// InsertVertex in the paper
static void insert_cavity_vertex_helper(MutableTriangleTopology& mesh, RawField<const Perturbed2,VertexId> X,
RawField<bool,VertexId> marked, const HalfedgeId vw) {
// If wv is a boundary edge, or we're already Delaunay and properly oriented, we're done
const auto wv = mesh.reverse(vw);
if (mesh.is_boundary(wv))
return;
const auto u = mesh.opposite(vw),
v = mesh.src(vw),
w = mesh.dst(vw),
x = mesh.opposite(wv);
const auto Xu = X[u],
Xv = X[v],
Xw = X[w],
Xx = X[x];
const bool in = incircle(Xu,Xv,Xw,Xx);
if (!in && triangle_oriented(Xu,Xv,Xw))
return;
// Flip edge and recurse
const auto xu = mesh.flip_edge(wv);
assert(mesh.vertices(xu)==vec(x,u));
const auto vx = mesh.prev(xu),
xw = mesh.next(mesh.reverse(xu)); // Grab this now before the recursive call changes uvx
insert_cavity_vertex_helper(mesh,X,marked,vx),
insert_cavity_vertex_helper(mesh,X,marked,xw);
if (!in)
marked[u] = marked[v] = marked[w] = marked[x] = true;
}
static void insert_cavity_vertex(MutableTriangleTopology& mesh, RawField<const Perturbed2,VertexId> X,
RawField<bool,VertexId> marked,
const VertexId u, const VertexId v, const VertexId w) {
#ifndef NDEBUG
{
const auto vw = mesh.halfedge(v);
assert(mesh.is_boundary(vw) && mesh.dst(vw)==w);
}
#endif
mesh.add_face(vec(u,v,w));
const auto vw = mesh.prev(mesh.reverse(mesh.halfedge(u)));
assert(mesh.vertices(vw)==vec(v,w));
insert_cavity_vertex_helper(mesh,X,marked,vw);
}
void decimate_inplace(MutableTriangleTopology& mesh, RawField<TV,VertexId> X,
const T distance, const T max_angle, const int min_vertices, const T boundary_distance) {
if (mesh.n_vertices() <= min_vertices)
return;
const T area = sqr(distance);
const T sign_sqr_min_cos = sign_sqr(max_angle > .99*pi ? -1 : cos(max_angle));
// Finds the best edge to collapse v along. Returns (q(e),dst(e)).
const auto best_collapse = [&mesh,X](const VertexId v) {
Quadric q = compute_quadric(mesh,X,v);
// Find the best edge, ignoring normal constraints
T min_q = inf;
HalfedgeId min_e;
for (const auto e : mesh.outgoing(v)) {
const T qx = q(X[mesh.dst(e)]);
if (min_q > qx) {
min_q = qx;
min_e = e;
}
}
return tuple(min_q,mesh.dst(min_e));
};
// Initialize quadrics and heap
Heap heap(mesh.n_vertices_);
for (const auto v : mesh.vertices()) {
const auto qe = best_collapse(v);
if (qe.x <= area)
heap.inv_heap[v] = heap.heap.append(tuple(v,qe.x,qe.y));
}
heap.make();
// Update the quadric information for a vertex
const auto update = [&heap,best_collapse,area](const VertexId v) {
const auto qe = best_collapse(v);
if (qe.x <= area)
heap.set(v,qe.x,qe.y);
else
heap.erase(v);
};
// Repeatedly collapse the best vertex
while (heap.size()) {
const auto v = heap.pop();
// Do these vertices still exist?
if (mesh.valid(v.x) && mesh.valid(v.y)) {
const auto e = mesh.halfedge(v.x,v.y);
// Is the collapse invalid?
if (e.valid() && mesh.is_collapse_safe(e)) {
const auto vs = mesh.src(e),
vd = mesh.dst(e);
const auto xs = X[vs],
xd = X[vd];
// Are we moving a boundary vertex too far from its two boundary lines?
{
const auto b = mesh.halfedge(vs);
if (mesh.is_boundary(b)) {
const auto x0 = X[mesh.dst(b)],
x1 = X[mesh.src(mesh.prev(b))];
if ( line_point_distance(simplex(xs,x0),xd) > boundary_distance
|| line_point_distance(simplex(xs,x1),xd) > boundary_distance)
goto bad;
}
}
// Do the normals change too much?
if (sign_sqr_min_cos > -1)
for (const auto ee : mesh.outgoing(vs))
if (e!=ee && !mesh.is_boundary(ee)) {
const auto v2 = mesh.opposite(ee);
if (v2 != vd) {
const auto x1 = X[mesh.dst(ee)],
x2 = X[v2];
const auto n0 = cross(x2-x1,xs-x1),
n1 = cross(x2-x1,xd-x1);
if (sign_sqr(dot(n0,n1)) < sign_sqr_min_cos*sqr_magnitude(n0)*sqr_magnitude(n1))
goto bad;
}
}
// Collapse vs onto vd, then update the heap
mesh.unsafe_collapse(e);
if (mesh.n_vertices() <= min_vertices)
break;
update(vd);
for (const auto e : mesh.outgoing(vd))
update(mesh.dst(e));
}
}
bad:;
}
}