Eyestate to World Coordinates
@@ -184,19 +192,14 @@ The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/dat
 # scene camera in the IMU coordinate system. Here, we express that
 # position in millimeters.
 scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
-
-eyeball_centers_in_imu = transform_scene_to_imu(
-    eyeball_centers.T, translation=scene_camera_position_in_imu,
-)
-eyeball_centers_in_world = transform_imu_to_world(
-    eyeball_centers_in_imu.T, imu_quaternions,
+eyeball_centers_in_world = transform_scene_to_world(
+    eyeball_centers, imu_quaternions, translation=scene_camera_position_in_imu,
 )
 
-optical_axes_in_imu = transform_scene_to_imu(
-    optical_axes.T,
-)
-optical_axes_in_world = transform_imu_to_world(
-    optical_axes_in_imu.T, imu_quaternions,
+# The optical axes are unit vectors originating at the eyeball centers,
+# so there is no need to translate them.
+optical_axes_in_world = transform_scene_to_world(
+    optical_axes, imu_quaternions,
 )
 ```
 

From 86962832bcd93feee93734bf8ec45f1afef16b5b Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Fri, 23 Aug 2024 00:23:08 +0200
Subject: [PATCH 30/54] simplify homogeneous coordinates part a bit; fix typo

---
 alpha-lab/imu-transformations/index.md | 20 +++++++++-----------
 1 file changed, 9 insertions(+), 11 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index a1301ade6..a6a246bbf 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -94,7 +94,7 @@ To facilitate the comparison, it can be useful to represent these data streams i
 We can use data from the IMU to transform gaze from scene camera coordinates to world coordinates. This is facilitated by the `transform_scene_to_imu` and `spherical_to_cartesian_scene` functions:
 
 ```python
-def transform_scene_to_imu(coords_in_scene, translation=None):
+def transform_scene_to_imu(coords_in_scene, translation=np.zeros((3,))):
     imu_scene_rotation_diff = np.deg2rad(-90 - 12)
     scene_to_imu = np.array(
         [
@@ -112,16 +112,14 @@ def transform_scene_to_imu(coords_in_scene, translation=None):
         ]
     )
 
-    if translation is not None:
-        scene_to_imu_homogeneous = np.zeros((4, 4))
-        scene_to_imu_homogeneous[:3, :3] = scene_to_imu
-        scene_to_imu_homogeneous[:3, 3] = translation
+    scene_to_imu_homogeneous = np.zeros((4, 4))
+    scene_to_imu_homogeneous[:3, :3] = scene_to_imu
+    scene_to_imu_homogeneous[:3, 3] = translation
+    scene_to_imu_homogeneous[3, 3] = 1.0
 
-        coords_in_scene_homogeneous = cv2.convertPointsToHomogeneous(coords_in_scene)
-        coords_in_imu_homogeneous = scene_to_imu_homogeneous @ coords_in_scene_homogeneous.reshape(-1, 4).T
-        coords_in_imu = cv2.convertPointsFromHomogeneous(coords_in_imu_homogeneous.T).reshape(-1, 3)
-    else:
-        coords_in_imu = (scene_to_imu @ coords_in_scene.T).T
+    coords_in_scene_homogeneous = cv2.convertPointsToHomogeneous(coords_in_scene)
+    coords_in_imu_homogeneous = scene_to_imu_homogeneous @ coords_in_scene_homogeneous.reshape(-1, 4).T
+    coords_in_imu = cv2.convertPointsFromHomogeneous(coords_in_imu_homogeneous.T).reshape(-1, 3)
 
     return coords_in_imu
 
@@ -161,7 +159,7 @@ def spherical_to_cartesian_scene(elevations, azimuths):
 Putting these together, we can build a general `transform_scene_to_world` function:
 
 ```python
-def transform_scene_to_world(coords_in_scene, imu_quaternions, translation=None):
+def transform_scene_to_world(coords_in_scene, imu_quaternions, translation=np.zeros((3,))):
     coords_in_imu = transform_scene_to_imu(coords_in_scene, translation)
     return transform_imu_to_world(coords_in_imu, imu_quaternions)
 ```

From ea482c2abfc60ace83c3149e769a79bab2d2fa7f Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Fri, 23 Aug 2024 09:51:08 +0200
Subject: [PATCH 31/54] unproject 2d gaze rather than convert 3d gaze from
 spherical to cartesian

---
 alpha-lab/imu-transformations/index.md | 86 +++++++++++++-------------
 1 file changed, 43 insertions(+), 43 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index a6a246bbf..428d73f96 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -69,7 +69,7 @@ An alternative representation of IMU orientation data is a heading vector that p
 ```python
 heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
 headings_in_world = transform_imu_to_world(
-    heading_neutral_in_imu_coords, imu_quaternions,
+    heading_neutral_in_imu_coords, imu_quaternions
 )
 ```
 
@@ -79,19 +79,19 @@ The IMU’s acceleration data are specified in its local coordinate system. Some
 
 ```python
 accelerations_in_world = transform_imu_to_world(
-    imu_accelerations, imu_quaternions,
+    imu_accelerations, imu_quaternions
 )
 ```
 
-## Gaze to World Coordinates
+## Scene to World Coordinates
 
 Neon simultaneously records gaze and IMU data, making it possible to study the relationship between head and eye movements.
 
 To facilitate the comparison, it can be useful to represent these data streams in the same coordinate system. An important step is accounting for [the fixed 102 degree rotation offset between the scene camera and IMU coordinate systems](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data), as depicted below.
 
-![Diagrams showing the 102 degree rotation offset between the IMU and scene camera coordinate systems.](./imu-scene_camera_offset-black.png)
+![Diagrams showing the fixed 102 degree rotation offset between the IMU and scene camera coordinate systems.](./imu-scene_camera_offset-black.png)
 
-We can use data from the IMU to transform gaze from scene camera coordinates to world coordinates. This is facilitated by the `transform_scene_to_imu` and `spherical_to_cartesian_scene` functions:
+We can use data from the IMU to transform gaze in scene camera coordinates to world coordinates. We proceed by building a `transform_scene_to_imu` function:
 
 ```python
 def transform_scene_to_imu(coords_in_scene, translation=np.zeros((3,))):
@@ -122,41 +122,9 @@ def transform_scene_to_imu(coords_in_scene, translation=np.zeros((3,))):
     coords_in_imu = cv2.convertPointsFromHomogeneous(coords_in_imu_homogeneous.T).reshape(-1, 3)
 
     return coords_in_imu
-
-
-def spherical_to_cartesian_scene(elevations, azimuths):
-    """
-    Convert Neon's spherical representation of 3D gaze to Cartesian coordinates.
-    """
-
-    elevations_rad = np.deg2rad(elevations)
-    azimuths_rad = np.deg2rad(azimuths)
-
-    # Elevation of 0 in Neon system corresponds to Y = 0, but
-    # an elevation of 0 in traditional spherical coordinates would
-    # correspond to Y = 1, so we convert elevation to the
-    # more traditional format.
-    elevations_rad += np.pi / 2
-
-    # Azimuth of 0 in Neon system corresponds to X = 0, but
-    # an azimuth of 0 in traditional spherical coordinates would
-    # correspond to X = 1. Also, azimuth to the right in Neon is
-    # more positive, whereas it is more negative in traditional
-    # spherical coordiantes. So, we convert azimuth to the more
-    # traditional format.
-    azimuths_rad *= -1.0
-    azimuths_rad += np.pi / 2
-
-    return np.array(
-        [
-            np.sin(elevations_rad) * np.cos(azimuths_rad),
-            np.cos(elevations_rad),
-            np.sin(elevations_rad) * np.sin(azimuths_rad),
-        ]
-    ).T
 ```
 
-Putting these together, we can build a general `transform_scene_to_world` function:
+Putting together the `transform_scene_to_imu` and `transform_imu_to_world` functions, we can build a general `transform_scene_to_world` function:
 
 ```python
 def transform_scene_to_world(coords_in_scene, imu_quaternions, translation=np.zeros((3,))):
@@ -164,15 +132,47 @@ def transform_scene_to_world(coords_in_scene, imu_quaternions, translation=np.ze
     return transform_imu_to_world(coords_in_imu, imu_quaternions)
 ```
 
-Now we have the tools to convert 3D gaze data to world coordinates:
+Now we have the tools to convert 3D gaze data to world coordinates, but first, we need to convert 2D gaze to 3D by undistorting and unprojecting the image coordinates:
+
+```python
+def unproject_2d_gaze(gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients):
+    """
+    Transform the 2D gaze values from Neon to 3D gaze vectors.
+    
+    See the associated Alpha Labs article for a tutorial on undistorting:
+    https://docs.pupil-labs.com/alpha-lab/undistort/
+    """
+
+    gaze_points_2d_undist = cv2.undistortPoints(gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients)
+    gaze_vectors_3d = cv2.convertPointsToHomogeneous(gaze_points_2d_undist)
+    gaze_vectors_3d /= np.linalg.norm(gaze_vectors_3d, axis=2)[:, np.newaxis]
+    
+    return gaze_vectors_3d
+```
+
+Putting all of this together, we can convert 2D gaze data in scene image coordinates to 3D gaze data in world coordinates:
 
 ```python
 # The gaze data and the IMU quaternions should be sampled at the
 # same timestamps. You can linearly interpolate the IMU data to
 # ensure this.
 
-cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevations, gaze_azimuths)
-cart_gazes_in_world = transform_scene_to_world(cart_gazes_in_scene, imu_quaternions)
+# I'm not sure if we want to show how to load the camera calib data
+#
+# scene_camera_info = []
+# with open(rec_dir + "scene_camera.json") as f:
+#     scene_camera_info = json.load(f)
+#
+# scene_camera_matrix = np.array(scene_camera_info["camera_matrix"])
+# scene_distortion_coefficients = np.array(scene_camera_info["distortion_coefficients"])
+
+gaze_points_2d = gaze[["gaze x [px]", "gaze y [px]"]].to_numpy()
+cart_gazes_in_scene = undistort_and_unproject(
+    gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients
+)
+cart_gazes_in_world = transform_scene_to_world(
+    cart_gazes_in_scene, imu_quaternions
+)
 ```
 
 ## Eyestate to World Coordinates
@@ -191,13 +191,13 @@ The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/dat
 # position in millimeters.
 scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
 eyeball_centers_in_world = transform_scene_to_world(
-    eyeball_centers, imu_quaternions, translation=scene_camera_position_in_imu,
+    eyeball_centers, imu_quaternions, translation=scene_camera_position_in_imu
 )
 
 # The optical axes are unit vectors originating at the eyeball centers,
 # so there is no need to translate them.
 optical_axes_in_world = transform_scene_to_world(
-    optical_axes, imu_quaternions,
+    optical_axes, imu_quaternions
 )
 ```
 

From a895e51b7f8edcbdc169758115db11fdb5bd10c4 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Fri, 23 Aug 2024 09:52:22 +0200
Subject: [PATCH 32/54] revert section title change

---
 alpha-lab/imu-transformations/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 428d73f96..9d9ad6e27 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -83,7 +83,7 @@ accelerations_in_world = transform_imu_to_world(
 )
 ```
 
-## Scene to World Coordinates
+## Gaze to World Coordinates
 
 Neon simultaneously records gaze and IMU data, making it possible to study the relationship between head and eye movements.
 

From e36c96935f80b1d13065596ea8cf4c8d70757a66 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Fri, 23 Aug 2024 13:28:52 +0200
Subject: [PATCH 33/54] implement new suggestions: link to calib loading doc,
 re-order sections, have 2d->world and 3d->world sections for gaze, compare
 homogeneous and explicit translation approaches, add eyestate convienence
 function, some formatting cleanup

---
 alpha-lab/imu-transformations/index.md | 204 ++++++++++++++++---------
 1 file changed, 135 insertions(+), 69 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 9d9ad6e27..52c8b7651 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -35,7 +35,7 @@ This guide contains various transformation functions that can assist when analys
 
 While working through this guide, it can be helpful to try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems. For example, some of the code in this article is [adapted from it](https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279).
 
-## Transform to World Coordinates
+## IMU to World Coordinates
 
 One of the key steps when dealing with the IMU is the transformation that takes coordinates in the local IMU coordinate system to their corresponding coordinates in the world coordinate system. The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function, defined below, will be used throughout this article.
 
@@ -60,41 +60,25 @@ def transform_imu_to_world(imu_coordinates, imu_quaternions):
         ])
 ```
 
-Now that we have the `transform_imu_to_world` function, let's use it!
+Now that we have the `transform_imu_to_world` function, let's build on it!
 
-## Obtain IMU Heading Vectors
-
-An alternative representation of IMU orientation data is a heading vector that points outwards from the center of the IMU. Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity:
-
-```python
-heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
-headings_in_world = transform_imu_to_world(
-    heading_neutral_in_imu_coords, imu_quaternions
-)
-```
+## Scene Camera to World Coordinates
 
-## Acceleration in World
+Neon simultaneously records data in the scene camera and IMU coordinate systems, making it possible to study the relationship between head and eye movements.
 
-The IMU’s acceleration data are specified in its local coordinate system. Sometimes, it can be useful to have the acceleration data specified in the world coordinate system instead:
+To facilitate the comparison, it can be useful to represent these data streams in the same coordinate system. An important step is accounting for [the fixed 102 degree rotation offset between the scene camera and IMU coordinate systems](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data), as depicted below.
 
-```python
-accelerations_in_world = transform_imu_to_world(
-    imu_accelerations, imu_quaternions
-)
-```
+![Diagrams showing the fixed 102 degree rotation offset between the IMU and scene camera coordinate systems.](./imu-scene_camera_offset-black.png)
 
-## Gaze to World Coordinates
+We can use data from the IMU to transform gaze in scene camera coordinates to world coordinates. We proceed by building a `transform_scene_to_imu` function that handles the rotation between the two coordinate systems. It also accepts a `translation_in_imu` keyword argument to specify if points should be shifted in the IMU system. This will be relevant when converting 3D eyestate to world coordinates.
 
-Neon simultaneously records gaze and IMU data, making it possible to study the relationship between head and eye movements.
+- ***Rob note:*** Keep this next bit if we decide to use homogeneous coordinates:
 
-To facilitate the comparison, it can be useful to represent these data streams in the same coordinate system. An important step is accounting for [the fixed 102 degree rotation offset between the scene camera and IMU coordinate systems](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data), as depicted below.
+We implement the rotation and translation as [a single affine transformation](https://en.wikipedia.org/wiki/Affine_transformation):
 
-![Diagrams showing the fixed 102 degree rotation offset between the IMU and scene camera coordinate systems.](./imu-scene_camera_offset-black.png)
-
-We can use data from the IMU to transform gaze in scene camera coordinates to world coordinates. We proceed by building a `transform_scene_to_imu` function:
 
 ```python
-def transform_scene_to_imu(coords_in_scene, translation=np.zeros((3,))):
+def transform_scene_to_imu(coords_in_scene, translation_in_imu=np.zeros((3,))):
     imu_scene_rotation_diff = np.deg2rad(-90 - 12)
     scene_to_imu = np.array(
         [
@@ -112,35 +96,123 @@ def transform_scene_to_imu(coords_in_scene, translation=np.zeros((3,))):
         ]
     )
 
-    scene_to_imu_homogeneous = np.zeros((4, 4))
-    scene_to_imu_homogeneous[:3, :3] = scene_to_imu
-    scene_to_imu_homogeneous[:3, 3] = translation
-    scene_to_imu_homogeneous[3, 3] = 1.0
+    # The homogeneous approach, for comparison
+    #
+    # scene_to_imu_homogeneous = np.zeros((4, 4))
+    # scene_to_imu_homogeneous[:3, :3] = scene_to_imu
+    # scene_to_imu_homogeneous[:3, 3] = translation_in_imu
+    # scene_to_imu_homogeneous[3, 3] = 1.0
+
+    # coords_in_scene_homogeneous = cv2.convertPointsToHomogeneous(coords_in_scene)
+    # coords_in_imu_homogeneous = scene_to_imu_homogeneous @ coords_in_scene_homogeneous.reshape(-1, 4).T
+    # coords_in_imu = cv2.convertPointsFromHomogeneous(coords_in_imu_homogeneous.T).reshape(-1, 3)
+
+    # Rob-Blob approach
+    coords_in_imu = scene_to_imu @ coords_in_scene.T
 
-    coords_in_scene_homogeneous = cv2.convertPointsToHomogeneous(coords_in_scene)
-    coords_in_imu_homogeneous = scene_to_imu_homogeneous @ coords_in_scene_homogeneous.reshape(-1, 4).T
-    coords_in_imu = cv2.convertPointsFromHomogeneous(coords_in_imu_homogeneous.T).reshape(-1, 3)
+    coords_in_imu[0, :] += translation_in_imu[0]
+    coords_in_imu[1, :] += translation_in_imu[1]
+    coords_in_imu[2, :] += translation_in_imu[2]
 
-    return coords_in_imu
+    return coords_in_imu.T
 ```
 
-Putting together the `transform_scene_to_imu` and `transform_imu_to_world` functions, we can build a general `transform_scene_to_world` function:
+Putting together the `transform_scene_to_imu` and `transform_imu_to_world` functions, we can build a composite `transform_scene_to_world` function:
 
 ```python
-def transform_scene_to_world(coords_in_scene, imu_quaternions, translation=np.zeros((3,))):
-    coords_in_imu = transform_scene_to_imu(coords_in_scene, translation)
+def transform_scene_to_world(coords_in_scene, imu_quaternions, translation_in_imu=np.zeros((3,))):
+    coords_in_imu = transform_scene_to_imu(coords_in_scene, translation_in_imu)
     return transform_imu_to_world(coords_in_imu, imu_quaternions)
 ```
 
-Now we have the tools to convert 3D gaze data to world coordinates, but first, we need to convert 2D gaze to 3D by undistorting and unprojecting the image coordinates:
+We can now apply this function to some data!
+
+## Eyestate to World Coordinates
+
+The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) provided by Neon are aligned with the scene camera coordinate system. This means we can use the `transform_scene_to_world` function above to reconstruct the pose of the eyes in the world coordinate system. We just need to consider that the scene camera is displaced a bit from the IMU.
+Since the eyeball center and optical axis values provided by the 3D eyestate estimates are intrinsically linked, we provide the `eyestate_to_world` function to simplify doing the conversions:
+
+```python
+# The 3D eyestate data and the IMU quaternions should be sampled
+# at the same timestamps. You can linearly interpolate the IMU data,
+# for example.
+
+def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions)
+    """
+    The eyeball_centers and optical_axes inputs are for the same eye.
+    """
+
+    # The eyeball centers are specified in terms of their distance from
+    # the center of the scene camera, so to accurately convert them to
+    # world coordinates, we need to account for the position of the
+    # scene camera in the IMU coordinate system. Here, we express that
+    # position in millimeters.
+    scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
+    eyeball_centers_in_world = transform_scene_to_world(
+        eyeball_centers, imu_quaternions, translation=scene_camera_position_in_imu
+    )
+
+    # The optical axes are unit vectors originating at the eyeball centers,
+    # so they should not be to translated.
+    optical_axes_in_world = transform_scene_to_world(
+        optical_axes, imu_quaternions
+    )
+
+    return eyeball_centers_in_world, optical_axes_in_world
+```
+
+## 3D Gaze to World Coordinates
+
+Neon provides 3D gaze data in the scene camera system in terms of [spherical coordinates (i.e., `azimuth/elevation [deg]`)](https://docs.pupil-labs.com/neon/data-collection/data-format/#gaze-csv). The `transform_scene_to_world` function above expects 3D Cartesian coordinates, so to convert spherical 3D gaze to world coordinates, we will need the `spherical_to_cartesian_scene` function:
+
+```python
+def spherical_to_cartesian_scene(elevations, azimuths):
+    """
+    Convert Neon's spherical representation of 3D gaze to Cartesian coordinates.
+    """
+
+    elevations_rad = np.deg2rad(elevations)
+    azimuths_rad = np.deg2rad(azimuths)
+
+    # Elevation of 0 in Neon system corresponds to Y = 0, but
+    # an elevation of 0 in traditional spherical coordinates would
+    # correspond to Y = 1, so we convert elevation to the
+    # more traditional format.
+    elevations_rad += np.pi / 2
+
+    # Azimuth of 0 in Neon system corresponds to X = 0, but
+    # an azimuth of 0 in traditional spherical coordinates would
+    # correspond to X = 1. Also, azimuth to the right in Neon is
+    # more positive, whereas it is more negative in traditional
+    # spherical coordiantes. So, we convert azimuth to the more
+    # traditional format.
+    azimuths_rad *= -1.0
+    azimuths_rad += np.pi / 2
+
+    return np.array(
+        [
+            np.sin(elevations_rad) * np.cos(azimuths_rad),
+            np.cos(elevations_rad),
+            np.sin(elevations_rad) * np.sin(azimuths_rad),
+        ]
+    ).T
+```
+
+Now we have the tools to convert 3D gaze data to world coordinates:
+
+```python
+cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevation_resampled, gaze_azimuth_resampled)
+cart_gazes_in_world = transform_scene_to_world(cart_gazes_in_scene, quaternions_resampled)
+```
+
+## 2D Gaze to World Coordinates
+
+If you are starting from [the 2D gaze values in scene image coordinates (i.e., `gaze x/y [px]`)](https://docs.pupil-labs.com/neon/data-collection/data-format/#gaze-csv), then you will need to first [undistort](https://docs.pupil-labs.com/alpha-lab/undistort/) and unproject those points to obtain the corresponding 3D gaze vectors. Note that this requires [loading the scene camera calibration data](https://docs-staging.pupil-labs.com/alpha-lab/undistort/#reading-from-the-cloud-download-json-file).
 
 ```python
 def unproject_2d_gaze(gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients):
     """
     Transform the 2D gaze values from Neon to 3D gaze vectors.
-    
-    See the associated Alpha Labs article for a tutorial on undistorting:
-    https://docs.pupil-labs.com/alpha-lab/undistort/
     """
 
     gaze_points_2d_undist = cv2.undistortPoints(gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients)
@@ -150,22 +222,13 @@ def unproject_2d_gaze(gaze_points_2d, scene_camera_matrix, scene_distortion_coef
     return gaze_vectors_3d
 ```
 
-Putting all of this together, we can convert 2D gaze data in scene image coordinates to 3D gaze data in world coordinates:
+Then, we can use the functions from [the previous section](#3d-gaze-to-world-coordinates) to convert 2D gaze to 3D world coordinates:
 
 ```python
 # The gaze data and the IMU quaternions should be sampled at the
 # same timestamps. You can linearly interpolate the IMU data to
 # ensure this.
 
-# I'm not sure if we want to show how to load the camera calib data
-#
-# scene_camera_info = []
-# with open(rec_dir + "scene_camera.json") as f:
-#     scene_camera_info = json.load(f)
-#
-# scene_camera_matrix = np.array(scene_camera_info["camera_matrix"])
-# scene_distortion_coefficients = np.array(scene_camera_info["distortion_coefficients"])
-
 gaze_points_2d = gaze[["gaze x [px]", "gaze y [px]"]].to_numpy()
 cart_gazes_in_scene = undistort_and_unproject(
     gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients
@@ -175,29 +238,26 @@ cart_gazes_in_world = transform_scene_to_world(
 )
 ```
 
-## Eyestate to World Coordinates
+## Obtain IMU Heading Vectors
 
-The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) provided by Neon are aligned with the scene camera coordinate system. To reconstruct the pose of the eyes in the world coordinate system, we need to also consider that the scene camera coordinate system is positioned downwards and a bit back from the IMU:
+An alternative representation of IMU orientation data is a heading vector that points outwards from the center of the IMU. It can be useful to compare this heading vector with the 3D gaze vectors in world coordinates.
 
 ```python
-# The 3D eyestate data and the IMU quaternions should be sampled
-# at the same timestamps. You can linearly interpolate the IMU data
-# to ensure this.
-
-# The eyeball centers are specified in terms of their distance from
-# the center of the scene camera, so to accurately convert them to
-# world coordinates, we need to account for the position of the
-# scene camera in the IMU coordinate system. Here, we express that
-# position in millimeters.
-scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
-eyeball_centers_in_world = transform_scene_to_world(
-    eyeball_centers, imu_quaternions, translation=scene_camera_position_in_imu
+heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
+headings_in_world = transform_imu_to_world(
+    heading_neutral_in_imu_coords, imu_quaternions
 )
+```
+
+Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
 
-# The optical axes are unit vectors originating at the eyeball centers,
-# so there is no need to translate them.
-optical_axes_in_world = transform_scene_to_world(
-    optical_axes, imu_quaternions
+## IMU Acceleration in World
+
+The IMU’s acceleration data are specified in its local coordinate system. If you want to understand how the observer is accelerating through their environment, then it can be easier to have the acceleration data specified in the world coordinate system:
+
+```python
+accelerations_in_world = transform_imu_to_world(
+    imu_accelerations, imu_quaternions
 )
 ```
 
@@ -241,6 +301,12 @@ def cartesian_to_spherical_world(world_points_3d):
     return elevation, azimuth
 ```
 
+## Application Example
+
+Below, we present a video showing how some of the functions in this article where used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://gist.github.com/rennis250/8a684ea1e2f92c79fa2104b7a0f30e20).
+
+- ***Rob Note***: Video currently [here](https://drive.google.com/file/d/1UhZ9GAHacVQ8jLwkI7bLZHykeF7OpOwa/view?usp=sharing)
+
 ::: tip
 Need assistance with the IMU code in this article? Or do you have something more custom in mind? Reach out to us via email at [info@pupil-labs.com](mailto:info@pupil-labs.com), on our [Discord server](https://pupil-labs.com/chat/), or visit our [Support Page](https://pupil-labs.com/products/support/) for dedicated support options.
 :::

From 449f70751bcf3c0e618e2b64891f3eedeaf9335d Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Fri, 23 Aug 2024 13:29:04 +0200
Subject: [PATCH 34/54] shorter section title

---
 alpha-lab/imu-transformations/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 52c8b7651..b9fb66539 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -62,7 +62,7 @@ def transform_imu_to_world(imu_coordinates, imu_quaternions):
 
 Now that we have the `transform_imu_to_world` function, let's build on it!
 
-## Scene Camera to World Coordinates
+## Scene to World Coordinates
 
 Neon simultaneously records data in the scene camera and IMU coordinate systems, making it possible to study the relationship between head and eye movements.
 

From ca01fbb558643cd55b2c395770fd913dfcb1db37 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Mon, 2 Sep 2024 13:29:17 +0200
Subject: [PATCH 35/54] - implement remaining changes for IMU alpha labs
 article - expand explanation of IMU coordinate systems in docs - update
 images in IMU docs

---
 alpha-lab/imu-transformations/index.md        | 121 ++++++++----------
 .../data-streams/imu-pitch-yaw-roll-black.png | Bin 0 -> 183939 bytes
 .../imu-scene_camera_offset-black.png         | Bin 0 -> 145548 bytes
 neon/data-collection/data-streams/index.md    |  28 ++--
 4 files changed, 75 insertions(+), 74 deletions(-)
 create mode 100644 neon/data-collection/data-streams/imu-pitch-yaw-roll-black.png
 create mode 100644 neon/data-collection/data-streams/imu-scene_camera_offset-black.png

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index b9fb66539..a22152cc0 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -27,17 +27,14 @@ import TagLinks from '@components/TagLinks.vue'
 <TagLinks :tags="$frontmatter.tags" />
 
 ::: tip
-Want to compare IMU and gaze data in the same coordinate system to better understand how people coordinate 
-head and eye movements? The transformation functions in this tutorial will show you how!
+Want to compare IMU and gaze data in the same coordinate system to better understand how people coordinate head and eye movements? The transformation functions in this tutorial will show you how!
 :::
 
-This guide contains various transformation functions that can assist when analysing Neon's IMU data.
-
-While working through this guide, it can be helpful to try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems. For example, some of the code in this article is [adapted from it](https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279).
+This guide contains various transformation functions that can assist when working with Neon's IMU data.
 
 ## IMU to World Coordinates
 
-One of the key steps when dealing with the IMU is the transformation that takes coordinates in the local IMU coordinate system to their corresponding coordinates in the world coordinate system. The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function, defined below, will be used throughout this article.
+One of the key steps when dealing with the IMU is the transformation that takes coordinates in the local IMU coordinate system to their corresponding coordinates in [the world coordinate system](http://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data). The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function, defined below, will be used throughout this article.
 
 Note that the origin of the IMU coordinate system is the same as the origin of the world coordinate system.
 
@@ -60,7 +57,31 @@ def transform_imu_to_world(imu_coordinates, imu_quaternions):
         ])
 ```
 
-Now that we have the `transform_imu_to_world` function, let's build on it!
+Now that we have the `transform_imu_to_world` function, let's use it!
+
+## Obtain IMU Heading Vectors
+
+An alternative representation of IMU orientation data is a heading vector that points outwards from the center of the IMU. It can be useful to compare this heading vector with the 3D gaze vectors in world coordinates.
+
+```python
+def imu_heading_in_world(imu_quaternions):
+    heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
+    return transform_imu_to_world(
+        heading_neutral_in_imu_coords, imu_quaternions
+    )
+```
+
+Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
+
+## IMU Acceleration in World
+
+The IMU’s acceleration data are specified in its local coordinate system. If you want to understand how the observer is accelerating through their environment, then it can be easier to have the acceleration data specified in the world coordinate system:
+
+```python
+accelerations_in_world = transform_imu_to_world(
+    imu_accelerations, imu_quaternions
+)
+```
 
 ## Scene to World Coordinates
 
@@ -72,10 +93,6 @@ To facilitate the comparison, it can be useful to represent these data streams i
 
 We can use data from the IMU to transform gaze in scene camera coordinates to world coordinates. We proceed by building a `transform_scene_to_imu` function that handles the rotation between the two coordinate systems. It also accepts a `translation_in_imu` keyword argument to specify if points should be shifted in the IMU system. This will be relevant when converting 3D eyestate to world coordinates.
 
-- ***Rob note:*** Keep this next bit if we decide to use homogeneous coordinates:
-
-We implement the rotation and translation as [a single affine transformation](https://en.wikipedia.org/wiki/Affine_transformation):
-
 
 ```python
 def transform_scene_to_imu(coords_in_scene, translation_in_imu=np.zeros((3,))):
@@ -96,18 +113,6 @@ def transform_scene_to_imu(coords_in_scene, translation_in_imu=np.zeros((3,))):
         ]
     )
 
-    # The homogeneous approach, for comparison
-    #
-    # scene_to_imu_homogeneous = np.zeros((4, 4))
-    # scene_to_imu_homogeneous[:3, :3] = scene_to_imu
-    # scene_to_imu_homogeneous[:3, 3] = translation_in_imu
-    # scene_to_imu_homogeneous[3, 3] = 1.0
-
-    # coords_in_scene_homogeneous = cv2.convertPointsToHomogeneous(coords_in_scene)
-    # coords_in_imu_homogeneous = scene_to_imu_homogeneous @ coords_in_scene_homogeneous.reshape(-1, 4).T
-    # coords_in_imu = cv2.convertPointsFromHomogeneous(coords_in_imu_homogeneous.T).reshape(-1, 3)
-
-    # Rob-Blob approach
     coords_in_imu = scene_to_imu @ coords_in_scene.T
 
     coords_in_imu[0, :] += translation_in_imu[0]
@@ -125,7 +130,7 @@ def transform_scene_to_world(coords_in_scene, imu_quaternions, translation_in_im
     return transform_imu_to_world(coords_in_imu, imu_quaternions)
 ```
 
-We can now apply this function to some data!
+You can now use this function to transform data in scene camera coordinates to world coordinates. Head to the [Application Example](#application-example) to see how!
 
 ## Eyestate to World Coordinates
 
@@ -134,26 +139,25 @@ Since the eyeball center and optical axis values provided by the 3D eyestate est
 
 ```python
 # The 3D eyestate data and the IMU quaternions should be sampled
-# at the same timestamps. You can linearly interpolate the IMU data,
-# for example.
+# at the same timestamps. You can linearly interpolate the IMU data.
+# See the Application Example:
+# http://docs.pupil-labs.com/alpha-lab/imu-transformations/#application-example
 
-def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions)
+def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
     """
     The eyeball_centers and optical_axes inputs are for the same eye.
     """
 
-    # The eyeball centers are specified in terms of their distance from
-    # the center of the scene camera, so to accurately convert them to
-    # world coordinates, we need to account for the position of the
-    # scene camera in the IMU coordinate system. Here, we express that
-    # position in millimeters.
+    # The eyeball centers are specified relative to the center of the scene
+    # camera, so we need to account for the position of the scene camera in
+    # the IMU coordinate system. Here, we express that position in millimeters.
     scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
     eyeball_centers_in_world = transform_scene_to_world(
         eyeball_centers, imu_quaternions, translation=scene_camera_position_in_imu
     )
 
     # The optical axes are unit vectors originating at the eyeball centers,
-    # so they should not be to translated.
+    # so they should not be translated.
     optical_axes_in_world = transform_scene_to_world(
         optical_axes, imu_quaternions
     )
@@ -163,7 +167,7 @@ def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions)
 
 ## 3D Gaze to World Coordinates
 
-Neon provides 3D gaze data in the scene camera system in terms of [spherical coordinates (i.e., `azimuth/elevation [deg]`)](https://docs.pupil-labs.com/neon/data-collection/data-format/#gaze-csv). The `transform_scene_to_world` function above expects 3D Cartesian coordinates, so to convert spherical 3D gaze to world coordinates, we will need the `spherical_to_cartesian_scene` function:
+Neon provides 3D gaze data in [spherical coordinates (i.e., `azimuth/elevation [deg]`)](https://docs.pupil-labs.com/neon/data-collection/data-format/#gaze-csv). The `transform_scene_to_world` function above expects 3D Cartesian coordinates, so to convert spherical 3D gaze to world coordinates, we will need the `spherical_to_cartesian_scene` function:
 
 ```python
 def spherical_to_cartesian_scene(elevations, azimuths):
@@ -201,8 +205,9 @@ def spherical_to_cartesian_scene(elevations, azimuths):
 Now we have the tools to convert 3D gaze data to world coordinates:
 
 ```python
-cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevation_resampled, gaze_azimuth_resampled)
-cart_gazes_in_world = transform_scene_to_world(cart_gazes_in_scene, quaternions_resampled)
+def gaze_3d_to_world(gaze_elevation, gaze_azimuth, imu_quaternions):
+    cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevation, gaze_azimuth)
+    return transform_scene_to_world(cart_gazes_in_scene, imu_quaternions)
 ```
 
 ## 2D Gaze to World Coordinates
@@ -228,37 +233,17 @@ Then, we can use the functions from [the previous section](#3d-gaze-to-world-coo
 # The gaze data and the IMU quaternions should be sampled at the
 # same timestamps. You can linearly interpolate the IMU data to
 # ensure this.
+# See the Application Example:
+# http://docs.pupil-labs.com/alpha-lab/imu-transformations/#application-example
 
 gaze_points_2d = gaze[["gaze x [px]", "gaze y [px]"]].to_numpy()
-cart_gazes_in_scene = undistort_and_unproject(
-    gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients
-)
-cart_gazes_in_world = transform_scene_to_world(
-    cart_gazes_in_scene, imu_quaternions
-)
-```
-
-## Obtain IMU Heading Vectors
-
-An alternative representation of IMU orientation data is a heading vector that points outwards from the center of the IMU. It can be useful to compare this heading vector with the 3D gaze vectors in world coordinates.
-
-```python
-heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])
-headings_in_world = transform_imu_to_world(
-    heading_neutral_in_imu_coords, imu_quaternions
-)
-```
-
-Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
-
-## IMU Acceleration in World
-
-The IMU’s acceleration data are specified in its local coordinate system. If you want to understand how the observer is accelerating through their environment, then it can be easier to have the acceleration data specified in the world coordinate system:
-
-```python
-accelerations_in_world = transform_imu_to_world(
-    imu_accelerations, imu_quaternions
-)
+def gaze_2d_to_world(gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients, imu_quaternions):
+    cart_gazes_in_scene = undistort_and_unproject(
+        gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients
+    )
+    return transform_scene_to_world(
+        cart_gazes_in_scene, imu_quaternions
+    )
 ```
 
 ## World Spherical Coordinates
@@ -305,7 +290,11 @@ def cartesian_to_spherical_world(world_points_3d):
 
 Below, we present a video showing how some of the functions in this article where used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://gist.github.com/rennis250/8a684ea1e2f92c79fa2104b7a0f30e20).
 
-- ***Rob Note***: Video currently [here](https://drive.google.com/file/d/1UhZ9GAHacVQ8jLwkI7bLZHykeF7OpOwa/view?usp=sharing)
+<Youtube src="uFNsKCJAxyo"/>
+
+## Related content
+
+While working through this guide, it can be helpful to try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems. For example, some of the code in this article is [adapted from it](https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279).
 
 ::: tip
 Need assistance with the IMU code in this article? Or do you have something more custom in mind? Reach out to us via email at [info@pupil-labs.com](mailto:info@pupil-labs.com), on our [Discord server](https://pupil-labs.com/chat/), or visit our [Support Page](https://pupil-labs.com/products/support/) for dedicated support options.
diff --git a/neon/data-collection/data-streams/imu-pitch-yaw-roll-black.png b/neon/data-collection/data-streams/imu-pitch-yaw-roll-black.png
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diff --git a/neon/data-collection/data-streams/index.md b/neon/data-collection/data-streams/index.md
index bae4c18b2..dbb2b8a11 100644
--- a/neon/data-collection/data-streams/index.md
+++ b/neon/data-collection/data-streams/index.md
@@ -82,22 +82,34 @@ Audio recording is disabled in the Neon Companion app by default and can be enab
 ## Movement (IMU Data)
 
 <Badge>Real-time</Badge><Badge>Pupil Cloud</Badge><Badge>Neon Player</Badge>
-The Neon module is equipped with a 9-DoF [inertial measurement unit](https://invensense.tdk.com/products/motion-tracking/9-axis/icm-20948/) (IMU) featuring an accelerometer, gyroscope, and magnetometer. The accelerometer and gyroscope measure linear acceleration and angular velocity, respectively, and are provided as raw values.
+The Neon module is equipped with a 9-DoF [inertial measurement unit](https://invensense.tdk.com/products/motion-tracking/9-axis/icm-20948/) (IMU) featuring an accelerometer, gyroscope, and magnetometer. The accelerometer and gyroscope measure linear acceleration and angular velocity, respectively, and are provided as raw values. Acceleration is reported in terms of [g-force](https://en.wikipedia.org/wiki/G-force), which differs from free-fall acceleration due to gravity. For instance, when standing upright on Earth, acceleration along the Z-axis would be +1g, with 0g on the X and Y axes.
 
-A fusion engine also combines these values with magnetometer readings to estimate the module's absolute orientation relative to magnetic north and gravity as a quaternion. Note that in order to obtain precise absolute yaw readings, the magnetometer needs to be [calibrated](/data-collection/calibrating-the-imu/).
+A fusion engine also combines these values with magnetometer readings to estimate the module's absolute orientation in the world coordinate system as a quaternion:
 
-The IMU is located in the top bar of the module and is sampled at 110 Hz. Its coordinate system is oriented with the x-axis pointing to the right, the y-axis pointing in front, and the z-axis pointing upwards.
+- World y-axis: Points towards magnetic north.
+- World z-axis: Points upwards, opposite gravity.
+- World x-axis: Points rightwards, defined by the cross-product of Y and Z.
+
+Note that in order to obtain precise absolute yaw readings, the magnetometer needs to be [calibrated](/data-collection/calibrating-the-imu/).
+
+The IMU is located in the top bar of the module and is sampled at 110 Hz. Its local coordinate system is oriented with the x-axis pointing to the right, the y-axis pointing in front, and the z-axis pointing upwards.
 
 ![IMU Coordinate System](./imu-xyz-black.jpg)
 
-When relating data from the IMU to things visible in the scene camera, it may be necessary to align their respective 3D coordinate systems. The IMU's coordinate system is rotated by 102° around the x-axis in relation to the scene camera's coordinate system.
+When relating data from the IMU to things visible in the scene camera, it may be necessary to align their respective 3D coordinate systems. The IMU's coordinate system is rotated by 102° around the x-axis in relation to the scene camera's coordinate system. See our [IMU Transformations article](https://docs.pupil-labs.com/alpha-lab/imu-transformations/) for a guide.
 
-![IMU Scene Camera](./imu-scene_camera-black.jpg)
+![IMU Scene Camera](./imu-scene_camera_offset-black.png)
 
 ### Euler Angles
 
-When exporting recordings from Pupil Cloud or Neon Player the IMU's orientation in Euler angles (i.e. roll, pitch, and yaw) is also available.
+When exporting recordings from Pupil Cloud or Neon Player the IMU's orientation in Euler angles (i.e. pitch, yaw, and roll) is also available:
+
+- Pitch is defined as a rotation around the world x-axis with a value range of -90° to +90°. Wearing Neon upright and looking parallel to the horizon roughly corresponds to 0° pitch. Backward tilt is positive, forward is negative.
+- Yaw is a rotation around the world z-axis with a value range of -180° to +180°. With a calibrated IMU, a yaw of 0° indicates alignment with magnetic north. Leftward turn is positive, rightward is negative.
+- Roll is a rotations around the world y-axis with a value range of -180° to +180°. Wearing Neon upright with a neutral head pose roughly corresponds to a roll of 0°. Rightward tilt is positive, leftward is negative.
 
-Pitch is defined as a rotation around the x-axis with a value range of -90° to +90°. Yaw and roll are rotations around the y- and z-axis, respectively, with value ranges of -180° to +180°.
+::: tip
+Neon may sit differently on each wearer’s face, causing deviations from a neutral orientation even when facing magnetic north.
+:::
 
-![IMU Pitch, Yaw, Roll](./imu-pitch_yaw_roll-black.jpg)
+![IMU Pitch, Yaw, Roll](./imu-pitch-yaw-roll-black.png)

From ba04e2d21b086e34118bce386223ea51aa47679a Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Mon, 2 Sep 2024 13:33:58 +0200
Subject: [PATCH 36/54] fix youtube preview link

---
 alpha-lab/imu-transformations/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index a22152cc0..48a85f780 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -290,7 +290,7 @@ def cartesian_to_spherical_world(world_points_3d):
 
 Below, we present a video showing how some of the functions in this article where used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://gist.github.com/rennis250/8a684ea1e2f92c79fa2104b7a0f30e20).
 
-<Youtube src="uFNsKCJAxyo"/>
+<Youtube src="lbmyBgpS2OE"/>
 
 ## Related content
 

From 7f5a73f300500e743af460f093f8274e969baa52 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Mon, 2 Sep 2024 13:57:12 +0200
Subject: [PATCH 37/54] formatting

---
 alpha-lab/imu-transformations/index.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 48a85f780..6eb35c8bc 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -59,7 +59,7 @@ def transform_imu_to_world(imu_coordinates, imu_quaternions):
 
 Now that we have the `transform_imu_to_world` function, let's use it!
 
-## Obtain IMU Heading Vectors
+### Obtain IMU Heading Vectors
 
 An alternative representation of IMU orientation data is a heading vector that points outwards from the center of the IMU. It can be useful to compare this heading vector with the 3D gaze vectors in world coordinates.
 
@@ -73,7 +73,7 @@ def imu_heading_in_world(imu_quaternions):
 
 Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
 
-## IMU Acceleration in World
+### IMU Acceleration in World
 
 The IMU’s acceleration data are specified in its local coordinate system. If you want to understand how the observer is accelerating through their environment, then it can be easier to have the acceleration data specified in the world coordinate system:
 

From 57762e4a6a82aee9c2c7326b23ffffcd89828f39 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Mon, 2 Sep 2024 14:07:09 +0200
Subject: [PATCH 38/54] formatting

---
 alpha-lab/imu-transformations/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 6eb35c8bc..846e37296 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -294,7 +294,7 @@ Below, we present a video showing how some of the functions in this article wher
 
 ## Related content
 
-While working through this guide, it can be helpful to try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems. For example, some of the code in this article is [adapted from it](https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279).
+It can also be helpful to try out our IMU visualization utility, [plimu](https://github.com/pupil-labs/plimu). This can assist in understanding the IMU data and the various coordinate systems. For example, some of the code in this article is [adapted from it](https://github.com/pupil-labs/plimu/blob/8b94302982363b203dddea2b15f43c6da60e787e/src/pupil_labs/plimu/visualizer.py#L274-L279).
 
 ::: tip
 Need assistance with the IMU code in this article? Or do you have something more custom in mind? Reach out to us via email at [info@pupil-labs.com](mailto:info@pupil-labs.com), on our [Discord server](https://pupil-labs.com/chat/), or visit our [Support Page](https://pupil-labs.com/products/support/) for dedicated support options.

From 47781974e64e70cd7cfe17f6e28a3b05ac674faa Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Mon, 2 Sep 2024 16:11:21 +0200
Subject: [PATCH 39/54] compress imu docs a bit, some tweaks for alpha labs
 article

---
 alpha-lab/imu-transformations/index.md     |  2 +-
 neon/data-collection/data-streams/index.md | 10 ++--------
 2 files changed, 3 insertions(+), 9 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 846e37296..7090574f2 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -34,7 +34,7 @@ This guide contains various transformation functions that can assist when workin
 
 ## IMU to World Coordinates
 
-One of the key steps when dealing with the IMU is the transformation that takes coordinates in the local IMU coordinate system to their corresponding coordinates in [the world coordinate system](http://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data). The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function, defined below, will be used throughout this article.
+One of the key steps when working with IMU data is the transformation that takes coordinates in the local IMU coordinate system to their corresponding coordinates in [the world coordinate system](http://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data). The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function, defined below, will be used throughout this article.
 
 Note that the origin of the IMU coordinate system is the same as the origin of the world coordinate system.
 
diff --git a/neon/data-collection/data-streams/index.md b/neon/data-collection/data-streams/index.md
index dbb2b8a11..fb186441c 100644
--- a/neon/data-collection/data-streams/index.md
+++ b/neon/data-collection/data-streams/index.md
@@ -82,15 +82,9 @@ Audio recording is disabled in the Neon Companion app by default and can be enab
 ## Movement (IMU Data)
 
 <Badge>Real-time</Badge><Badge>Pupil Cloud</Badge><Badge>Neon Player</Badge>
-The Neon module is equipped with a 9-DoF [inertial measurement unit](https://invensense.tdk.com/products/motion-tracking/9-axis/icm-20948/) (IMU) featuring an accelerometer, gyroscope, and magnetometer. The accelerometer and gyroscope measure linear acceleration and angular velocity, respectively, and are provided as raw values. Acceleration is reported in terms of [g-force](https://en.wikipedia.org/wiki/G-force), which differs from free-fall acceleration due to gravity. For instance, when standing upright on Earth, acceleration along the Z-axis would be +1g, with 0g on the X and Y axes.
+The Neon module is equipped with a 9-DoF [inertial measurement unit](https://invensense.tdk.com/products/motion-tracking/9-axis/icm-20948/) (IMU) featuring an accelerometer, gyroscope, and magnetometer. The accelerometer and gyroscope measure linear acceleration and angular velocity, respectively, and are provided as raw values.
 
-A fusion engine also combines these values with magnetometer readings to estimate the module's absolute orientation in the world coordinate system as a quaternion:
-
-- World y-axis: Points towards magnetic north.
-- World z-axis: Points upwards, opposite gravity.
-- World x-axis: Points rightwards, defined by the cross-product of Y and Z.
-
-Note that in order to obtain precise absolute yaw readings, the magnetometer needs to be [calibrated](/data-collection/calibrating-the-imu/).
+A fusion engine also combines these values with magnetometer readings to estimate the module's absolute orientation relative to magnetic north (positive world y-axis), gravity (negative world z-axis), and a rightward pointing vector (positive world x-axis) as a quaternion. We refer to this as the world coordinate system. It is important to note that this is not the same as the local IMU coordinate system.
 
 The IMU is located in the top bar of the module and is sampled at 110 Hz. Its local coordinate system is oriented with the x-axis pointing to the right, the y-axis pointing in front, and the z-axis pointing upwards.
 

From 2ab1b5352fd6e9fa917924671698af2c1d39760c Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Mon, 2 Sep 2024 16:25:54 +0200
Subject: [PATCH 40/54] add tip about pos/neg rotations

---
 neon/data-collection/data-streams/index.md | 4 ++++
 1 file changed, 4 insertions(+)

diff --git a/neon/data-collection/data-streams/index.md b/neon/data-collection/data-streams/index.md
index fb186441c..7fdac4f17 100644
--- a/neon/data-collection/data-streams/index.md
+++ b/neon/data-collection/data-streams/index.md
@@ -94,6 +94,10 @@ When relating data from the IMU to things visible in the scene camera, it may be
 
 ![IMU Scene Camera](./imu-scene_camera_offset-black.png)
 
+::: tip
+Note that leftward rotations for the yaw values reported by the IMU are positive, whereas leftward rotations about the y-axis of the scene camera and 3D eye state coordinate systems are negative.
+:::
+
 ### Euler Angles
 
 When exporting recordings from Pupil Cloud or Neon Player the IMU's orientation in Euler angles (i.e. pitch, yaw, and roll) is also available:

From 9b154a08bda1824d5d1a4e9f8310dc255951c9a3 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Mon, 2 Sep 2024 16:29:34 +0200
Subject: [PATCH 41/54] compress tip about pos/neg leftward rotations

---
 neon/data-collection/data-streams/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/neon/data-collection/data-streams/index.md b/neon/data-collection/data-streams/index.md
index 7fdac4f17..2e1bbbaf5 100644
--- a/neon/data-collection/data-streams/index.md
+++ b/neon/data-collection/data-streams/index.md
@@ -95,7 +95,7 @@ When relating data from the IMU to things visible in the scene camera, it may be
 ![IMU Scene Camera](./imu-scene_camera_offset-black.png)
 
 ::: tip
-Note that leftward rotations for the yaw values reported by the IMU are positive, whereas leftward rotations about the y-axis of the scene camera and 3D eye state coordinate systems are negative.
+Note that leftward rotations for the yaw values reported by the IMU are positive, whereas leftward rotations about the y-axis of the scene camera coordinate system are negative.
 :::
 
 ### Euler Angles

From 8817a78a9870434671b6e604e35360bea2cd37d5 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Mon, 2 Sep 2024 16:55:11 +0200
Subject: [PATCH 42/54] find a better place for leftward pos/neg rotations

---
 neon/data-collection/data-streams/index.md | 6 +-----
 1 file changed, 1 insertion(+), 5 deletions(-)

diff --git a/neon/data-collection/data-streams/index.md b/neon/data-collection/data-streams/index.md
index 2e1bbbaf5..cb0945ae3 100644
--- a/neon/data-collection/data-streams/index.md
+++ b/neon/data-collection/data-streams/index.md
@@ -94,10 +94,6 @@ When relating data from the IMU to things visible in the scene camera, it may be
 
 ![IMU Scene Camera](./imu-scene_camera_offset-black.png)
 
-::: tip
-Note that leftward rotations for the yaw values reported by the IMU are positive, whereas leftward rotations about the y-axis of the scene camera coordinate system are negative.
-:::
-
 ### Euler Angles
 
 When exporting recordings from Pupil Cloud or Neon Player the IMU's orientation in Euler angles (i.e. pitch, yaw, and roll) is also available:
@@ -107,7 +103,7 @@ When exporting recordings from Pupil Cloud or Neon Player the IMU's orientation
 - Roll is a rotations around the world y-axis with a value range of -180° to +180°. Wearing Neon upright with a neutral head pose roughly corresponds to a roll of 0°. Rightward tilt is positive, leftward is negative.
 
 ::: tip
-Neon may sit differently on each wearer’s face, causing deviations from a neutral orientation even when facing magnetic north.
+Neon may sit differently on each wearer’s face, causing deviations from a neutral orientation even when facing magnetic north. Also, leftward rotations for the yaw values reported by the IMU are positive, whereas leftward rotations about the y-axis of the scene camera coordinate system are negative.
 :::
 
 ![IMU Pitch, Yaw, Roll](./imu-pitch-yaw-roll-black.png)

From 38b8cf1539e90635327ee01255a84d69e36d4d6a Mon Sep 17 00:00:00 2001
From: Marc Tonsen <mar@pupil-labs.com>
Date: Tue, 3 Sep 2024 10:20:46 +0200
Subject: [PATCH 43/54] Iterated on content

---
 alpha-lab/imu-transformations/index.md | 142 ++++++-------------------
 1 file changed, 30 insertions(+), 112 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 7090574f2..9425e1854 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -26,17 +26,13 @@ import TagLinks from '@components/TagLinks.vue'
 
 <TagLinks :tags="$frontmatter.tags" />
 
-::: tip
-Want to compare IMU and gaze data in the same coordinate system to better understand how people coordinate head and eye movements? The transformation functions in this tutorial will show you how!
-:::
-
-This guide contains various transformation functions that can assist when working with Neon's IMU data.
+This guide contains various transformation functions that help with relating [Neon's IMU data](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data) with other data streams.
 
-## IMU to World Coordinates
+## Rotation between the IMU and the World
 
-One of the key steps when working with IMU data is the transformation that takes coordinates in the local IMU coordinate system to their corresponding coordinates in [the world coordinate system](http://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data). The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function, defined below, will be used throughout this article.
+The IMU data includes a description of how the IMU is rotated in relation to the world. Concretely, the IMU data contains quaternions that define a rotation transformation between the [the world coordinate system](http://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data) and the IMU's local coordinate system at different points in time. 
 
-Note that the origin of the IMU coordinate system is the same as the origin of the world coordinate system.
+The `transform_imu_to_world` function below demonstrates how to use these quaternions to transform data from the IMU's local coordinate system to the world coordinate system.
 
 ```python
 from scipy.spatial.transform import Rotation as R
@@ -57,11 +53,11 @@ def transform_imu_to_world(imu_coordinates, imu_quaternions):
         ])
 ```
 
-Now that we have the `transform_imu_to_world` function, let's use it!
+### Example: Heading Vectors in World Coordinates
 
-### Obtain IMU Heading Vectors
+The `transform_imu_to_world` function can be used to calculate heading vectors of the IMU in world coordinates. The heading vector essentially describes the direction the IMU is facing. If we imagine the IMU inside the Neon module while it is worn on sombody's head, the heading vector describes the direction the wearer's face is pointing.
 
-An alternative representation of IMU orientation data is a heading vector that points outwards from the center of the IMU. It can be useful to compare this heading vector with the 3D gaze vectors in world coordinates.
+The "forward-facing axis" is the y-axis, so we can calculate the heading vector by transforming the `(0, 1, 0)` vector.
 
 ```python
 def imu_heading_in_world(imu_quaternions):
@@ -71,11 +67,13 @@ def imu_heading_in_world(imu_quaternions):
     )
 ```
 
-Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
+::: tip
+Neutral orientation (i.e. an identity rotation in the quaternion) of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
+:::
 
-### IMU Acceleration in World
+### Example: Acceleration in World Coordinates
 
-The IMU’s acceleration data are specified in its local coordinate system. If you want to understand how the observer is accelerating through their environment, then it can be easier to have the acceleration data specified in the world coordinate system:
+The IMU’s translational acceleration data is given in the IMU's local coordinate system. To understand how the observer is accelerating through the world it can be helpful to transform the data into the world coordinate system:
 
 ```python
 accelerations_in_world = transform_imu_to_world(
@@ -85,17 +83,16 @@ accelerations_in_world = transform_imu_to_world(
 
 ## Scene to World Coordinates
 
-Neon simultaneously records data in the scene camera and IMU coordinate systems, making it possible to study the relationship between head and eye movements.
+A lot of the data generated by Neon is provided in the scene camera's coordinate system, including e.g. gaze, fixation, and eye state data. This coordinate system is **not** equal to the IMU's coordinate system! There is a translation between them (simply because there is a physical distance between the camera and the IMU in the module) and also a rotation (because of how the scene camera's coordinate system is defined).
 
-To facilitate the comparison, it can be useful to represent these data streams in the same coordinate system. An important step is accounting for [the fixed 102 degree rotation offset between the scene camera and IMU coordinate systems](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data), as depicted below.
+The rotation is a 102 degree rotation around the x-axis of the IMU coordinate system and the translation is along the vector `(0.0 mm, -1.3 mm, -6.62 mm)`.
 
 ![Diagrams showing the fixed 102 degree rotation offset between the IMU and scene camera coordinate systems.](./imu-scene_camera_offset-black.png)
 
-We can use data from the IMU to transform gaze in scene camera coordinates to world coordinates. We proceed by building a `transform_scene_to_imu` function that handles the rotation between the two coordinate systems. It also accepts a `translation_in_imu` keyword argument to specify if points should be shifted in the IMU system. This will be relevant when converting 3D eyestate to world coordinates.
-
+We can define a `transform_scene_to_imu` function that handles the rotation between the two coordinate systems. 
 
 ```python
-def transform_scene_to_imu(coords_in_scene, translation_in_imu=np.zeros((3,))):
+def transform_scene_to_imu(coords_in_scene, translation_in_imu=np.array([0.0, -1.3, -6.62])):
     imu_scene_rotation_diff = np.deg2rad(-90 - 12)
     scene_to_imu = np.array(
         [
@@ -122,27 +119,25 @@ def transform_scene_to_imu(coords_in_scene, translation_in_imu=np.zeros((3,))):
     return coords_in_imu.T
 ```
 
-Putting together the `transform_scene_to_imu` and `transform_imu_to_world` functions, we can build a composite `transform_scene_to_world` function:
+Combining the `transform_scene_to_imu` function with the `transform_imu_to_world` function allows us to go all the way from scene camera coordinate system to world coordinate system
 
 ```python
-def transform_scene_to_world(coords_in_scene, imu_quaternions, translation_in_imu=np.zeros((3,))):
+def transform_scene_to_world(coords_in_scene, imu_quaternions, translation_in_imu=np.array([0.0, -1.3, -6.62])):
     coords_in_imu = transform_scene_to_imu(coords_in_scene, translation_in_imu)
     return transform_imu_to_world(coords_in_imu, imu_quaternions)
 ```
 
-You can now use this function to transform data in scene camera coordinates to world coordinates. Head to the [Application Example](#application-example) to see how!
+### Example: Eyestate in World Coordinates
 
-## Eyestate to World Coordinates
+The `transform_scene_to_world` function allows us easily convert [eye state data](](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states)) given in scene camera coordinates to world coordinates.
 
-The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) provided by Neon are aligned with the scene camera coordinate system. This means we can use the `transform_scene_to_world` function above to reconstruct the pose of the eyes in the world coordinate system. We just need to consider that the scene camera is displaced a bit from the IMU.
-Since the eyeball center and optical axis values provided by the 3D eyestate estimates are intrinsically linked, we provide the `eyestate_to_world` function to simplify doing the conversions:
+::: warning
+Note, to do this right in practice you need to make sure you sample the quaternions and eye state data from the same timestamps. Since both data streams are generated independently and do not share the same set of timestamps, this is a challenge in itself. 
 
-```python
-# The 3D eyestate data and the IMU quaternions should be sampled
-# at the same timestamps. You can linearly interpolate the IMU data.
-# See the Application Example:
-# http://docs.pupil-labs.com/alpha-lab/imu-transformations/#application-example
+We are glossing over this here, but one possible solution to this is interpolating the IMU data to match the timestamps of the eye state data, which is demonstrated [here](http://docs.pupil-labs.com/alpha-lab/imu-transformations/#application-example).
+:::
 
+```python
 def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
     """
     The eyeball_centers and optical_axes inputs are for the same eye.
@@ -165,9 +160,8 @@ def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
     return eyeball_centers_in_world, optical_axes_in_world
 ```
 
-## 3D Gaze to World Coordinates
-
-Neon provides 3D gaze data in [spherical coordinates (i.e., `azimuth/elevation [deg]`)](https://docs.pupil-labs.com/neon/data-collection/data-format/#gaze-csv). The `transform_scene_to_world` function above expects 3D Cartesian coordinates, so to convert spherical 3D gaze to world coordinates, we will need the `spherical_to_cartesian_scene` function:
+### Example: 3D Gaze Direction in World Coordinates
+Neon provides 3D gaze directions in [spherical coordinates (i.e., `azimuth/elevation [deg]`)](https://docs.pupil-labs.com/neon/data-collection/data-format/#gaze-csv). The `transform_scene_to_world` function above expects 3D Cartesian coordinates, so we need to convert the data first.
 
 ```python
 def spherical_to_cartesian_scene(elevations, azimuths):
@@ -202,93 +196,17 @@ def spherical_to_cartesian_scene(elevations, azimuths):
     ).T
 ```
 
-Now we have the tools to convert 3D gaze data to world coordinates:
+Now we can transform the data to world coordinates. Since we are dealing with 3D directions, rather than 3D points here, it does not make sense to apply the translation that we used in the `transform_scene_to_world` function above. We are thus setting it to zero here.
 
 ```python
 def gaze_3d_to_world(gaze_elevation, gaze_azimuth, imu_quaternions):
     cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevation, gaze_azimuth)
-    return transform_scene_to_world(cart_gazes_in_scene, imu_quaternions)
-```
-
-## 2D Gaze to World Coordinates
-
-If you are starting from [the 2D gaze values in scene image coordinates (i.e., `gaze x/y [px]`)](https://docs.pupil-labs.com/neon/data-collection/data-format/#gaze-csv), then you will need to first [undistort](https://docs.pupil-labs.com/alpha-lab/undistort/) and unproject those points to obtain the corresponding 3D gaze vectors. Note that this requires [loading the scene camera calibration data](https://docs-staging.pupil-labs.com/alpha-lab/undistort/#reading-from-the-cloud-download-json-file).
-
-```python
-def unproject_2d_gaze(gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients):
-    """
-    Transform the 2D gaze values from Neon to 3D gaze vectors.
-    """
-
-    gaze_points_2d_undist = cv2.undistortPoints(gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients)
-    gaze_vectors_3d = cv2.convertPointsToHomogeneous(gaze_points_2d_undist)
-    gaze_vectors_3d /= np.linalg.norm(gaze_vectors_3d, axis=2)[:, np.newaxis]
-    
-    return gaze_vectors_3d
-```
-
-Then, we can use the functions from [the previous section](#3d-gaze-to-world-coordinates) to convert 2D gaze to 3D world coordinates:
-
-```python
-# The gaze data and the IMU quaternions should be sampled at the
-# same timestamps. You can linearly interpolate the IMU data to
-# ensure this.
-# See the Application Example:
-# http://docs.pupil-labs.com/alpha-lab/imu-transformations/#application-example
-
-gaze_points_2d = gaze[["gaze x [px]", "gaze y [px]"]].to_numpy()
-def gaze_2d_to_world(gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients, imu_quaternions):
-    cart_gazes_in_scene = undistort_and_unproject(
-        gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients
-    )
-    return transform_scene_to_world(
-        cart_gazes_in_scene, imu_quaternions
-    )
-```
-
-## World Spherical Coordinates
-
-When studying head orientation and gaze orientation as observers navigate a 3D environment, it can be useful to know how much these quantities deviate from pointing at a given landmark or direction. For instance, you might want to know when someone’s gaze or heading deviates from parallel with the horizon. This can be simplified by converting world points from Cartesian to spherical coordinates. The [Euler angles from the IMU](https://docs.pupil-labs.com/neon/data-collection/data-streams/#euler-angles) are already in a compatible format. For gaze data in world coordinates, the `cartesian_to_spherical_world` function below will do the necessary transformation. When wearing Neon normally, an elevation and azimuth of 0 degrees corresponds to a neutral orientation: i.e., aimed at magnetic North and parallel to the horizon.
-
-```python
-def cartesian_to_spherical_world(world_points_3d):
-    """
-    Convert points in 3D Cartesian world coordinates to spherical coordinates.
-    
-    For elevation:
-      - Neutral orientation = 0 (i.e., parallel with horizon)
-      - Upwards is positive
-      - Downwards is negative
-
-    For azimuth:
-      - Neutral orientation = 0 (i.e., aligned with magnetic North)
-      - Leftwards is positive
-      - Rightwards is negative
-    """
-
-    x = world_points_3d[:, 0]
-    y = world_points_3d[:, 1]
-    z = world_points_3d[:, 2]
-
-    radii = np.sqrt(x**2 + y**2 + z**2)
-
-    elevation = -(np.arccos(z / radii) - np.pi / 2)
-    azimuth = np.arctan2(y, x) - np.pi / 2
-
-    # Keep all azimuth values in the range of [-180, 180] to remain
-    # consistent with the yaw orientation values provided by the IMU.
-    azimuth[azimuth < -np.pi] += 2 * np.pi
-    azimuth[azimuth > np.pi] -= 2 * np.pi
-
-    elevation = np.rad2deg(elevation)
-    azimuth = np.rad2deg(azimuth)
-
-    return elevation, azimuth
+    return transform_scene_to_world(cart_gazes_in_scene, imu_quaternions, translation_in_imu=np.zeros(3))
 ```
 
 ## Application Example
 
-Below, we present a video showing how some of the functions in this article where used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://gist.github.com/rennis250/8a684ea1e2f92c79fa2104b7a0f30e20).
+Below, we present a video showing how some of the functions in this article were used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://gist.github.com/rennis250/8a684ea1e2f92c79fa2104b7a0f30e20).
 
 <Youtube src="lbmyBgpS2OE"/>
 

From 548ae4d3e4ca9560e10c8e4fdc94f6037d3bb140 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Tue, 3 Sep 2024 11:50:25 +0200
Subject: [PATCH 44/54] remove tip about neon positioning and leftward pos/neg
 rotations

---
 neon/data-collection/data-streams/index.md | 4 ----
 1 file changed, 4 deletions(-)

diff --git a/neon/data-collection/data-streams/index.md b/neon/data-collection/data-streams/index.md
index cb0945ae3..4c3312943 100644
--- a/neon/data-collection/data-streams/index.md
+++ b/neon/data-collection/data-streams/index.md
@@ -102,8 +102,4 @@ When exporting recordings from Pupil Cloud or Neon Player the IMU's orientation
 - Yaw is a rotation around the world z-axis with a value range of -180° to +180°. With a calibrated IMU, a yaw of 0° indicates alignment with magnetic north. Leftward turn is positive, rightward is negative.
 - Roll is a rotations around the world y-axis with a value range of -180° to +180°. Wearing Neon upright with a neutral head pose roughly corresponds to a roll of 0°. Rightward tilt is positive, leftward is negative.
 
-::: tip
-Neon may sit differently on each wearer’s face, causing deviations from a neutral orientation even when facing magnetic north. Also, leftward rotations for the yaw values reported by the IMU are positive, whereas leftward rotations about the y-axis of the scene camera coordinate system are negative.
-:::
-
 ![IMU Pitch, Yaw, Roll](./imu-pitch-yaw-roll-black.png)

From 739440c3afde25074b003854a5767d2dd54ddcb3 Mon Sep 17 00:00:00 2001
From: Marc Tonsen <mar@pupil-labs.com>
Date: Tue, 3 Sep 2024 10:20:46 +0200
Subject: [PATCH 45/54] Iterated on content

---
 alpha-lab/imu-transformations/index.md | 142 ++++++-------------------
 1 file changed, 30 insertions(+), 112 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 7090574f2..9425e1854 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -26,17 +26,13 @@ import TagLinks from '@components/TagLinks.vue'
 
 <TagLinks :tags="$frontmatter.tags" />
 
-::: tip
-Want to compare IMU and gaze data in the same coordinate system to better understand how people coordinate head and eye movements? The transformation functions in this tutorial will show you how!
-:::
-
-This guide contains various transformation functions that can assist when working with Neon's IMU data.
+This guide contains various transformation functions that help with relating [Neon's IMU data](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data) with other data streams.
 
-## IMU to World Coordinates
+## Rotation between the IMU and the World
 
-One of the key steps when working with IMU data is the transformation that takes coordinates in the local IMU coordinate system to their corresponding coordinates in [the world coordinate system](http://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data). The quaternion values provided by the IMU can be used to convert between the two coordinate systems. The `transform_imu_to_world` function, defined below, will be used throughout this article.
+The IMU data includes a description of how the IMU is rotated in relation to the world. Concretely, the IMU data contains quaternions that define a rotation transformation between the [the world coordinate system](http://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data) and the IMU's local coordinate system at different points in time. 
 
-Note that the origin of the IMU coordinate system is the same as the origin of the world coordinate system.
+The `transform_imu_to_world` function below demonstrates how to use these quaternions to transform data from the IMU's local coordinate system to the world coordinate system.
 
 ```python
 from scipy.spatial.transform import Rotation as R
@@ -57,11 +53,11 @@ def transform_imu_to_world(imu_coordinates, imu_quaternions):
         ])
 ```
 
-Now that we have the `transform_imu_to_world` function, let's use it!
+### Example: Heading Vectors in World Coordinates
 
-### Obtain IMU Heading Vectors
+The `transform_imu_to_world` function can be used to calculate heading vectors of the IMU in world coordinates. The heading vector essentially describes the direction the IMU is facing. If we imagine the IMU inside the Neon module while it is worn on sombody's head, the heading vector describes the direction the wearer's face is pointing.
 
-An alternative representation of IMU orientation data is a heading vector that points outwards from the center of the IMU. It can be useful to compare this heading vector with the 3D gaze vectors in world coordinates.
+The "forward-facing axis" is the y-axis, so we can calculate the heading vector by transforming the `(0, 1, 0)` vector.
 
 ```python
 def imu_heading_in_world(imu_quaternions):
@@ -71,11 +67,13 @@ def imu_heading_in_world(imu_quaternions):
     )
 ```
 
-Neutral orientation of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
+::: tip
+Neutral orientation (i.e. an identity rotation in the quaternion) of the IMU would correspond to a heading vector that points at magnetic North and that is oriented perpendicular to the line of gravity.
+:::
 
-### IMU Acceleration in World
+### Example: Acceleration in World Coordinates
 
-The IMU’s acceleration data are specified in its local coordinate system. If you want to understand how the observer is accelerating through their environment, then it can be easier to have the acceleration data specified in the world coordinate system:
+The IMU’s translational acceleration data is given in the IMU's local coordinate system. To understand how the observer is accelerating through the world it can be helpful to transform the data into the world coordinate system:
 
 ```python
 accelerations_in_world = transform_imu_to_world(
@@ -85,17 +83,16 @@ accelerations_in_world = transform_imu_to_world(
 
 ## Scene to World Coordinates
 
-Neon simultaneously records data in the scene camera and IMU coordinate systems, making it possible to study the relationship between head and eye movements.
+A lot of the data generated by Neon is provided in the scene camera's coordinate system, including e.g. gaze, fixation, and eye state data. This coordinate system is **not** equal to the IMU's coordinate system! There is a translation between them (simply because there is a physical distance between the camera and the IMU in the module) and also a rotation (because of how the scene camera's coordinate system is defined).
 
-To facilitate the comparison, it can be useful to represent these data streams in the same coordinate system. An important step is accounting for [the fixed 102 degree rotation offset between the scene camera and IMU coordinate systems](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data), as depicted below.
+The rotation is a 102 degree rotation around the x-axis of the IMU coordinate system and the translation is along the vector `(0.0 mm, -1.3 mm, -6.62 mm)`.
 
 ![Diagrams showing the fixed 102 degree rotation offset between the IMU and scene camera coordinate systems.](./imu-scene_camera_offset-black.png)
 
-We can use data from the IMU to transform gaze in scene camera coordinates to world coordinates. We proceed by building a `transform_scene_to_imu` function that handles the rotation between the two coordinate systems. It also accepts a `translation_in_imu` keyword argument to specify if points should be shifted in the IMU system. This will be relevant when converting 3D eyestate to world coordinates.
-
+We can define a `transform_scene_to_imu` function that handles the rotation between the two coordinate systems. 
 
 ```python
-def transform_scene_to_imu(coords_in_scene, translation_in_imu=np.zeros((3,))):
+def transform_scene_to_imu(coords_in_scene, translation_in_imu=np.array([0.0, -1.3, -6.62])):
     imu_scene_rotation_diff = np.deg2rad(-90 - 12)
     scene_to_imu = np.array(
         [
@@ -122,27 +119,25 @@ def transform_scene_to_imu(coords_in_scene, translation_in_imu=np.zeros((3,))):
     return coords_in_imu.T
 ```
 
-Putting together the `transform_scene_to_imu` and `transform_imu_to_world` functions, we can build a composite `transform_scene_to_world` function:
+Combining the `transform_scene_to_imu` function with the `transform_imu_to_world` function allows us to go all the way from scene camera coordinate system to world coordinate system
 
 ```python
-def transform_scene_to_world(coords_in_scene, imu_quaternions, translation_in_imu=np.zeros((3,))):
+def transform_scene_to_world(coords_in_scene, imu_quaternions, translation_in_imu=np.array([0.0, -1.3, -6.62])):
     coords_in_imu = transform_scene_to_imu(coords_in_scene, translation_in_imu)
     return transform_imu_to_world(coords_in_imu, imu_quaternions)
 ```
 
-You can now use this function to transform data in scene camera coordinates to world coordinates. Head to the [Application Example](#application-example) to see how!
+### Example: Eyestate in World Coordinates
 
-## Eyestate to World Coordinates
+The `transform_scene_to_world` function allows us easily convert [eye state data](](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states)) given in scene camera coordinates to world coordinates.
 
-The [3D eyestate estimates](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) provided by Neon are aligned with the scene camera coordinate system. This means we can use the `transform_scene_to_world` function above to reconstruct the pose of the eyes in the world coordinate system. We just need to consider that the scene camera is displaced a bit from the IMU.
-Since the eyeball center and optical axis values provided by the 3D eyestate estimates are intrinsically linked, we provide the `eyestate_to_world` function to simplify doing the conversions:
+::: warning
+Note, to do this right in practice you need to make sure you sample the quaternions and eye state data from the same timestamps. Since both data streams are generated independently and do not share the same set of timestamps, this is a challenge in itself. 
 
-```python
-# The 3D eyestate data and the IMU quaternions should be sampled
-# at the same timestamps. You can linearly interpolate the IMU data.
-# See the Application Example:
-# http://docs.pupil-labs.com/alpha-lab/imu-transformations/#application-example
+We are glossing over this here, but one possible solution to this is interpolating the IMU data to match the timestamps of the eye state data, which is demonstrated [here](http://docs.pupil-labs.com/alpha-lab/imu-transformations/#application-example).
+:::
 
+```python
 def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
     """
     The eyeball_centers and optical_axes inputs are for the same eye.
@@ -165,9 +160,8 @@ def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
     return eyeball_centers_in_world, optical_axes_in_world
 ```
 
-## 3D Gaze to World Coordinates
-
-Neon provides 3D gaze data in [spherical coordinates (i.e., `azimuth/elevation [deg]`)](https://docs.pupil-labs.com/neon/data-collection/data-format/#gaze-csv). The `transform_scene_to_world` function above expects 3D Cartesian coordinates, so to convert spherical 3D gaze to world coordinates, we will need the `spherical_to_cartesian_scene` function:
+### Example: 3D Gaze Direction in World Coordinates
+Neon provides 3D gaze directions in [spherical coordinates (i.e., `azimuth/elevation [deg]`)](https://docs.pupil-labs.com/neon/data-collection/data-format/#gaze-csv). The `transform_scene_to_world` function above expects 3D Cartesian coordinates, so we need to convert the data first.
 
 ```python
 def spherical_to_cartesian_scene(elevations, azimuths):
@@ -202,93 +196,17 @@ def spherical_to_cartesian_scene(elevations, azimuths):
     ).T
 ```
 
-Now we have the tools to convert 3D gaze data to world coordinates:
+Now we can transform the data to world coordinates. Since we are dealing with 3D directions, rather than 3D points here, it does not make sense to apply the translation that we used in the `transform_scene_to_world` function above. We are thus setting it to zero here.
 
 ```python
 def gaze_3d_to_world(gaze_elevation, gaze_azimuth, imu_quaternions):
     cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevation, gaze_azimuth)
-    return transform_scene_to_world(cart_gazes_in_scene, imu_quaternions)
-```
-
-## 2D Gaze to World Coordinates
-
-If you are starting from [the 2D gaze values in scene image coordinates (i.e., `gaze x/y [px]`)](https://docs.pupil-labs.com/neon/data-collection/data-format/#gaze-csv), then you will need to first [undistort](https://docs.pupil-labs.com/alpha-lab/undistort/) and unproject those points to obtain the corresponding 3D gaze vectors. Note that this requires [loading the scene camera calibration data](https://docs-staging.pupil-labs.com/alpha-lab/undistort/#reading-from-the-cloud-download-json-file).
-
-```python
-def unproject_2d_gaze(gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients):
-    """
-    Transform the 2D gaze values from Neon to 3D gaze vectors.
-    """
-
-    gaze_points_2d_undist = cv2.undistortPoints(gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients)
-    gaze_vectors_3d = cv2.convertPointsToHomogeneous(gaze_points_2d_undist)
-    gaze_vectors_3d /= np.linalg.norm(gaze_vectors_3d, axis=2)[:, np.newaxis]
-    
-    return gaze_vectors_3d
-```
-
-Then, we can use the functions from [the previous section](#3d-gaze-to-world-coordinates) to convert 2D gaze to 3D world coordinates:
-
-```python
-# The gaze data and the IMU quaternions should be sampled at the
-# same timestamps. You can linearly interpolate the IMU data to
-# ensure this.
-# See the Application Example:
-# http://docs.pupil-labs.com/alpha-lab/imu-transformations/#application-example
-
-gaze_points_2d = gaze[["gaze x [px]", "gaze y [px]"]].to_numpy()
-def gaze_2d_to_world(gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients, imu_quaternions):
-    cart_gazes_in_scene = undistort_and_unproject(
-        gaze_points_2d, scene_camera_matrix, scene_distortion_coefficients
-    )
-    return transform_scene_to_world(
-        cart_gazes_in_scene, imu_quaternions
-    )
-```
-
-## World Spherical Coordinates
-
-When studying head orientation and gaze orientation as observers navigate a 3D environment, it can be useful to know how much these quantities deviate from pointing at a given landmark or direction. For instance, you might want to know when someone’s gaze or heading deviates from parallel with the horizon. This can be simplified by converting world points from Cartesian to spherical coordinates. The [Euler angles from the IMU](https://docs.pupil-labs.com/neon/data-collection/data-streams/#euler-angles) are already in a compatible format. For gaze data in world coordinates, the `cartesian_to_spherical_world` function below will do the necessary transformation. When wearing Neon normally, an elevation and azimuth of 0 degrees corresponds to a neutral orientation: i.e., aimed at magnetic North and parallel to the horizon.
-
-```python
-def cartesian_to_spherical_world(world_points_3d):
-    """
-    Convert points in 3D Cartesian world coordinates to spherical coordinates.
-    
-    For elevation:
-      - Neutral orientation = 0 (i.e., parallel with horizon)
-      - Upwards is positive
-      - Downwards is negative
-
-    For azimuth:
-      - Neutral orientation = 0 (i.e., aligned with magnetic North)
-      - Leftwards is positive
-      - Rightwards is negative
-    """
-
-    x = world_points_3d[:, 0]
-    y = world_points_3d[:, 1]
-    z = world_points_3d[:, 2]
-
-    radii = np.sqrt(x**2 + y**2 + z**2)
-
-    elevation = -(np.arccos(z / radii) - np.pi / 2)
-    azimuth = np.arctan2(y, x) - np.pi / 2
-
-    # Keep all azimuth values in the range of [-180, 180] to remain
-    # consistent with the yaw orientation values provided by the IMU.
-    azimuth[azimuth < -np.pi] += 2 * np.pi
-    azimuth[azimuth > np.pi] -= 2 * np.pi
-
-    elevation = np.rad2deg(elevation)
-    azimuth = np.rad2deg(azimuth)
-
-    return elevation, azimuth
+    return transform_scene_to_world(cart_gazes_in_scene, imu_quaternions, translation_in_imu=np.zeros(3))
 ```
 
 ## Application Example
 
-Below, we present a video showing how some of the functions in this article where used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://gist.github.com/rennis250/8a684ea1e2f92c79fa2104b7a0f30e20).
+Below, we present a video showing how some of the functions in this article were used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://gist.github.com/rennis250/8a684ea1e2f92c79fa2104b7a0f30e20).
 
 <Youtube src="lbmyBgpS2OE"/>
 

From 42af1c1b51b669f9d7044349e07d0163ddc68fa3 Mon Sep 17 00:00:00 2001
From: Marc Tonsen <mar@pupil-labs.com>
Date: Tue, 3 Sep 2024 13:25:00 +0200
Subject: [PATCH 46/54] Bring back the spehrical world coordinates section

---
 alpha-lab/imu-transformations/index.md | 43 ++++++++++++++++++++++++++
 1 file changed, 43 insertions(+)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 9425e1854..0628c1986 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -204,6 +204,49 @@ def gaze_3d_to_world(gaze_elevation, gaze_azimuth, imu_quaternions):
     return transform_scene_to_world(cart_gazes_in_scene, imu_quaternions, translation_in_imu=np.zeros(3))
 ```
 
+## World Spherical Coordinates
+Using the transformations introduced above, we can transform various data into cartesian world coordinates. For some things it is more intuitive to have the data in spherical coordinates though. For instance, you might want to know when someone’s gaze or heading deviates from parallel with the horizon, i.e. if they are looking/facing upwards or downwards. 
+
+Converting data into spherical world coordinates makes these things obvious. When wearing Neon, an elevation and azimuth of 0 degrees corresponds to a neutral orientation: i.e., aimed at magnetic North and parallel to the horizon. A positive elevation corresponds to looking upwards, and a negative elevation corresponds to looking downwards.
+
+The [Euler angles from the IMU](https://docs.pupil-labs.com/neon/data-collection/data-streams/#euler-angles) are already in a compatible format. For gaze data in world coordinates, the `cartesian_to_spherical_world` function below will do the necessary transformation. 
+
+```python
+def cartesian_to_spherical_world(world_points_3d):
+    """
+    Convert points in 3D Cartesian world coordinates to spherical coordinates.
+    
+    For elevation:
+      - Neutral orientation = 0 (i.e., parallel with horizon)
+      - Upwards is positive
+      - Downwards is negative
+
+    For azimuth:
+      - Neutral orientation = 0 (i.e., aligned with magnetic North)
+      - Leftwards is positive
+      - Rightwards is negative
+    """
+
+    x = world_points_3d[:, 0]
+    y = world_points_3d[:, 1]
+    z = world_points_3d[:, 2]
+
+    radii = np.sqrt(x**2 + y**2 + z**2)
+
+    elevation = -(np.arccos(z / radii) - np.pi / 2)
+    azimuth = np.arctan2(y, x) - np.pi / 2
+
+    # Keep all azimuth values in the range of [-180, 180] to remain
+    # consistent with the yaw orientation values provided by the IMU.
+    azimuth[azimuth < -np.pi] += 2 * np.pi
+    azimuth[azimuth > np.pi] -= 2 * np.pi
+
+    elevation = np.rad2deg(elevation)
+    azimuth = np.rad2deg(azimuth)
+
+    return elevation, azimuth
+```
+
 ## Application Example
 
 Below, we present a video showing how some of the functions in this article were used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://gist.github.com/rennis250/8a684ea1e2f92c79fa2104b7a0f30e20).

From 3c95c094c3087d270d80815e0c392c82f3505e22 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Tue, 3 Sep 2024 14:30:17 +0200
Subject: [PATCH 47/54] transform_scene_to_world incorporates scene camera
 offset, so adapt eyestate_to_world to accomodate that

---
 alpha-lab/imu-transformations/index.md | 7 +++----
 1 file changed, 3 insertions(+), 4 deletions(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 0628c1986..6d4c80451 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -89,7 +89,7 @@ The rotation is a 102 degree rotation around the x-axis of the IMU coordinate sy
 
 ![Diagrams showing the fixed 102 degree rotation offset between the IMU and scene camera coordinate systems.](./imu-scene_camera_offset-black.png)
 
-We can define a `transform_scene_to_imu` function that handles the rotation between the two coordinate systems. 
+We can define a `transform_scene_to_imu` function that handles the rotation between the two coordinate systems.
 
 ```python
 def transform_scene_to_imu(coords_in_scene, translation_in_imu=np.array([0.0, -1.3, -6.62])):
@@ -146,15 +146,14 @@ def eyestate_to_world(eyeball_centers, optical_axes, imu_quaternions):
     # The eyeball centers are specified relative to the center of the scene
     # camera, so we need to account for the position of the scene camera in
     # the IMU coordinate system. Here, we express that position in millimeters.
-    scene_camera_position_in_imu = np.array([0.0, -1.3, -6.62])
     eyeball_centers_in_world = transform_scene_to_world(
-        eyeball_centers, imu_quaternions, translation=scene_camera_position_in_imu
+        eyeball_centers, imu_quaternions
     )
 
     # The optical axes are unit vectors originating at the eyeball centers,
     # so they should not be translated.
     optical_axes_in_world = transform_scene_to_world(
-        optical_axes, imu_quaternions
+        optical_axes, imu_quaternions, translation=np.zeros(3)
     )
 
     return eyeball_centers_in_world, optical_axes_in_world

From 273fae453bae0d84d0befa716106dc6133601f7b Mon Sep 17 00:00:00 2001
From: Marc Tonsen <mar@pupil-labs.com>
Date: Tue, 3 Sep 2024 14:45:08 +0200
Subject: [PATCH 48/54] Fix link

---
 alpha-lab/imu-transformations/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 6d4c80451..4370d8709 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -129,7 +129,7 @@ def transform_scene_to_world(coords_in_scene, imu_quaternions, translation_in_im
 
 ### Example: Eyestate in World Coordinates
 
-The `transform_scene_to_world` function allows us easily convert [eye state data](](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states)) given in scene camera coordinates to world coordinates.
+The `transform_scene_to_world` function allows us easily convert [eye state data](https://docs.pupil-labs.com/neon/data-collection/data-streams/#_3d-eye-states) given in scene camera coordinates to world coordinates.
 
 ::: warning
 Note, to do this right in practice you need to make sure you sample the quaternions and eye state data from the same timestamps. Since both data streams are generated independently and do not share the same set of timestamps, this is a challenge in itself. 

From e1fd6cb2a76c564384dd6d4ba758e24eacb40a6d Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Tue, 3 Sep 2024 16:06:15 +0200
Subject: [PATCH 49/54] remove audio from youtube video

---
 alpha-lab/imu-transformations/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 4370d8709..e4e3a242f 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -250,7 +250,7 @@ def cartesian_to_spherical_world(world_points_3d):
 
 Below, we present a video showing how some of the functions in this article were used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://gist.github.com/rennis250/8a684ea1e2f92c79fa2104b7a0f30e20).
 
-<Youtube src="lbmyBgpS2OE"/>
+<Youtube src="hdZpnX9XDs0"/>
 
 ## Related content
 

From 67bcd4ec858f32746cdbd0058873ec4ffed3a360 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Miguel=20Garc=C3=ADa=20Garc=C3=ADa?=
 <mikelgg93@users.noreply.github.com>
Date: Wed, 4 Sep 2024 08:48:06 +0200
Subject: [PATCH 50/54] data-format: Fixed typo on directly and slight improve
 the text.

Modified https://arc.net/l/quote/nbgwfwmn to fix a typo on direclty
---
 neon/data-collection/data-format/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/neon/data-collection/data-format/index.md b/neon/data-collection/data-format/index.md
index 67a625d6a..c7d263cc1 100644
--- a/neon/data-collection/data-format/index.md
+++ b/neon/data-collection/data-format/index.md
@@ -1,7 +1,7 @@
 # Recording Format
 The page describes the data format when downloading Neon recordings from Pupil Cloud in the "Timeseries Data" and "Timeseries Data + Scene Video" formats. In this format the Data from the Neon Companion is augmented by adding gaze and eye state estimates whenever they had not been computed in realtime. Futhermore fixations and blink data as well as some IMU transformations are computed.
 
-When downloading Native Recording Data from cloud or direclty [extracting it via usb from the companion device](/data-collection/transfer-recordings-via-usb/), use the [pl-neon-recording](https://github.com/pupil-labs/pl-neon-recording) python library to read and access the data. This data format will contain all video data as well as all data that had been computed in realtime on the companion device.
+When downloading Native Recording Data from the Cloud or directly [extracting it via USB from the companion device](/data-collection/transfer-recordings-via-usb/), you can use the [pl-neon-recording](https://github.com/pupil-labs/pl-neon-recording) python library to read and access the data or load it onto [Neon Player](/neon-player). This data format will contain all video data as well as all data that has been computed in realtime on the companion device.
 
 
 ## Recording Folders

From 1b432f9ce5fb8898f9c5102b7b191cd057d0f502 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Wed, 4 Sep 2024 13:43:16 +0200
Subject: [PATCH 51/54] update youtube link

---
 alpha-lab/imu-transformations/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index e4e3a242f..0d68833a6 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -250,7 +250,7 @@ def cartesian_to_spherical_world(world_points_3d):
 
 Below, we present a video showing how some of the functions in this article were used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://gist.github.com/rennis250/8a684ea1e2f92c79fa2104b7a0f30e20).
 
-<Youtube src="hdZpnX9XDs0"/>
+<Youtube src="n0SjQzTqu3U"/>
 
 ## Related content
 

From fdd9a4c75e4903561afa6b6a0ff3e4b96a68b412 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Miguel=20Garc=C3=ADa=20Garc=C3=ADa?= <mikelgg93@gmail.com>
Date: Wed, 4 Sep 2024 14:32:53 +0200
Subject: [PATCH 52/54] Fixed link

---
 neon/data-collection/data-format/index.md | 232 +++++++++++-----------
 1 file changed, 115 insertions(+), 117 deletions(-)

diff --git a/neon/data-collection/data-format/index.md b/neon/data-collection/data-format/index.md
index c7d263cc1..7fe76fdfb 100644
--- a/neon/data-collection/data-format/index.md
+++ b/neon/data-collection/data-format/index.md
@@ -1,172 +1,170 @@
 # Recording Format
-The page describes the data format when downloading Neon recordings from Pupil Cloud in the "Timeseries Data" and "Timeseries Data + Scene Video" formats. In this format the Data from the Neon Companion is augmented by adding gaze and eye state estimates whenever they had not been computed in realtime. Futhermore fixations and blink data as well as some IMU transformations are computed.
 
-When downloading Native Recording Data from the Cloud or directly [extracting it via USB from the companion device](/data-collection/transfer-recordings-via-usb/), you can use the [pl-neon-recording](https://github.com/pupil-labs/pl-neon-recording) python library to read and access the data or load it onto [Neon Player](/neon-player). This data format will contain all video data as well as all data that has been computed in realtime on the companion device.
+The page describes the data format when downloading Neon recordings from Pupil Cloud in the "Timeseries Data" and "Timeseries Data + Scene Video" formats. In this format the Data from the Neon Companion is augmented by adding gaze and eye state estimates whenever they had not been computed in realtime. Futhermore fixations and blink data as well as some IMU transformations are computed.
 
+When downloading Native Recording Data from the Cloud or directly [extracting it via USB from the companion device](/data-collection/transfer-recordings-via-usb/), you can use the [pl-neon-recording](https://github.com/pupil-labs/pl-neon-recording) python library to read and access the data or load it onto [Neon Player](/neon-player/). This data format will contain all video data as well as all data that has been computed in realtime on the companion device.
 
 ## Recording Folders
+
 The export contains one folder per recording following this naming scheme:
-```<recording name>-<start of recording ID>```
+`<recording name>-<start of recording ID>`
 The files included in every folder are described in the following.
 
 ## info.json
-This file contains meta-information on the recording.
 
+This file contains meta-information on the recording.
 
-| Field                     | Description | 
-| ------------------------- | -------- | 
-| **android_device_id**     | Unique identifier of the Android device used as Companion.     |
-| **android_device_model**  | Model name of the Companion device.     |
-| **android_device_name**   | Device name of the Companion device.     |
-| **app_version**           | Version of the Neon Companion app used to make the recording. |
-| **calib_version**         | Version of the offset correction used by the Neon Companion app. |
-| **data_format_version**   | Version of the data format used by the Neon Companion app.     |
-| **duration**              | Duration of the recording in nanoseconds|
-| **firmware_version**      | Version numbers of the firmware and FPGA. |
-| **frame_id**              | Number identifying the type of frame used for this recording. |
-| **frame_name**            | Name of the frame used for this recording. |
-| **gaze_offset**           | Gaze offset applied to this recording using the offset correction. Values are in pixels.|
-| **module_serial_number**  | Serial number of the Neon module used for the recording. This number is encoded in the QR code on the back of the Neon module. |
-| **os_version**            | Version of the Android OS that was installed on the recording Companion device. |
-| **pipeline_version**      | Version of the gaze estimation pipeline used by the Neon Companion app. |
-| **recording_id**          | Unique identifier of the recording. |
-| **start_time**            | Timestamp of when the recording was started. Given as UTC timestamp in nanoseconds. |
-| **template_data**         | Data regarding the selected template for the recording as well as the response values. |
-| **wearer_id**             | Unique identifier of the wearer selected for this recording. |
-| **wearer_name**           | Name of the wearer selected for this recording. |
-| **workspace_id**          | The ID of the Pupil Cloud workspace this recording has been assigned to. |
-
+| Field                    | Description                                                                                                                    |
+| ------------------------ | ------------------------------------------------------------------------------------------------------------------------------ |
+| **android_device_id**    | Unique identifier of the Android device used as Companion.                                                                     |
+| **android_device_model** | Model name of the Companion device.                                                                                            |
+| **android_device_name**  | Device name of the Companion device.                                                                                           |
+| **app_version**          | Version of the Neon Companion app used to make the recording.                                                                  |
+| **calib_version**        | Version of the offset correction used by the Neon Companion app.                                                               |
+| **data_format_version**  | Version of the data format used by the Neon Companion app.                                                                     |
+| **duration**             | Duration of the recording in nanoseconds                                                                                       |
+| **firmware_version**     | Version numbers of the firmware and FPGA.                                                                                      |
+| **frame_id**             | Number identifying the type of frame used for this recording.                                                                  |
+| **frame_name**           | Name of the frame used for this recording.                                                                                     |
+| **gaze_offset**          | Gaze offset applied to this recording using the offset correction. Values are in pixels.                                       |
+| **module_serial_number** | Serial number of the Neon module used for the recording. This number is encoded in the QR code on the back of the Neon module. |
+| **os_version**           | Version of the Android OS that was installed on the recording Companion device.                                                |
+| **pipeline_version**     | Version of the gaze estimation pipeline used by the Neon Companion app.                                                        |
+| **recording_id**         | Unique identifier of the recording.                                                                                            |
+| **start_time**           | Timestamp of when the recording was started. Given as UTC timestamp in nanoseconds.                                            |
+| **template_data**        | Data regarding the selected template for the recording as well as the response values.                                         |
+| **wearer_id**            | Unique identifier of the wearer selected for this recording.                                                                   |
+| **wearer_name**          | Name of the wearer selected for this recording.                                                                                |
+| **workspace_id**         | The ID of the Pupil Cloud workspace this recording has been assigned to.                                                       |
 
 **Scene Video**
 Scene video is contained in a file following the following naming scheme:
-```<beginning of section ID>_<section start time>-<section end time>.mp4```
+`<beginning of section ID>_<section start time>-<section end time>.mp4`
 
 ## scene_camera.json
-This file contains the camera intrinsics of the used scene camera. The values are determined via calibration of every camera during manufacturing.
-
-| Field | Description | 
-| -------- | -------- | 
-| **camera_matrix** | The camera matrix of the scene camera.     |
-| **dist_coefs** | The distortion coefficients of the scene camera. The order of the values is `(k1, k2, p1, p2, k3, k4, k5, k6)` following [OpenCV's distortion model](https://docs.opencv.org/4.x/d9/d0c/group__calib3d.html#ga3207604e4b1a1758aa66acb6ed5aa65d). |
-| **serial_number** | Serial number of Neon module used for the recording. This number is encoded in the QR code on the back of the Neon module.      |
-| **version** | The version of the intrinsics data format.     |
 
+This file contains the camera intrinsics of the used scene camera. The values are determined via calibration of every camera during manufacturing.
 
+| Field             | Description                                                                                                                                                                                                                                      |
+| ----------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
+| **camera_matrix** | The camera matrix of the scene camera.                                                                                                                                                                                                           |
+| **dist_coefs**    | The distortion coefficients of the scene camera. The order of the values is `(k1, k2, p1, p2, k3, k4, k5, k6)` following [OpenCV's distortion model](https://docs.opencv.org/4.x/d9/d0c/group__calib3d.html#ga3207604e4b1a1758aa66acb6ed5aa65d). |
+| **serial_number** | Serial number of Neon module used for the recording. This number is encoded in the QR code on the back of the Neon module.                                                                                                                       |
+| **version**       | The version of the intrinsics data format.                                                                                                                                                                                                       |
 
 ## world_timestamps.csv
+
 This file contains the timestamps of every world video frame.
 
-| Field | Description | 
-| -------- | -------- | 
-| **section id** | Unique identifier of the corresponding section.     |
-| **recording id** | Unique identifier of the recording this sample belongs to.     |
+| Field              | Description                                                    |
+| ------------------ | -------------------------------------------------------------- |
+| **section id**     | Unique identifier of the corresponding section.                |
+| **recording id**   | Unique identifier of the recording this sample belongs to.     |
 | **timestamp [ns]** | UTC timestamp in nanoseconds of the corresponding world frame. |
 
-
 ## events.csv
-This file contains [event](/data-collection/events/) data for all recordings. It contains both event annotations from Pupil Cloud and real-time recording events.
 
-| Field | Description | 
-| -------- | -------- | 
-| **recording id** | Unique identifier of the recording this event belongs to.     |
-| **timestamp [ns]** | UTC timestamp of the event. |
-| **name** | Name of the event.     |
-| **type** | Type of the event. Possible values: cloud, recording     |
+This file contains [event](/data-collection/events/) data for all recordings. It contains both event annotations from Pupil Cloud and real-time recording events.
 
+| Field              | Description                                               |
+| ------------------ | --------------------------------------------------------- |
+| **recording id**   | Unique identifier of the recording this event belongs to. |
+| **timestamp [ns]** | UTC timestamp of the event.                               |
+| **name**           | Name of the event.                                        |
+| **type**           | Type of the event. Possible values: cloud, recording      |
 
 ## gaze.csv
-This file contains [gaze](/data-collection/data-streams/#gaze) data in world camera coordinates.
-
 
-| Field | Description | 
-| -------- | -------- | 
-| **section id** | Unique identifier of the corresponding section.     |
-| **recording id** | Unique identifier of the recording this sample belongs to.     |
-| **timestamp&nbsp;[ns]** | UTC timestamp in nanoseconds of the sample. Equal to the timestamp of the eye video frame this sample was generated with.     |
-| **gaze x [px]** | Float value representing the x-coordinate of the mapped gaze point in world camera pixel coordinates.
-| **gaze y [px]** | Same as "gaze x [px]" but for the y-coordinate.     |
-| **worn** | These values indicate whether Neon has been worn by a subject at the respective point in time. `1.0` indicates that it has been worn, while `0.0` indicates that it has not been worn.    |
-| **fixation id** | If this gaze sample belongs to a fixation event, this is the corresponding id of the fixation. Otherwise, this field is empty.     |
-| **blink id** | If this gaze samples belongs to a blink event, this is the corresponding id of the blink. Otherwise this field is empty.     |
-| **azimuth [deg]** | The [azimuth](https://en.wikipedia.org/wiki/Horizontal_coordinate_system) of the gaze ray in relation to the scene camera in degrees.     |
-| **elevation [deg]** | The [elevation](https://en.wikipedia.org/wiki/Horizontal_coordinate_system) of the gaze ray in relation to the scene camera in degrees.     |
+This file contains [gaze](/data-collection/data-streams/#gaze) data in world camera coordinates.
 
+| Field                   | Description                                                                                                                                                                            |
+| ----------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
+| **section id**          | Unique identifier of the corresponding section.                                                                                                                                        |
+| **recording id**        | Unique identifier of the recording this sample belongs to.                                                                                                                             |
+| **timestamp&nbsp;[ns]** | UTC timestamp in nanoseconds of the sample. Equal to the timestamp of the eye video frame this sample was generated with.                                                              |
+| **gaze x [px]**         | Float value representing the x-coordinate of the mapped gaze point in world camera pixel coordinates.                                                                                  |
+| **gaze y [px]**         | Same as "gaze x [px]" but for the y-coordinate.                                                                                                                                        |
+| **worn**                | These values indicate whether Neon has been worn by a subject at the respective point in time. `1.0` indicates that it has been worn, while `0.0` indicates that it has not been worn. |
+| **fixation id**         | If this gaze sample belongs to a fixation event, this is the corresponding id of the fixation. Otherwise, this field is empty.                                                         |
+| **blink id**            | If this gaze samples belongs to a blink event, this is the corresponding id of the blink. Otherwise this field is empty.                                                               |
+| **azimuth [deg]**       | The [azimuth](https://en.wikipedia.org/wiki/Horizontal_coordinate_system) of the gaze ray in relation to the scene camera in degrees.                                                  |
+| **elevation [deg]**     | The [elevation](https://en.wikipedia.org/wiki/Horizontal_coordinate_system) of the gaze ray in relation to the scene camera in degrees.                                                |
 
 ## fixations.csv
+
 This file contains [fixations](/data-collection/data-streams/#fixations-saccades) detected in the gaze data stream.
 The corresponding gaze samples that belong to each fixation can be determined from the `gaze.csv` file using the `fixation id` field.
 
-
-| Field | Description | 
-| -------- | -------- | 
-| **section id** | Unique identifier of the corresponding section.     |
-| **recording id** | Unique identifier of the recording this sample belongs to.     |
-| **fixation id** | Identifier of the fixation. The counter starts at the beginning of the recording.     |
-| **start&nbsp;timestamp&nbsp;[ns]** | UTC timestamp in nanoseconds of the start of the fixation.     |
-| **end timestamp [ns]** | UTC timestamp in nanoseconds of the end of the fixation.     |
-| **duration [ms]** | Duration of the fixation in milliseconds.     |
-| **fixation x [px]** | Float value representing the x-coordinate of the fixation in world camera pixel coordinates. This position is the average of all gaze samples within the fixation.     |
-| **fixation y [px]** | Same as "fixation x [px]" but for the y-coordinate.     |
-| **azimuth [deg]** | The [azimuth](https://en.wikipedia.org/wiki/Horizontal_coordinate_system) of the gaze ray corresponding to the fixation location in relation to the scene camera in degrees.     |
-| **elevation [deg]** | The [elevation](https://en.wikipedia.org/wiki/Horizontal_coordinate_system) of the gaze ray corresponding to the fixation location in relation to the scene camera in degrees.     |
+| Field                              | Description                                                                                                                                                                    |
+| ---------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
+| **section id**                     | Unique identifier of the corresponding section.                                                                                                                                |
+| **recording id**                   | Unique identifier of the recording this sample belongs to.                                                                                                                     |
+| **fixation id**                    | Identifier of the fixation. The counter starts at the beginning of the recording.                                                                                              |
+| **start&nbsp;timestamp&nbsp;[ns]** | UTC timestamp in nanoseconds of the start of the fixation.                                                                                                                     |
+| **end timestamp [ns]**             | UTC timestamp in nanoseconds of the end of the fixation.                                                                                                                       |
+| **duration [ms]**                  | Duration of the fixation in milliseconds.                                                                                                                                      |
+| **fixation x [px]**                | Float value representing the x-coordinate of the fixation in world camera pixel coordinates. This position is the average of all gaze samples within the fixation.             |
+| **fixation y [px]**                | Same as "fixation x [px]" but for the y-coordinate.                                                                                                                            |
+| **azimuth [deg]**                  | The [azimuth](https://en.wikipedia.org/wiki/Horizontal_coordinate_system) of the gaze ray corresponding to the fixation location in relation to the scene camera in degrees.   |
+| **elevation [deg]**                | The [elevation](https://en.wikipedia.org/wiki/Horizontal_coordinate_system) of the gaze ray corresponding to the fixation location in relation to the scene camera in degrees. |
 
 ## saccades.csv
-This file contains [saccades](/data-collection/data-streams/#fixations-saccades) detected by the fixation detector.
 
+This file contains [saccades](/data-collection/data-streams/#fixations-saccades) detected by the fixation detector.
 
-| Field | Description | 
-| -------- | -------- | 
-| **section id** | Unique identifier of the corresponding section.     |
-| **recording id** | Unique identifier of the recording this sample belongs to.     |
-| **saccade id** | Identifier of the saccade. The counter starts at the beginning of the recording.     |
-| **start&nbsp;timestamp&nbsp;[ns]** | UTC timestamp in nanoseconds of the start of the saccade.     |
-| **end&nbsp;timestamp&nbsp;[ns]** | UTC timestamp in nanoseconds of the end of the saccade.     |
-| **duration [ms]** | Duration of the saccade in milliseconds.     |
-| **amplitude [px]** | Float value representing the amplitude of the saccade in world camera pixel coordinates. |
-| **amplitude [deg]** | Float value representing the amplitude of the saccade in degrees of visual angle. |
+| Field                              | Description                                                                                             |
+| ---------------------------------- | ------------------------------------------------------------------------------------------------------- |
+| **section id**                     | Unique identifier of the corresponding section.                                                         |
+| **recording id**                   | Unique identifier of the recording this sample belongs to.                                              |
+| **saccade id**                     | Identifier of the saccade. The counter starts at the beginning of the recording.                        |
+| **start&nbsp;timestamp&nbsp;[ns]** | UTC timestamp in nanoseconds of the start of the saccade.                                               |
+| **end&nbsp;timestamp&nbsp;[ns]**   | UTC timestamp in nanoseconds of the end of the saccade.                                                 |
+| **duration [ms]**                  | Duration of the saccade in milliseconds.                                                                |
+| **amplitude [px]**                 | Float value representing the amplitude of the saccade in world camera pixel coordinates.                |
+| **amplitude [deg]**                | Float value representing the amplitude of the saccade in degrees of visual angle.                       |
 | **mean&nbsp;velocity&nbsp;[px/s]** | Float value representing the mean velocity of the saccade in world camera pixel coordinates per second. |
 | **peak&nbsp;velocity&nbsp;[px/s]** | Float value representing the peak velocity of the saccade in world camera pixel coordinates per second. |
 
-
 ## 3d_eye_states.csv
-This file contains [3D eye states](/data-collection/data-streams/#_3d-eye-states) as well as [pupil diameter](/data-collection/data-streams/#pupil-diameters) data.
 
+This file contains [3D eye states](/data-collection/data-streams/#_3d-eye-states) as well as [pupil diameter](/data-collection/data-streams/#pupil-diameters) data.
 
-| Field                     | Description | 
-| ------------------------- | -------- | 
-| **section id**            | Unique identifier of the corresponding section.     |
-| **recording id**          | Unique identifier of the recording this sample belongs to.     |
-| **timestamp [ns]**        | UTC timestamp in nanoseconds of the sample. Equal to the timestamp of the eye video frame this sample was generated with. |
-| **pupil&nbsp;diameter&nbsp;left&nbsp;[mm]**   | Physical diameter of the pupil of the left eye. |
-| **pupil&nbsp;diameter&nbsp;right&nbsp;[mm]**   | Physical diameter of the pupil of the right eye. |
-| **eye&nbsp;ball&nbsp;center&nbsp;left&nbsp;x&nbsp;[mm]**<br /> **eye ball center left y [mm]**<br /> **eye ball center left z [mm]**<br /> **eye&nbsp;ball&nbsp;center&nbsp;right&nbsp;x&nbsp;[mm]**<br /> **eye&nbsp;ball&nbsp;center&nbsp;right&nbsp;y&nbsp;[mm]**<br /> **eye ball center right z [mm]** | Location of left and right eye ball centers in millimeters in relation to the scene camera of the Neon module. For details on the coordinate systems see [here](/data-collection/data-streams/#_3d-eye-states). |
-| **optical axis left x**<br /> **optical axis left y**<br /> **optical axis left z**<br /> **optical axis right x**<br /> **optical axis right y**<br /> **optical axis right z** | Directional vector describing the optical axis of the left and right eye, i.e. the vector pointing from eye ball center to pupil center of the respective eye. For details on the coordinate systems see [here](/data-collection/data-streams/#_3d-eye-states). |
+| Field                                                                                                                                                                                                                                                                                                       | Description                                                                                                                                                                                                                                                     |
+| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
+| **section id**                                                                                                                                                                                                                                                                                              | Unique identifier of the corresponding section.                                                                                                                                                                                                                 |
+| **recording id**                                                                                                                                                                                                                                                                                            | Unique identifier of the recording this sample belongs to.                                                                                                                                                                                                      |
+| **timestamp [ns]**                                                                                                                                                                                                                                                                                          | UTC timestamp in nanoseconds of the sample. Equal to the timestamp of the eye video frame this sample was generated with.                                                                                                                                       |
+| **pupil&nbsp;diameter&nbsp;left&nbsp;[mm]**                                                                                                                                                                                                                                                                 | Physical diameter of the pupil of the left eye.                                                                                                                                                                                                                 |
+| **pupil&nbsp;diameter&nbsp;right&nbsp;[mm]**                                                                                                                                                                                                                                                                | Physical diameter of the pupil of the right eye.                                                                                                                                                                                                                |
+| **eye&nbsp;ball&nbsp;center&nbsp;left&nbsp;x&nbsp;[mm]**<br /> **eye ball center left y [mm]**<br /> **eye ball center left z [mm]**<br /> **eye&nbsp;ball&nbsp;center&nbsp;right&nbsp;x&nbsp;[mm]**<br /> **eye&nbsp;ball&nbsp;center&nbsp;right&nbsp;y&nbsp;[mm]**<br /> **eye ball center right z [mm]** | Location of left and right eye ball centers in millimeters in relation to the scene camera of the Neon module. For details on the coordinate systems see [here](/data-collection/data-streams/#_3d-eye-states).                                                 |
+| **optical axis left x**<br /> **optical axis left y**<br /> **optical axis left z**<br /> **optical axis right x**<br /> **optical axis right y**<br /> **optical axis right z**                                                                                                                            | Directional vector describing the optical axis of the left and right eye, i.e. the vector pointing from eye ball center to pupil center of the respective eye. For details on the coordinate systems see [here](/data-collection/data-streams/#_3d-eye-states). |
 
 ## blinks.csv
+
 This file contains [blinks](/data-collection/data-streams/#blinks) detected in the eye video.
 The corresponding gaze samples that belong to each blink can be determined from the `gaze.csv` file using the `blink id` field.
 
-
-| Field | Description | 
-| -------- | -------- | 
-| **section id** | Unique identifier of the corresponding section.     |
-| **recording id** | Unique identifier of the recording this sample belongs to.     |
-| **blink id** | Identifier of the blink. The counter starts at the beginning of the recording.     |
-| **start&nbsp;timestamp&nbsp;[ns]** | UTC timestamp in nanoseconds of the start of the blink.     |
-| **end timestamp [ns]** | UTC timestamp in nanoseconds of the end of the blink.     |
-| **duration [ms]** | Duration of the blink in milliseconds.     |
+| Field                              | Description                                                                    |
+| ---------------------------------- | ------------------------------------------------------------------------------ |
+| **section id**                     | Unique identifier of the corresponding section.                                |
+| **recording id**                   | Unique identifier of the recording this sample belongs to.                     |
+| **blink id**                       | Identifier of the blink. The counter starts at the beginning of the recording. |
+| **start&nbsp;timestamp&nbsp;[ns]** | UTC timestamp in nanoseconds of the start of the blink.                        |
+| **end timestamp [ns]**             | UTC timestamp in nanoseconds of the end of the blink.                          |
+| **duration [ms]**                  | Duration of the blink in milliseconds.                                         |
 
 ## imu.csv
+
 This file contains data recorded by the integrated [IMU](/data-collection/data-streams/#movement-imu-data) (inertial measurement unit).
 
-| Field | Description | 
-| -------- | -------- | 
-| **section id** | Unique identifier of the corresponding section.     |
-| **recording id** | Unique identifier of the recording this sample belongs to.     |
-| **timestamp [ns]** | UTC timestamp in nanoseconds of the sample.    |
-| **gyro x [deg/s]**<br />**gyro y [deg/s]**<br />**gyro z [deg/s]** | Rotation speed around x, y or z-axis respectively in degrees/s.    |
-| **acceleration&nbsp;x&nbsp;[g]**<br />**acceleration&nbsp;y&nbsp;[g]**<br />**acceleration z [g]** | Translational acceleration (in terms of [g-force](https://en.m.wikipedia.org/wiki/G-force)) along the x, y or z-axis respectively. Note `1 g = 9.80665 m/s^2`.|
-| **roll [deg]** | Drift-free estimate of the roll (head tilt from side to side) in degrees. The output range is -180 to +180 degrees. Added in version 2 of this enrichment.    |
-| **pitch [deg]** | Drift-free estimate of the pitch (head tilt from front to back) in degrees. The output range is -90 to +90 degrees. Added in version 2 of this enrichment. |
-| **yaw [deg]** | Drift-free estimate of the yaw (horizontal head rotation) in degrees. The output range is -180 to +180 degrees. Added in version 2 of this enrichment. |
-| **quaternion w**<br />**quaternion x**<br />**quaternion y**<br />**quaternion z** | Quaternion describing the rotation of the Neon module. |
+| Field                                                                                              | Description                                                                                                                                                    |
+| -------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------- |
+| **section id**                                                                                     | Unique identifier of the corresponding section.                                                                                                                |
+| **recording id**                                                                                   | Unique identifier of the recording this sample belongs to.                                                                                                     |
+| **timestamp [ns]**                                                                                 | UTC timestamp in nanoseconds of the sample.                                                                                                                    |
+| **gyro x [deg/s]**<br />**gyro y [deg/s]**<br />**gyro z [deg/s]**                                 | Rotation speed around x, y or z-axis respectively in degrees/s.                                                                                                |
+| **acceleration&nbsp;x&nbsp;[g]**<br />**acceleration&nbsp;y&nbsp;[g]**<br />**acceleration z [g]** | Translational acceleration (in terms of [g-force](https://en.m.wikipedia.org/wiki/G-force)) along the x, y or z-axis respectively. Note `1 g = 9.80665 m/s^2`. |
+| **roll [deg]**                                                                                     | Drift-free estimate of the roll (head tilt from side to side) in degrees. The output range is -180 to +180 degrees. Added in version 2 of this enrichment.     |
+| **pitch [deg]**                                                                                    | Drift-free estimate of the pitch (head tilt from front to back) in degrees. The output range is -90 to +90 degrees. Added in version 2 of this enrichment.     |
+| **yaw [deg]**                                                                                      | Drift-free estimate of the yaw (horizontal head rotation) in degrees. The output range is -180 to +180 degrees. Added in version 2 of this enrichment.         |
+| **quaternion w**<br />**quaternion x**<br />**quaternion y**<br />**quaternion z**                 | Quaternion describing the rotation of the Neon module.                                                                                                         |

From babf8bd7a05feb6e49d41c6baf58141723ce5476 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Miguel=20Garc=C3=ADa=20Garc=C3=ADa?= <mikelgg93@gmail.com>
Date: Wed, 4 Sep 2024 14:33:38 +0200
Subject: [PATCH 53/54] Bump Node version to 22, as 18 is soon to be deprecated
 in GH Actions

---
 .github/workflows/PR-checks.yml               |  6 +--
 .github/workflows/build-deploy-production.yml | 51 +++++++++----------
 .github/workflows/build-deploy-staging.yml    | 46 ++++++++---------
 .github/workflows/build-only.yml              | 41 ++++++++-------
 4 files changed, 71 insertions(+), 73 deletions(-)

diff --git a/.github/workflows/PR-checks.yml b/.github/workflows/PR-checks.yml
index ec237fd3e..115e0788c 100644
--- a/.github/workflows/PR-checks.yml
+++ b/.github/workflows/PR-checks.yml
@@ -9,7 +9,7 @@ on:
 
 jobs:
   build:
-    name: 📦 ${{ matrix.component }} build 
+    name: 📦 ${{ matrix.component }} build
     runs-on: ubuntu-latest
     strategy:
       fail-fast: false
@@ -21,7 +21,7 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./${{ matrix.component }}
@@ -101,7 +101,7 @@ jobs:
       - uses: actions/checkout@v3
         with:
           fetch-depth: 2
-      - name: 'Check Grammar'
+      - name: "Check Grammar"
         id: cspell
         uses: streetsidesoftware/cspell-action@v5
         with:
diff --git a/.github/workflows/build-deploy-production.yml b/.github/workflows/build-deploy-production.yml
index 682612ab0..2da959989 100644
--- a/.github/workflows/build-deploy-production.yml
+++ b/.github/workflows/build-deploy-production.yml
@@ -16,7 +16,7 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./neon
@@ -32,13 +32,13 @@ jobs:
       - name: Make build
         working-directory: ./neon
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: neon
           path: neon/.vitepress/dist
-  
+
   invisible:
     name: Build Invisible
     runs-on: ubuntu-latest
@@ -49,12 +49,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./invisible
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -65,13 +65,13 @@ jobs:
       - name: Make build
         working-directory: ./invisible
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: invisible
           path: invisible/.vitepress/dist
-  
+
   core:
     name: Build Core
     runs-on: ubuntu-latest
@@ -82,12 +82,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./core
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -98,13 +98,13 @@ jobs:
       - name: Make build
         working-directory: ./core
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: core
           path: core/.vitepress/dist
-  
+
   alpha-lab:
     name: Build Alpha Lab
     runs-on: ubuntu-latest
@@ -115,12 +115,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./alpha-lab
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -131,7 +131,7 @@ jobs:
       - name: Make build
         working-directory: ./alpha-lab
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
@@ -148,12 +148,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./landing-page
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -164,19 +164,19 @@ jobs:
       - name: Make build
         working-directory: ./landing-page
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: landing-page
           path: landing-page/.vitepress/dist
-          
+
   debugging-info:
     name: Upload debugging info
     runs-on: ubuntu-latest
 
     steps:
-      - uses: actions/checkout@v3         
+      - uses: actions/checkout@v3
       - name: Write commit SHA to gitcommit.txt
         shell: bash
         run: |
@@ -203,13 +203,13 @@ jobs:
         shell: bash
         run: |
           du -ah .
-      
+
       - name: Download gitcommit
         uses: actions/download-artifact@v3
         with:
           name: gitcommit
           path: gitcommit.txt
-      
+
       - name: Download Neon
         uses: actions/download-artifact@v3
         with:
@@ -221,13 +221,13 @@ jobs:
         with:
           name: invisible
           path: invisible/
-      
+
       - name: Download Core
         uses: actions/download-artifact@v3
         with:
           name: core
           path: core/
-      
+
       - name: Download AlphaLab
         uses: actions/download-artifact@v3
         with:
@@ -248,7 +248,6 @@ jobs:
           remote_host: ${{ secrets.REMOTE_HOST}}
           remote_user: ${{ secrets.REMOTE_USER }}
           remote_key: ${{ secrets.DEPLOY_KEY }}
-  
 
   deploy-production:
     name: Deploy to production
@@ -272,13 +271,13 @@ jobs:
         with:
           name: invisible
           path: invisible/
-      
+
       - name: Download Core
         uses: actions/download-artifact@v3
         with:
           name: core
           path: core/
-      
+
       - name: Download AlphaLab
         uses: actions/download-artifact@v3
         with:
diff --git a/.github/workflows/build-deploy-staging.yml b/.github/workflows/build-deploy-staging.yml
index 92b61a2bf..9b6fdf33b 100644
--- a/.github/workflows/build-deploy-staging.yml
+++ b/.github/workflows/build-deploy-staging.yml
@@ -14,7 +14,7 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./neon
@@ -30,13 +30,13 @@ jobs:
       - name: Make build
         working-directory: ./neon
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: neon
           path: neon/.vitepress/dist
-  
+
   invisible:
     name: Build Invisible
     runs-on: ubuntu-latest
@@ -47,12 +47,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./invisible
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -63,13 +63,13 @@ jobs:
       - name: Make build
         working-directory: ./invisible
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: invisible
           path: invisible/.vitepress/dist
- 
+
   core:
     name: Build Core
     runs-on: ubuntu-latest
@@ -80,12 +80,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./core
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -96,13 +96,13 @@ jobs:
       - name: Make build
         working-directory: ./core
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: core
           path: core/.vitepress/dist
-  
+
   alpha-lab:
     name: Build Alpha Lab
     runs-on: ubuntu-latest
@@ -113,12 +113,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./alpha-lab
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -129,7 +129,7 @@ jobs:
       - name: Make build
         working-directory: ./alpha-lab
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
@@ -146,12 +146,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./landing-page
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -162,19 +162,19 @@ jobs:
       - name: Make build
         working-directory: ./landing-page
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: landing-page
           path: landing-page/.vitepress/dist
-          
+
   debugging-info:
     name: Upload debugging info
     runs-on: ubuntu-latest
 
     steps:
-      - uses: actions/checkout@v3         
+      - uses: actions/checkout@v3
       - name: Write commit SHA to gitcommit.txt
         shell: bash
         run: |
@@ -201,13 +201,13 @@ jobs:
         shell: bash
         run: |
           du -ah .
-      
+
       - name: Download gitcommit
         uses: actions/download-artifact@v3
         with:
           name: gitcommit
           path: gitcommit.txt
-      
+
       - name: Download Neon
         uses: actions/download-artifact@v3
         with:
@@ -219,13 +219,13 @@ jobs:
         with:
           name: invisible
           path: invisible/
-      
+
       - name: Download Core
         uses: actions/download-artifact@v3
         with:
           name: core
           path: core/
-      
+
       - name: Download AlphaLab
         uses: actions/download-artifact@v3
         with:
diff --git a/.github/workflows/build-only.yml b/.github/workflows/build-only.yml
index cab837c5e..c587819f9 100644
--- a/.github/workflows/build-only.yml
+++ b/.github/workflows/build-only.yml
@@ -1,7 +1,6 @@
 name: Build only
 
-on:
-  pull_request
+on: pull_request
 
 jobs:
   neon:
@@ -14,7 +13,7 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./neon
@@ -30,13 +29,13 @@ jobs:
       - name: Make build
         working-directory: ./neon
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: neon
           path: neon/.vitepress/dist
-  
+
   invisible:
     name: Build Invisible
     runs-on: ubuntu-latest
@@ -47,12 +46,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./invisible
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -63,13 +62,13 @@ jobs:
       - name: Make build
         working-directory: ./invisible
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: invisible
           path: invisible/.vitepress/dist
- 
+
   core:
     name: Build Core
     runs-on: ubuntu-latest
@@ -80,12 +79,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./core
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -96,13 +95,13 @@ jobs:
       - name: Make build
         working-directory: ./core
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: core
           path: core/.vitepress/dist
-  
+
   alpha-lab:
     name: Build Alpha Lab
     runs-on: ubuntu-latest
@@ -113,12 +112,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./alpha-lab
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -129,7 +128,7 @@ jobs:
       - name: Make build
         working-directory: ./alpha-lab
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
@@ -146,12 +145,12 @@ jobs:
       - name: Use Node.js
         uses: actions/setup-node@v3
         with:
-          node-version: 18
+          node-version: 22
 
       - name: Install dependencies
         working-directory: ./landing-page
         run: npm install
-      
+
       # Copy node modules to provide dependencies for shared components in root folder
       - name: Copy node_modules
         shell: bash
@@ -162,19 +161,19 @@ jobs:
       - name: Make build
         working-directory: ./landing-page
         run: npm run docs:build
-      
+
       - name: Upload
         uses: actions/upload-artifact@v3
         with:
           name: landing-page
           path: landing-page/.vitepress/dist
-          
+
   debugging-info:
     name: Upload debugging info
     runs-on: ubuntu-latest
 
     steps:
-      - uses: actions/checkout@v3         
+      - uses: actions/checkout@v3
       - name: Write commit SHA to gitcommit.txt
         shell: bash
         run: |

From 4d58143948c083d360093df7409c3a1e0e6b57e2 Mon Sep 17 00:00:00 2001
From: Robert Ennis <rob@pupil-labs.com>
Date: Thu, 5 Sep 2024 14:25:24 +0200
Subject: [PATCH 54/54] keep code example alongside the article

---
 .../imu_heading_visualization.ipynb           | 549 ++++++++++++++++++
 alpha-lab/imu-transformations/index.md        |   4 +-
 2 files changed, 552 insertions(+), 1 deletion(-)
 create mode 100644 alpha-lab/imu-transformations/imu_heading_visualization.ipynb

diff --git a/alpha-lab/imu-transformations/imu_heading_visualization.ipynb b/alpha-lab/imu-transformations/imu_heading_visualization.ipynb
new file mode 100644
index 000000000..d8eb6cf9f
--- /dev/null
+++ b/alpha-lab/imu-transformations/imu_heading_visualization.ipynb
@@ -0,0 +1,549 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import json\n",
+    "\n",
+    "import cv2\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from scipy.spatial.transform import Rotation as R\n",
+    "\n",
+    "\n",
+    "def transform_imu_to_world(imu_coordinates, imu_quaternions):\n",
+    "    # This array contains a timeseries of transformation matrices,\n",
+    "    # as calculated from the IMU's timeseries of quaternions values.\n",
+    "    imu_to_world_matrices = R.from_quat(imu_quaternions).as_matrix()\n",
+    "\n",
+    "    if np.ndim(imu_coordinates) == 1:\n",
+    "        return imu_to_world_matrices @ imu_coordinates\n",
+    "    else:\n",
+    "        return np.array(\n",
+    "            [\n",
+    "                imu_to_world @ imu_coord\n",
+    "                for imu_to_world, imu_coord in zip(\n",
+    "                    imu_to_world_matrices, imu_coordinates\n",
+    "                )\n",
+    "            ]\n",
+    "        )\n",
+    "\n",
+    "\n",
+    "def transform_scene_to_imu(\n",
+    "    coords_in_scene, translation_in_imu=np.array([0.0, -1.3, -6.62])\n",
+    "):\n",
+    "    imu_scene_rotation_diff = np.deg2rad(-90 - 12)\n",
+    "    scene_to_imu = np.array(\n",
+    "        [\n",
+    "            [1.0, 0.0, 0.0],\n",
+    "            [\n",
+    "                0.0,\n",
+    "                np.cos(imu_scene_rotation_diff),\n",
+    "                -np.sin(imu_scene_rotation_diff),\n",
+    "            ],\n",
+    "            [\n",
+    "                0.0,\n",
+    "                np.sin(imu_scene_rotation_diff),\n",
+    "                np.cos(imu_scene_rotation_diff),\n",
+    "            ],\n",
+    "        ]\n",
+    "    )\n",
+    "\n",
+    "    coords_in_imu = scene_to_imu @ coords_in_scene.T\n",
+    "\n",
+    "    coords_in_imu[0, :] += translation_in_imu[0]\n",
+    "    coords_in_imu[1, :] += translation_in_imu[1]\n",
+    "    coords_in_imu[2, :] += translation_in_imu[2]\n",
+    "\n",
+    "    return coords_in_imu.T\n",
+    "\n",
+    "\n",
+    "def spherical_to_cartesian_scene(elevations, azimuths):\n",
+    "    \"\"\"\n",
+    "    Convert Neon's spherical representation of 3D gaze to Cartesian coordinates.\n",
+    "    \"\"\"\n",
+    "\n",
+    "    elevations_rad = np.deg2rad(elevations)\n",
+    "    azimuths_rad = np.deg2rad(azimuths)\n",
+    "\n",
+    "    # Elevation of 0 in Neon system corresponds to Y = 0, but\n",
+    "    # an elevation of 0 in traditional spherical coordinates would\n",
+    "    # correspond to Y = 1, so we convert elevation to the\n",
+    "    # more traditional format.\n",
+    "    elevations_rad += np.pi / 2\n",
+    "\n",
+    "    # Azimuth of 0 in Neon system corresponds to X = 0, but\n",
+    "    # an azimuth of 0 in traditional spherical coordinates would\n",
+    "    # correspond to X = 1. Also, azimuth to the right in Neon is\n",
+    "    # more positive, whereas it is more negative in traditional\n",
+    "    # spherical coordiantes. So, we convert azimuth to the more\n",
+    "    # traditional format.\n",
+    "    azimuths_rad *= -1.0\n",
+    "    azimuths_rad += np.pi / 2\n",
+    "\n",
+    "    return np.array(\n",
+    "        [\n",
+    "            np.sin(elevations_rad) * np.cos(azimuths_rad),\n",
+    "            np.cos(elevations_rad),\n",
+    "            np.sin(elevations_rad) * np.sin(azimuths_rad),\n",
+    "        ]\n",
+    "    ).T\n",
+    "\n",
+    "\n",
+    "def transform_scene_to_world(\n",
+    "    coords_in_scene, imu_quaternions, translation_in_imu=np.array([0.0, -1.3, -6.62])\n",
+    "):\n",
+    "    coords_in_imu = transform_scene_to_imu(coords_in_scene, translation_in_imu)\n",
+    "    return transform_imu_to_world(coords_in_imu, imu_quaternions)\n",
+    "\n",
+    "\n",
+    "def gaze_3d_to_world(gaze_elevation, gaze_azimuth, imu_quaternions):\n",
+    "    cart_gazes_in_scene = spherical_to_cartesian_scene(gaze_elevation, gaze_azimuth)\n",
+    "    return transform_scene_to_world(\n",
+    "        cart_gazes_in_scene, imu_quaternions, translation_in_imu=np.zeros(3)\n",
+    "    )\n",
+    "\n",
+    "\n",
+    "def imu_heading_in_world(imu_quaternions):\n",
+    "    heading_neutral_in_imu_coords = np.array([0.0, 1.0, 0.0])\n",
+    "    return transform_imu_to_world(heading_neutral_in_imu_coords, imu_quaternions)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "rec_dir = \"./2024-08-19_14-20-57-48fd6838/\"\n",
+    "\n",
+    "gaze = pd.read_csv(rec_dir + \"gaze.csv\")\n",
+    "eye3d = pd.read_csv(rec_dir + \"3d_eye_states.csv\")\n",
+    "imu = pd.read_csv(rec_dir + \"imu.csv\")\n",
+    "world = pd.read_csv(rec_dir + \"world_timestamps.csv\")\n",
+    "\n",
+    "gaze_ts = gaze[\"timestamp [ns]\"]\n",
+    "imu_ts = imu[\"timestamp [ns]\"]\n",
+    "world_ts = world[\"timestamp [ns]\"]\n",
+    "\n",
+    "info = []\n",
+    "with open(rec_dir + \"info.json\", \"r\") as f:\n",
+    "    info = json.load(f)\n",
+    "\n",
+    "gaze[\"relative ts [s]\"] = (gaze_ts - info[\"start_time\"]) * 1e-9\n",
+    "imu[\"relative ts [s]\"] = (imu_ts - info[\"start_time\"]) * 1e-9\n",
+    "world[\"relative ts [s]\"] = (world_ts - info[\"start_time\"]) * 1e-9\n",
+    "\n",
+    "relative_demo_video_ts = np.arange(\n",
+    "    world[\"relative ts [s]\"].iloc[200], world[\"relative ts [s]\"].iloc[-100], 1/30\n",
+    ")\n",
+    "\n",
+    "# We have more gaze datapoints (sampled at 200Hz) than\n",
+    "# IMU datapoints (sampled at 110Hz). We also need to sample values at the\n",
+    "# framerate of the visualization video that we will make, so linearly\n",
+    "# interpolate the IMU and gaze datapoints to be congruent with each other\n",
+    "# and the video render.\n",
+    "quaternions_resampled = np.array(\n",
+    "    [\n",
+    "        np.interp(relative_demo_video_ts, imu[\"relative ts [s]\"], imu[\"quaternion x\"]),\n",
+    "        np.interp(relative_demo_video_ts, imu[\"relative ts [s]\"], imu[\"quaternion y\"]),\n",
+    "        np.interp(relative_demo_video_ts, imu[\"relative ts [s]\"], imu[\"quaternion z\"]),\n",
+    "        np.interp(relative_demo_video_ts, imu[\"relative ts [s]\"], imu[\"quaternion w\"]),\n",
+    "    ]\n",
+    ").T\n",
+    "\n",
+    "gaze_elevation_resampled = np.interp(relative_demo_video_ts, gaze[\"relative ts [s]\"], gaze[\"elevation [deg]\"])\n",
+    "gaze_azimuth_resampled = np.interp(relative_demo_video_ts, gaze[\"relative ts [s]\"], gaze[\"azimuth [deg]\"])\n",
+    "\n",
+    "# Now, we can apply the functions.\n",
+    "\n",
+    "headings_in_world = imu_heading_in_world(quaternions_resampled)\n",
+    "\n",
+    "cart_gazes_in_world = gaze_3d_to_world(\n",
+    "    gaze_elevation_resampled, gaze_azimuth_resampled, quaternions_resampled\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 1050x500 with 2 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.animation as animation\n",
+    "\n",
+    "fig, axs = plt.subplots(1, 2, figsize=(10.5, 5))\n",
+    "\n",
+    "# Overhead visualization\n",
+    "\n",
+    "axs[0].axis(\"square\")\n",
+    "axs[0].set_aspect(\"equal\", adjustable=\"box\")\n",
+    "axs[0].set_xlim(-1.29, 1.29)\n",
+    "axs[0].set_ylim(-1.29, 1.29)\n",
+    "\n",
+    "axs[0].axis(\"off\")\n",
+    "\n",
+    "circle = plt.Circle((0, 0), 1.0, color=\"black\", fill=False)\n",
+    "axs[0].add_patch(circle)\n",
+    "\n",
+    "axs[0].text(0, 1.11, \"N\", ha=\"center\", va=\"bottom\", fontsize=\"xx-large\")\n",
+    "axs[0].text(0, -1.11, \"S\", ha=\"center\", va=\"top\", fontsize=\"xx-large\")\n",
+    "axs[0].text(1.11, 0, \"E\", ha=\"left\", va=\"center\", fontsize=\"xx-large\")\n",
+    "axs[0].text(-1.11, 0, \"W\", ha=\"right\", va=\"center\", fontsize=\"xx-large\")\n",
+    "\n",
+    "axs[0].plot([0, 0], [1.0, 1.08], color=\"black\")\n",
+    "axs[0].plot([0, 0], [-1.0, -1.08], color=\"black\")\n",
+    "axs[0].plot([1.0, 1.08], [0, 0], color=\"black\")\n",
+    "axs[0].plot([-1.0, -1.08], [0, 0], color=\"black\")\n",
+    "\n",
+    "heading_quiver_overhead = axs[0].quiver(\n",
+    "    0,\n",
+    "    0,\n",
+    "    headings_in_world[0, 0],\n",
+    "    headings_in_world[0, 1],\n",
+    "    color=\"b\",\n",
+    "    label=\"IMU Heading\",\n",
+    "    scale=1,\n",
+    "    scale_units=\"xy\",\n",
+    "    angles=\"xy\",\n",
+    "    width=0.01,\n",
+    ")\n",
+    "\n",
+    "gaze_quiver_overhead = axs[0].quiver(\n",
+    "    0,\n",
+    "    0,\n",
+    "    cart_gazes_in_world[0, 0],\n",
+    "    cart_gazes_in_world[0, 1],\n",
+    "    color=\"r\",\n",
+    "    label=\"Gaze vector\",\n",
+    "    scale=1,\n",
+    "    scale_units=\"xy\",\n",
+    "    angles=\"xy\",\n",
+    "    width=0.01,\n",
+    ")\n",
+    "\n",
+    "axs[0].legend(loc=\"lower right\", prop={\"size\": 11})\n",
+    "\n",
+    "axs[0].set_title(\"Heading and Gaze in World - Overhead\", fontsize=\"xx-large\")\n",
+    "\n",
+    "\n",
+    "# Side profile visualization\n",
+    "\n",
+    "axs[1].axis(\"square\")\n",
+    "axs[1].set_aspect(\"equal\", adjustable=\"box\")\n",
+    "axs[1].set_xlim(-1.29, 1.29)\n",
+    "axs[1].set_ylim(-1.29, 1.29)\n",
+    "\n",
+    "axs[1].axis(\"off\")\n",
+    "\n",
+    "circle = plt.Circle((0, 0), 1.0, color=\"black\", fill=False)\n",
+    "axs[1].add_patch(circle)\n",
+    "\n",
+    "axs[1].plot([0, 0], [1.0, 1.08], color=\"black\")\n",
+    "axs[1].plot([0, 0], [-1.0, -1.08], color=\"black\")\n",
+    "\n",
+    "axs[1].text(0, 1.11, \"Sky\", ha=\"center\", va=\"bottom\", fontsize=\"xx-large\")\n",
+    "axs[1].text(0, -1.11, \"Earth\", ha=\"center\", va=\"top\", fontsize=\"xx-large\")\n",
+    "\n",
+    "axs[1].hlines(0, -1.0, 1.0, color=\"black\", linestyle=\"--\")\n",
+    "\n",
+    "heading_quiver_sideprofile = axs[1].quiver(\n",
+    "    0,\n",
+    "    0,\n",
+    "    headings_in_world[0, 1],\n",
+    "    headings_in_world[0, 2],\n",
+    "    color=\"b\",\n",
+    "    scale=1,\n",
+    "    scale_units=\"xy\",\n",
+    "    angles=\"xy\",\n",
+    "    width=0.01,\n",
+    ")\n",
+    "\n",
+    "gaze_quiver_sideprofile = axs[1].quiver(\n",
+    "    0,\n",
+    "    0,\n",
+    "    cart_gazes_in_world[0, 1],\n",
+    "    cart_gazes_in_world[0, 2],\n",
+    "    color=\"r\",\n",
+    "    scale=1,\n",
+    "    scale_units=\"xy\",\n",
+    "    angles=\"xy\",\n",
+    "    width=0.01,\n",
+    ")\n",
+    "\n",
+    "axs[1].set_title(\"Heading and Gaze in World - Side-profile\", fontsize=\"xx-large\")\n",
+    "\n",
+    "\n",
+    "def update(frame):\n",
+    "    global heading_quiver_overhead\n",
+    "    heading_quiver_overhead.remove()\n",
+    "    heading_quiver_overhead = axs[0].quiver(\n",
+    "        0,\n",
+    "        0,\n",
+    "        headings_in_world[frame, 0],\n",
+    "        headings_in_world[frame, 1],\n",
+    "        color=\"b\",\n",
+    "        label=\"IMU Heading\",\n",
+    "        scale=1,\n",
+    "        scale_units=\"xy\",\n",
+    "        angles=\"xy\",\n",
+    "        width=0.01,\n",
+    "    )\n",
+    "\n",
+    "    global gaze_quiver_overhead\n",
+    "    gaze_quiver_overhead.remove()\n",
+    "    gaze_quiver_overhead = axs[0].quiver(\n",
+    "        0,\n",
+    "        0,\n",
+    "        cart_gazes_in_world[frame, 0],\n",
+    "        cart_gazes_in_world[frame, 1],\n",
+    "        color=\"r\",\n",
+    "        label=\"Gaze vector\",\n",
+    "        scale=1,\n",
+    "        scale_units=\"xy\",\n",
+    "        angles=\"xy\",\n",
+    "        width=0.01,\n",
+    "    )\n",
+    "    \n",
+    "    global heading_quiver_sideprofile\n",
+    "    heading_quiver_sideprofile.remove()\n",
+    "    heading_quiver_sideprofile = axs[1].quiver(\n",
+    "        0,\n",
+    "        0,\n",
+    "        headings_in_world[frame, 1],\n",
+    "        headings_in_world[frame, 2],\n",
+    "        color=\"b\",\n",
+    "        scale=1,\n",
+    "        scale_units=\"xy\",\n",
+    "        angles=\"xy\",\n",
+    "        width=0.01,\n",
+    "    )\n",
+    "\n",
+    "    global gaze_quiver_sideprofile\n",
+    "    gaze_quiver_sideprofile.remove()\n",
+    "    gaze_quiver_sideprofile = axs[1].quiver(\n",
+    "        0,\n",
+    "        0,\n",
+    "        cart_gazes_in_world[frame, 1],\n",
+    "        cart_gazes_in_world[frame, 2],\n",
+    "        color=\"r\",\n",
+    "        scale=1,\n",
+    "        scale_units=\"xy\",\n",
+    "        angles=\"xy\",\n",
+    "        width=0.01,\n",
+    "    )\n",
+    "\n",
+    "    return\n",
+    "\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "\n",
+    "ani = animation.FuncAnimation(\n",
+    "    fig=fig, func=update, frames=len(relative_demo_video_ts), interval=33.3333333333\n",
+    ")\n",
+    "ani.save(\"imu_heading.mp4\", writer=\"ffmpeg\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "gaze_left_start = 140\n",
+    "head_left_start = 315\n",
+    "gaze_head_left_start = 515\n",
+    "gaze_left_end = 255\n",
+    "head_left_end = 460\n",
+    "gaze_head_left_end = 650\n",
+    "\n",
+    "\n",
+    "gaze_right_start = 1201\n",
+    "head_right_start = 1364\n",
+    "gaze_head_right_start = 1614\n",
+    "gaze_right_end = 1325\n",
+    "head_right_end = 1560\n",
+    "gaze_head_right_end = 1762\n",
+    "\n",
+    "\n",
+    "gaze_up_start = 690\n",
+    "head_up_start = 850\n",
+    "gaze_head_up_start = 1030\n",
+    "gaze_up_end = 805\n",
+    "head_up_end = 1005\n",
+    "gaze_head_up_end = 1160\n",
+    "\n",
+    "\n",
+    "gaze_down_start = 1828\n",
+    "head_down_start = 2028\n",
+    "gaze_head_down_start = 2218\n",
+    "gaze_down_end = 1931\n",
+    "head_down_end = 2155\n",
+    "gaze_head_down_end = 2356\n",
+    "\n",
+    "\n",
+    "free_viewing_start = 2415\n",
+    "free_viewing_end = len(world_ts[200:-100]) - 1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# The gaze+eye overlay video was made with the pl-neon-recording library:\n",
+    "# https://github.com/pupil-labs/pl-neon-recording\n",
+    "import pupil_labs.neon_recording as nr\n",
+    "\n",
+    "native_rec_dir = \"./native_2024-08-19_14-20-57-48fd6838/\"\n",
+    "recording = nr.load(native_rec_dir)\n",
+    "\n",
+    "\n",
+    "def overlay_image(img, img_overlay, x, y):\n",
+    "    \"\"\"Overlay `img_overlay` onto `img` at (x, y).\"\"\"\n",
+    "\n",
+    "    # Image ranges\n",
+    "    y1, y2 = max(0, y), min(img.shape[0], y + img_overlay.shape[0])\n",
+    "    x1, x2 = max(0, x), min(img.shape[1], x + img_overlay.shape[1])\n",
+    "\n",
+    "    # Overlay ranges\n",
+    "    y1o, y2o = max(0, -y), min(img_overlay.shape[0], img.shape[0] - y)\n",
+    "    x1o, x2o = max(0, -x), min(img_overlay.shape[1], img.shape[1] - x)\n",
+    "\n",
+    "    if y1 >= y2 or x1 >= x2 or y1o >= y2o or x1o >= x2o:\n",
+    "        return\n",
+    "\n",
+    "    img_crop = img[y1:y2, x1:x2]\n",
+    "    img_overlay_crop = img_overlay[y1o:y2o, x1o:x2o]\n",
+    "    img_crop[:] = img_overlay_crop\n",
+    "\n",
+    "\n",
+    "def overlay_text(img, x, y, text=\"hello!\"):\n",
+    "    text_background = np.zeros((110, 1600, 3), dtype=np.uint8)\n",
+    "    text_background[:] = (255, 255, 255)\n",
+    "\n",
+    "    overlay_image(img, text_background, 0, y-85)\n",
+    "\n",
+    "    text_position = (x, y)\n",
+    "    font = cv2.FONT_HERSHEY_SIMPLEX\n",
+    "    font_scale = 3\n",
+    "    color = (0, 0, 0)\n",
+    "    thickness = 4\n",
+    "    img = cv2.putText(img, text, text_position, font, font_scale, color, thickness)\n",
+    "    return img\n",
+    "\n",
+    "\n",
+    "def make_overlaid_video(recording, output_video_path, fps=30):\n",
+    "    video_writer = cv2.VideoWriter(\n",
+    "        str(output_video_path),\n",
+    "        cv2.VideoWriter_fourcc(*\"MJPG\"),\n",
+    "        fps,\n",
+    "        (recording.scene.width, recording.scene.height),\n",
+    "    )\n",
+    "\n",
+    "    output_timestamps = np.arange(\n",
+    "        world_ts.iloc[200] * 1e-9,\n",
+    "        world_ts.iloc[-100] * 1e-9,\n",
+    "        1 / 30,\n",
+    "    )\n",
+    "\n",
+    "    combined_data = zip(\n",
+    "        recording.scene.sample(output_timestamps),\n",
+    "        recording.eye.sample(output_timestamps),\n",
+    "        recording.gaze.sample(output_timestamps),\n",
+    "    )\n",
+    "\n",
+    "    frame_idx = 0\n",
+    "    for scene_frame, eye_frame, gaze_datum in combined_data:\n",
+    "        frame_idx += 1\n",
+    "        frame_pixels = scene_frame.bgr\n",
+    "        eye_pixels = cv2.cvtColor(eye_frame.gray, cv2.COLOR_GRAY2BGR)\n",
+    "        \n",
+    "        frame_pixels = cv2.circle(\n",
+    "            frame_pixels, (int(gaze_datum.x), int(gaze_datum.y)), 50, (0, 0, 255), 10\n",
+    "        )\n",
+    "\n",
+    "        overlay_image(frame_pixels, eye_pixels, 50, 50)\n",
+    "        \n",
+    "        if frame_idx >= gaze_left_start and frame_idx <= gaze_left_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Gaze left\")\n",
+    "        elif frame_idx >= head_left_start and frame_idx <= head_left_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Head left\")\n",
+    "        elif frame_idx >= gaze_head_left_start and frame_idx <= gaze_head_left_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Gaze and head left\")\n",
+    "        elif frame_idx >= gaze_right_start and frame_idx <= gaze_right_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Gaze right\")\n",
+    "        elif frame_idx >= head_right_start and frame_idx <= head_right_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Head right\")\n",
+    "        elif frame_idx >= gaze_head_right_start and frame_idx <= gaze_head_right_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Gaze and head right\")\n",
+    "        elif frame_idx >= gaze_up_start and frame_idx <= gaze_up_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Gaze up\")\n",
+    "        elif frame_idx >= head_up_start and frame_idx <= head_up_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Head up\")\n",
+    "        elif frame_idx >= gaze_head_up_start and frame_idx <= gaze_head_up_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Gaze and head up\")\n",
+    "        elif frame_idx >= gaze_down_start and frame_idx <= gaze_down_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Gaze down\")\n",
+    "        elif frame_idx >= head_down_start and frame_idx <= head_down_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Head down\")\n",
+    "        elif frame_idx >= gaze_head_down_start and frame_idx <= gaze_head_down_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Gaze and head down\")\n",
+    "        elif frame_idx >= free_viewing_start and frame_idx <= free_viewing_end:\n",
+    "            frame_pixels = overlay_text(frame_pixels, 100, 1100, \"Free viewing\")\n",
+    "        \n",
+    "        video_writer.write(frame_pixels)\n",
+    "\n",
+    "    video_writer.release()\n",
+    "\n",
+    "\n",
+    "make_overlaid_video(recording, \"eye-gaze-overlay-output-video.avi\")"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".venv",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.8"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/alpha-lab/imu-transformations/index.md b/alpha-lab/imu-transformations/index.md
index 0d68833a6..0d158e347 100644
--- a/alpha-lab/imu-transformations/index.md
+++ b/alpha-lab/imu-transformations/index.md
@@ -28,6 +28,8 @@ import TagLinks from '@components/TagLinks.vue'
 
 This guide contains various transformation functions that help with relating [Neon's IMU data](https://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data) with other data streams.
 
+As you work through this guide, you may want to check out the [Application Example](#application-example) to see the code in action.
+
 ## Rotation between the IMU and the World
 
 The IMU data includes a description of how the IMU is rotated in relation to the world. Concretely, the IMU data contains quaternions that define a rotation transformation between the [the world coordinate system](http://docs.pupil-labs.com/neon/data-collection/data-streams/#movement-imu-data) and the IMU's local coordinate system at different points in time. 
@@ -248,7 +250,7 @@ def cartesian_to_spherical_world(world_points_3d):
 
 ## Application Example
 
-Below, we present a video showing how some of the functions in this article were used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://gist.github.com/rennis250/8a684ea1e2f92c79fa2104b7a0f30e20).
+Below, we present a video showing how some of the functions in this article were used to visualize different combinations of head and eye movements in world coordinates. The code for producing the visualization [can be found here](https://github.com/pupil-labs/pupil-docs/tree/master/alpha-lab/imu-transformations/imu_heading_visualization.ipynb).
 
 <Youtube src="n0SjQzTqu3U"/>