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(Span<const Vector2> contour) {
  
      auto n = contour.size();
      const Vector2 *c = contour.data();
  
      real_t A = 0.0f;
  
      for (int p = n - 1, q = 0; q < n; p = q++) {
          A += c[p].cross(c[q]);
      }
      return A * 0.5f;
  }
  
  /*
       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 > 0.0f) && (bCROSScp > 0.0f) && (cCROSSap > 0.0f));
      } else {
          return ((aCROSSbp >= 0.0f) && (bCROSScp >= 0.0f) && (cCROSSap >= 0.0f));
      }
  };
  
  bool Triangulate::snip(Span<const Vector2> p_contour, int u, int v, int w, int n, Span<const int> V, bool relaxed) {
      int p;
      real_t Ax, Ay, Bx, By, Cx, Cy, Px, Py;
      const Vector2 *contour = p_contour.data();
  
      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(Span<const Vector2> contour, Vector<int> &result) {
      /* allocate and initialize list of Vertices in polygon */
  
      int n = contour.size();
      if (n < 3)
          return false;
  
      FixedVector<int,16,true> V;
      V.resize(n);
  
      /* we want a counter-clockwise polygon in V */
  
      if (0.0f < get_area(contour))
          for (int v = 0; v < n; v++)
              V[v] = v;
      else
          for (int v = 0; v < n; v++)
              V[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[s] = V[t];
  
              nv--;
  
              /* reset error detection counter */
              count = 2 * nv;
          }
      }
  
      return true;
  }
  