/// <summary>
/// Returns the entries of the cotangent-weighted Laplacian matrix in row-major order.
/// Based on symmetric derivation of the Laplace-Beltrami operator detailed in http://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Vallet08.pdf.
/// Also returns the barycentric dual area of each vertex.
/// Assumes triangle mesh.
/// </summary>
/// <param name="mesh"></param>
/// <param name="entriesOut"></param>
/// <param name="areasOut"></param>
public static void GetLaplacianMatrix(this Mesh mesh, double[] entriesOut, double[] areasOut)
{
var verts = mesh.Vertices;
int n = verts.Count;
double t = 1.0 / 6.0;
Array.Clear(entriesOut, 0, n * n);
Array.Clear(areasOut, 0, n);
// iterate faces to collect weights and vertex areas (upper triangular only)
foreach (MeshFace mf in mesh.Faces)
{
// circulate verts in face
for (int i = 0; i < 3; i++)
{
int i0 = mf[i];
int i1 = mf[(i + 1) % 3];
int i2 = mf[(i + 2) % 3];
Vec3d v0 = verts[i0] - verts[i2];
Vec3d v1 = verts[i1] - verts[i2];
// add to vertex area
double a = Vec3d.CrossProduct(v0, v1).Length;
areasOut[i0] += a * t;
// add to edge cotangent weights (assumes consistent face orientation)
if (i1 < i0)
{
entriesOut[i0 * n + i1] += 0.5 * v0 * v1 / a;
}
else
{
entriesOut[i1 * n + i0] += 0.5 * v0 * v1 / a;
}
}
}
// normalize weights with areas and sum along diagonals
for (int i = 0; i < n; i++)
{
int ii = i * n + i;
for (int j = i + 1; j < n; j++)
{
double w = entriesOut[i * n + j];
w /= Math.Sqrt(areasOut[i] * areasOut[j]);
// set symmetric entries
entriesOut[i * n + j] = entriesOut[j * n + i] = w;
// sum along diagonal entries
entriesOut[j * n + j] -= w;
entriesOut[ii] -= w;
}
}
}
