text: /*************************************************************************/
  /*  triangulate.cpp                                                      */
  /*************************************************************************/
  /*                       This file is part of:                           */
  /*                           GODOT ENGINE                                */
  /*                      https://godotengine.org                          */
  /*************************************************************************/
  /* Copyright (c) 2007-2019 Juan Linietsky, Ariel Manzur.                 */
  /* Copyright (c) 2014-2019 Godot Engine contributors (cf. AUTHORS.md)    */
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  #include "triangulate.h"
  
  real_t Triangulate::get_area(const Vector<Vector2> &contour) {
  
  	int n = contour.size();
  	const Vector2 *c = &contour[0];
  
  	real_t A = 0.0;
  
  	for (int p = n - 1, q = 0; q < n; p = q++) {
  		A += c[p].cross(c[q]);
  	}
  	return A * 0.5;
  }
  
  /*
       is_inside_triangle decides if a point P is Inside of the triangle
       defined by A, B, C.
     */
  
  bool Triangulate::is_inside_triangle(real_t Ax, real_t Ay,
  		real_t Bx, real_t By,
  		real_t Cx, real_t Cy,
  		real_t Px, real_t Py,
  		bool include_edges)
  
  {
  	real_t ax, ay, bx, by, cx, cy, apx, apy, bpx, bpy, cpx, cpy;
  	real_t cCROSSap, bCROSScp, aCROSSbp;
  
  	ax = Cx - Bx;
  	ay = Cy - By;
  	bx = Ax - Cx;
  	by = Ay - Cy;
  	cx = Bx - Ax;
  	cy = By - Ay;
  	apx = Px - Ax;
  	apy = Py - Ay;
  	bpx = Px - Bx;
  	bpy = Py - By;
  	cpx = Px - Cx;
  	cpy = Py - Cy;
  
  	aCROSSbp = ax * bpy - ay * bpx;
  	cCROSSap = cx * apy - cy * apx;
  	bCROSScp = bx * cpy - by * cpx;
  
  	if (include_edges) {
  		return ((aCROSSbp > 0.0) && (bCROSScp > 0.0) && (cCROSSap > 0.0));
  	} else {
  		return ((aCROSSbp >= 0.0) && (bCROSScp >= 0.0) && (cCROSSap >= 0.0));
  	}
  };
  
  bool Triangulate::snip(const Vector<Vector2> &p_contour, int u, int v, int w, int n, const Vector<int> &V, bool relaxed) {
  	int p;
  	real_t Ax, Ay, Bx, By, Cx, Cy, Px, Py;
  	const Vector2 *contour = &p_contour[0];
  
  	Ax = contour[V[u]].x;
  	Ay = contour[V[u]].y;
  
  	Bx = contour[V[v]].x;
  	By = contour[V[v]].y;
  
  	Cx = contour[V[w]].x;
  	Cy = contour[V[w]].y;
  
  	// It can happen that the triangulation ends up with three aligned vertices to deal with.
  	// In this scenario, making the check below strict may reject the possibility of
  	// forming a last triangle with these aligned vertices, preventing the triangulatiom
  	// from completing.
  	// To avoid that we allow zero-area triangles if all else failed.
  	float threshold = relaxed ? -CMP_EPSILON : CMP_EPSILON;
  
  	if (threshold > (((Bx - Ax) * (Cy - Ay)) - ((By - Ay) * (Cx - Ax)))) return false;
  
  	for (p = 0; p < n; p++) {
  		if ((p == u) || (p == v) || (p == w)) continue;
  		Px = contour[V[p]].x;
  		Py = contour[V[p]].y;
  		if (is_inside_triangle(Ax, Ay, Bx, By, Cx, Cy, Px, Py, relaxed)) return false;
  	}
  
  	return true;
  }
  
  bool Triangulate::triangulate(const Vector<Vector2> &contour, Vector<int> &result) {
  	/* allocate and initialize list of Vertices in polygon */
  
  	int n = contour.size();
  	if (n < 3) return false;
  
  	Vector<int> V;
  	V.resize(n);
  
  	/* we want a counter-clockwise polygon in V */
  
  	if (0.0 < get_area(contour))
  		for (int v = 0; v < n; v++)
  			V.write[v] = v;
  	else
  		for (int v = 0; v < n; v++)
  			V.write[v] = (n - 1) - v;
  
  	bool relaxed = false;
  
  	int nv = n;
  
  	/*  remove nv-2 Vertices, creating 1 triangle every time */
  	int count = 2 * nv; /* error detection */
  
  	for (int v = nv - 1; nv > 2;) {
  		/* if we loop, it is probably a non-simple polygon */
  		if (0 >= (count--)) {
  			if (relaxed) {
  				//** Triangulate: ERROR - probable bad polygon!
  				return false;
  			} else {
  				// There may be aligned vertices that the strict
  				// checks prevent from triangulating. In this situation
  				// we are better off adding flat triangles than
  				// failing, so we relax the checks and try one last
  				// round.
  				// Only relaxing the constraints as a last resort avoids
  				// degenerate triangles when they aren't necessary.
  				count = 2 * nv;
  				relaxed = true;
  			}
  		}
  
  		/* three consecutive vertices in current polygon, <u,v,w> */
  		int u = v;
  		if (nv <= u) u = 0; /* previous */
  		v = u + 1;
  		if (nv <= v) v = 0; /* new v    */
  		int w = v + 1;
  		if (nv <= w) w = 0; /* next     */
  
  		if (snip(contour, u, v, w, nv, V, relaxed)) {
  			int a, b, c, s, t;
  
  			/* true names of the vertices */
  			a = V[u];
  			b = V[v];
  			c = V[w];
  
  			/* output Triangle */
  			result.push_back(a);
  			result.push_back(b);
  			result.push_back(c);
  
  			/* remove v from remaining polygon */
  			for (s = v, t = v + 1; t < nv; s++, t++)
  				V.write[s] = V[t];
  
  			nv--;
  
  			/* reset error detection counter */
  			count = 2 * nv;
  		}
  	}
  
  	return true;
  }
  