Pick's Theorem Added

Author: alirezakhm

new_geometry/polygon.cpp

@@ -1,4 +1,8 @@
 // POLYGONS
+// Pick's Theorem
+// polygon without self-intersections & if all vertices have integer coordinates in 2D grid
+// Area = I + B/2 - 1. B = # of points lying on polygon sides == (sum of gcd(x_diff, y_diff) on edges)
+// I = number of points with integer coordinates lying strictly inside the polygon
 
 // Implementation
 // 3 points, entered in counter clockwise order, 0-based indexing
@@ -113,38 +117,28 @@ vdd convexHull(vdd P){
 
 
 //for finding the centroid of a polygon
-point compute2DPolygonCentroid(const std::vector<point> vertices) {
- point centroid;
- double signedArea = 0.0;
- double x0 = 0.0; // Current vertex X
- double y0 = 0.0; // Current vertex Y
- double x1 = 0.0; // Next vertex X
- double y1 = 0.0; // Next vertex Y
- double a = 0.0; // Partial signed area
- for (int i = 0; i < vertices.size() - 1; ++i) {
-
- x0 = vertices[i].x;
- y0 = vertices[i].y;
- x1 = vertices[i + 1].x;
- y1 = vertices[i + 1].y;
+dd compute2DPolygonCentroid(const vdd P) {
+ dd centroid;
+ double signedArea, x0, y0, x1, y1, a = 0.0;
+ for (int i = 0; i < P.size() - 1; ++i) {
+ x0, y0 = P[i].first, P[i].second;
+ x1, y1 = P[i + 1].first, P[i + 1].second;
  a = x0 * y1 - x1 * y0;
  signedArea += a;
- centroid.x += (x0 + x1) * a;
- centroid.y += (y0 + y1) * a;
-
+ centroid.first += (x0 + x1) * a;
+ centroid.second += (y0 + y1) * a;
  }
- x0 = vertices.back().x;
- y0 = vertices.back().y;
- x1 = vertices.front().x;
- y1 = vertices.front().y;
+ 
+ x0, y0 = P.back().first, P.back().second;
+ x1, y1 = P.front().first, P.front().second;
  a = x0 * y1 - x1 * y0;
  signedArea += a;
- centroid.x += (x0 + x1) * a;
- centroid.y += (y0 + y1) * a;
+ centroid.first += (x0 + x1) * a;
+ centroid.second += (y0 + y1) * a;
 
  signedArea *= 0.5;
- centroid.x /= (6.0 * signedArea);
- centroid.y /= (6.0 * signedArea);
+ centroid.first /= (6.0 * signedArea);
+ centroid.second /= (6.0 * signedArea);
 
  return centroid;
 }