// Compute cotan-weighted neighbour sum around a vertex.
// These weights are numerically unstable if any of the triangles are degenerate.
// We catch these problems and return input vertex as centroid
// http://www.geometry.caltech.edu/pubs/DMSB_III.pdf
public static Vector3d CotanCentroid(DMesh3 mesh, int v_i)
{
Vector3d vSum = Vector3d.Zero;
double wSum = 0;
Vector3d Vi = mesh.GetVertex(v_i);
int v_j = DMesh3.InvalidID, opp_v1 = DMesh3.InvalidID, opp_v2 = DMesh3.InvalidID;
int t1 = DMesh3.InvalidID, t2 = DMesh3.InvalidID;
bool bAborted = false;
foreach (int eid in mesh.GetVtxEdges(v_i))
{
opp_v2 = DMesh3.InvalidID;
mesh.GetVtxNbrhood(eid, v_i, ref v_j, ref opp_v1, ref opp_v2, ref t1, ref t2);
Vector3d Vj = mesh.GetVertex(v_j);
Vector3d Vo1 = mesh.GetVertex(opp_v1);
double cot_alpha_ij = MathUtil.VectorCot(
(Vi - Vo1).Normalized, (Vj - Vo1).Normalized);
if (cot_alpha_ij == 0)
{
bAborted = true;
break;
}
double w_ij = cot_alpha_ij;
if (opp_v2 != DMesh3.InvalidID)
{
Vector3d Vo2 = mesh.GetVertex(opp_v2);
double cot_beta_ij = MathUtil.VectorCot(
(Vi - Vo2).Normalized, (Vj - Vo2).Normalized);
if (cot_beta_ij == 0)
{
bAborted = true;
break;
}
w_ij += cot_beta_ij;
}
vSum += w_ij * Vj;
wSum += w_ij;
}
if (bAborted || Math.Abs(wSum) < MathUtil.ZeroTolerance)
{
return(Vi);
}
return(vSum / wSum);
}
// http://128.148.32.110/courses/cs224/papers/mean_value.pdf
public static Vector3d MeanValueCentroid(DMesh3 mesh, int v_i)
{
Vector3d vSum = Vector3d.Zero;
double wSum = 0;
Vector3d Vi = mesh.GetVertex(v_i);
int v_j = DMesh3.InvalidID, opp_v1 = DMesh3.InvalidID, opp_v2 = DMesh3.InvalidID;
int t1 = DMesh3.InvalidID, t2 = DMesh3.InvalidID;
foreach (int eid in mesh.GetVtxEdges(v_i))
{
opp_v2 = DMesh3.InvalidID;
mesh.GetVtxNbrhood(eid, v_i, ref v_j, ref opp_v1, ref opp_v2, ref t1, ref t2);
Vector3d Vj = mesh.GetVertex(v_j);
Vector3d vVj = (Vj - Vi);
double len_vVj = vVj.Normalize();
// [RMS] is this the right thing to do? if vertices are coincident,
// weight of this vertex should be very high!
if (len_vVj < MathUtil.ZeroTolerance)
{
continue;
}
Vector3d vVdelta = (mesh.GetVertex(opp_v1) - Vi).Normalized;
double w_ij = VectorTanHalfAngle(vVj, vVdelta);
if (opp_v2 != DMesh3.InvalidID)
{
Vector3d vVgamma = (mesh.GetVertex(opp_v2) - Vi).Normalized;
w_ij += VectorTanHalfAngle(vVj, vVgamma);
}
w_ij /= len_vVj;
vSum += w_ij * Vj;
wSum += w_ij;
}
return(vSum / wSum);
}