// from http://www.geometry.caltech.edu/pubs/DMSB_III.pdf
public static double VoronoiArea(DMesh3 mesh, int v_i)
{
double areaSum = 0;
Vector3d Vi = mesh.GetVertex(v_i);
foreach (int tid in mesh.VtxTrianglesItr(v_i))
{
Index3i t = mesh.GetTriangle(tid);
int ti = (t[0] == v_i) ? 0 : ((t[1] == v_i) ? 1 : 2);
Vector3d Vj = mesh.GetVertex(t[(ti + 1) % 3]);
Vector3d Vk = mesh.GetVertex(t[(ti + 2) % 3]);
if (MathUtil.IsObtuse(Vi, Vj, Vk))
{
Vector3d Vij = Vj - Vi;
Vector3d Vik = Vk - Vi;
Vij.Normalize(); Vik.Normalize();
double areaT = 0.5 * Vij.Cross(Vik).Length;
if (Vector3d.AngleR(Vij, Vik) > MathUtil.HalfPI)
{
areaSum += (areaT * 0.5); // obtuse at v_i
}
else
{
areaSum += (areaT * 0.25); // not obtuse
}
}
else
{
// voronoi area
Vector3d Vji = Vi - Vj;
double dist_ji = Vji.Normalize();
Vector3d Vki = Vi - Vk;
double dist_ki = Vki.Normalize();
Vector3d Vkj = (Vj - Vk).Normalized;
double cot_alpha_ij = MathUtil.VectorCot(Vki, Vkj);
double cot_alpha_ik = MathUtil.VectorCot(Vji, -Vkj);
areaSum += dist_ji * dist_ji * cot_alpha_ij * 0.125;
areaSum += dist_ki * dist_ki * cot_alpha_ik * 0.125;
}
}
return(areaSum);
}