✨ adds algorithm to find lattice points of a polygon

algorithms/geometry/polygon-lattice-points.cpp

@@ -0,0 +1,33 @@
+ll cross(ll x1, ll y1, ll x2, ll y2) {
+  return x1 * y2 - x2 * y1;
+}
+
+ll polygonArea(vector<pll>& pts) {
+  ll ats = 0;
+  for (int i = 2; i < len(pts); i++)
+    ats += cross(pts[i].first - pts[0].first,
+                 pts[i].second - pts[0].second,
+                 pts[i - 1].first - pts[0].first,
+                 pts[i - 1].second - pts[0].second);
+  return abs(ats / 2ll);
+}
+
+ll boundary(vector<pll>& pts) {
+  ll ats = pts.size();
+  for (int i = 0; i < len(pts); i++) {
+    ll deltax =
+      (pts[i].first - pts[(i + 1) % pts.size()].first);
+    ll deltay =
+      (pts[i].second - pts[(i + 1) % pts.size()].second);
+    ats += abs(__gcd(deltax, deltay)) - 1;
+  }
+  return ats;
+}
+
+pll latticePoints(vector<pll>& pts) {
+  ll bounds = boundary(pts);
+  ll area = polygonArea(pts);
+  ll inside = area + 1ll - bounds / 2ll;
+
+  return {inside, bounds};
+}

algorithms/geometry/polygon-lattice-points.tex

@@ -0,0 +1,5 @@
+\subsection{Polygon Lattice Points (Pick's Theorem)}
+
+Given a polygon with $N$ points finds the number of lattice points inside and on boundaries.
+Time : $O(N)$
+