Field<T,VertexHandle> mean_curvatures(const TriMesh& mesh) {
Field<T,VertexHandle> area(mesh.n_vertices()); // Actually 8*area
Field<TV,VertexHandle> normal(mesh.n_vertices());
Field<TV,VertexHandle> Hn(mesh.n_vertices()); // Actually 4*Hn
for (const auto f : mesh.face_handles()) {
const auto v = mesh.vertex_handles(f);
const TV x0 = mesh.point(v.x),
x1 = mesh.point(v.y),
x2 = mesh.point(v.z),
x01 = x1-x0,
x12 = x2-x1,
x20 = x0-x2;
const T cot0 = cot(x01,x20),
cot1 = cot(x01,x12),
cot2 = cot(x12,x20);
// Compute A_mixed as in Meyer et al.
const TV n = mesh.calc_face_normal(f);
if (cot0<=0 && cot1<=0 && cot2<=0) { // Voronoi case
const T area0 = cot0*sqr_magnitude(x12),
area1 = cot1*sqr_magnitude(x20),
area2 = cot2*sqr_magnitude(x01);
area[v.x] -= (area1+area2);
area[v.y] -= (area2+area0);
area[v.z] -= (area0+area1);
normal[v.x] -= (area1+area2)*n;
normal[v.y] -= (area2+area0)*n;
normal[v.z] -= (area0+area1)*n;
} else { // One of the triangles is obtuse
const T a = 2*mesh.area(f);
area[v.x] += (1+(cot0>0))*a;
area[v.y] += (1+(cot1>0))*a;
area[v.z] += (1+(cot2>0))*a;
normal[v.x] += (1+(cot0>0))*a*n;
normal[v.y] += (1+(cot1>0))*a*n;
normal[v.z] += (1+(cot2>0))*a*n;
}
// Accumulate mean curvature normal
Hn[v.x] += cot2*x01-cot1*x20;
Hn[v.y] += cot0*x12-cot2*x01;
Hn[v.z] += cot1*x20-cot0*x12;
}
Field<T,VertexHandle> H(mesh.n_vertices(),uninit);
for (const auto v : mesh.vertex_handles()) {
const TV n = normal[v];
H[v] = 2/area[v] * (mesh.is_boundary(v) ? dot(Hn[v],normalized(n))
: copysign(magnitude(Hn[v]),dot(Hn[v],n)));
}
return H;
}
Field<T,VertexHandle> gaussian_curvatures(const TriMesh& mesh) {
// Compute mixed areas
Field<T,VertexHandle> area(mesh.n_vertices()); // Actually 8*area
for (const auto f : mesh.face_handles()) {
const auto v = mesh.vertex_handles(f);
const TV x0 = mesh.point(v.x),
x1 = mesh.point(v.y),
x2 = mesh.point(v.z),
x01 = x1-x0,
x12 = x2-x1,
x20 = x0-x2;
const T cot0 = cot(x01,x20),
cot1 = cot(x01,x12),
cot2 = cot(x12,x20);
// Compute A_mixed as in Meyer et al.
if (cot0<=0 && cot1<=0 && cot2<=0) { // Voronoi case
const T area0 = cot0*sqr_magnitude(x12),
area1 = cot1*sqr_magnitude(x20),
area2 = cot2*sqr_magnitude(x01);
area[v.x] -= (area1+area2);
area[v.y] -= (area2+area0);
area[v.z] -= (area0+area1);
} else { // One of the triangles is obtuse
const T a = 2*mesh.area(f);
area[v.x] += (1+(cot0>0))*a;
area[v.y] += (1+(cot1>0))*a;
area[v.z] += (1+(cot2>0))*a;
}
}
// Compute curvatures
Field<T,VertexHandle> K(mesh.n_vertices(),uninit);
for (const auto v : mesh.vertex_handles()) {
const TV x = mesh.point(v);
T sum = 0;
for (auto e=mesh.cvoh_iter(v);e;++e)
if (!mesh.is_boundary(e)) {
const auto v0 = mesh.to_vertex_handle(e),
v1 = mesh.from_vertex_handle(mesh.prev_halfedge_handle(e));
sum += angle_between(mesh.point(v0)-x,mesh.point(v1)-x);
}
K[v] = 8*((1+!mesh.is_boundary(v))*pi-sum)/area(v);
}
return K;
}
