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);
}
GEODE_NEVER_INLINE static void add_constraint_edges(MutableTriangleTopology& mesh, RawField<const EV,VertexId> X,
RawArray<const Vector<int,2>> edges, const bool validate) {
if (!edges.size())
return;
IntervalScope scope;
Hashtable<Vector<VertexId,2>> constrained;
Array<VertexId> left_cavity, right_cavity; // List of vertices for both cavities
const auto random = new_<Random>(key+7);
for (int i=0;i<edges.size();i++) {
// Randomly choose an edge to ensure optimal time complexity
const auto edge = edges[int(random_permute(edges.size(),key+5,i))].sorted();
auto v0 = VertexId(edge.x),
v1 = VertexId(edge.y);
const auto vs = vec(v0,v1);
GEODE_ASSERT(mesh.valid(v0) && mesh.valid(v1));
{
// Check if the edge already exists in the triangulation. To ensure optimal complexity,
// we loop around both vertices interleaved so that our time is O(min(degree(v0),degree(v1))).
const auto s0 = mesh.halfedge(v0),
s1 = mesh.halfedge(v1);
{
auto e0 = s0,
e1 = s1;
do {
if (mesh.dst(e0)==v1 || mesh.dst(e1)==v0)
goto success; // The edge already exists, so there's nothing to be done.
e0 = mesh.left(e0);
e1 = mesh.left(e1);
} while (e0!=s0 && e1!=s1);
}
// Find a triangle touching v0 or v1 containing part of the v0-v1 segment.
// As above, we loop around both vertices interleaved.
auto e0 = s0;
{
auto e1 = s1;
if (mesh.is_boundary(e0)) e0 = mesh.left(e0);
if (mesh.is_boundary(e1)) e1 = mesh.left(e1);
const auto x0 = Perturbed2(v0.id,X[v0]),
x1 = Perturbed2(v1.id,X[v1]);
const auto e0d = mesh.dst(e0),
e1d = mesh.dst(e1);
bool e0o = triangle_oriented(x0,Perturbed2(e0d.id,X[e0d]),x1),
e1o = triangle_oriented(x1,Perturbed2(e1d.id,X[e1d]),x0);
for (;;) { // No need to check for an end condition, since we're guaranteed to terminate
const auto n0 = mesh.left(e0),
n1 = mesh.left(e1);
const auto n0d = mesh.dst(n0),
n1d = mesh.dst(n1);
const bool n0o = triangle_oriented(x0,Perturbed2(n0d.id,X[n0d]),x1),
n1o = triangle_oriented(x1,Perturbed2(n1d.id,X[n1d]),x0);
if (e0o && !n0o)
break;
if (e1o && !n1o) {
// Swap v0 with v1 and e0 with e1 so that our ray starts at v0
swap(v0,v1);
swap(e0,e1);
break;
}
e0 = n0;
e1 = n1;
e0o = n0o;
e1o = n1o;
}
}
// If we only need to walk one step, the retriangulation is a single edge flip
auto cut = mesh.reverse(mesh.next(e0));
if (mesh.dst(mesh.next(cut))==v1) {
if (constrained.contains(vec(mesh.src(cut),mesh.dst(cut)).sorted()))
throw DelaunayConstraintConflict(vec(v0.id,v1.id),vec(mesh.src(cut).id,mesh.dst(cut).id));
cut = mesh.flip_edge(cut);
goto success;
}
// Walk from v0 to v1, collecting the two cavities.
const auto x0 = Perturbed2(v0.id,X[v0]),
x1 = Perturbed2(v1.id,X[v1]);
right_cavity.copy(vec(v0,mesh.dst(cut)));
left_cavity .copy(vec(v0,mesh.src(cut)));
mesh.erase(mesh.face(e0));
for (;;) {
if (constrained.contains(vec(mesh.src(cut),mesh.dst(cut)).sorted()))
throw DelaunayConstraintConflict(vec(v0.id,v1.id),vec(mesh.src(cut).id,mesh.dst(cut).id));
const auto n = mesh.reverse(mesh.next(cut)),
p = mesh.reverse(mesh.prev(cut));
const auto v = mesh.src(n);
mesh.erase(mesh.face(cut));
if (v == v1) {
left_cavity.append(v);
right_cavity.append(v);
break;
} else if (triangle_oriented(x0,x1,Perturbed2(v.id,X[v]))) {
left_cavity.append(v);
cut = n;
} else {
right_cavity.append(v);
cut = p;
}
}
// Retriangulate both cavities
left_cavity.reverse();
cavity_delaunay(mesh,X,left_cavity,random),
cavity_delaunay(mesh,X,right_cavity,random);
}
success:
constrained.set(vs);
}
// If desired, check that the final mesh is constrained Delaunay
if (validate)
assert_delaunay("constrained delaunay validate: ",mesh,X,constrained);
}
// Retriangulate a cavity formed when a constraint edge is inserted, following Shewchuck and Brown.
// The cavity is defined by a counterclockwise list of vertices v[0] to v[m-1] as in Shewchuck and Brown, Figure 5.
static void cavity_delaunay(MutableTriangleTopology& parent_mesh, RawField<const EV,VertexId> X,
RawArray<const VertexId> cavity, Random& random) {
// Since the algorithm generates meshes which may be inconsistent with the outer mesh, and the cavity
// array may have duplicate vertices, we use a temporary mesh and then copy the triangles over when done.
// In the temporary, vertices are indexed consecutively from 0 to m-1.
const int m = cavity.size();
assert(m >= 3);
const auto mesh = new_<MutableTriangleTopology>();
Field<Perturbed2,VertexId> Xc(m,uninit);
for (const int i : range(m))
Xc.flat[i] = Perturbed2(cavity[i].id,X[cavity[i]]);
mesh->add_vertices(m);
const auto xs = Xc.flat[0],
xe = Xc.flat[m-1];
// Set up data structures for prev, next, pi in the paper
const Field<VertexId,VertexId> prev(m,uninit),
next(m,uninit);
for (int i=0;i<m-1;i++) {
next.flat[i] = VertexId(i+1);
prev.flat[i+1] = VertexId(i);
}
const Array<VertexId> pi_(m-2,uninit);
for (int i=1;i<=m-2;i++)
pi_[i-1] = VertexId(i);
#define PI(i) pi_[(i)-1]
// Randomly shuffle [1,m-2], subject to vertices closer to xs-xe than both their neighbors occurring later
for (int i=m-2;i>=2;i--) {
int j;
for (;;) {
j = random.uniform<int>(0,i)+1;
const auto pj = PI(j);
if (!( segment_directions_oriented(xe,xs,Xc[pj],Xc[prev[pj]])
&& segment_directions_oriented(xe,xs,Xc[pj],Xc[next[pj]])))
break;
}
swap(PI(i),PI(j));
// Remove PI(i) from the list
const auto pi = PI(i);
next[prev[pi]] = next[pi];
prev[next[pi]] = prev[pi];
}
// Add the first triangle
mesh->add_face(vec(VertexId(0),PI(1),VertexId(m-1)));
// Add remaining triangles, flipping to ensure Delaunay
const Field<bool,VertexId> marked(m);
Array<HalfedgeId> fan;
bool used_chew = false;
for (int i=2;i<m-1;i++) {
const auto pi = PI(i);
insert_cavity_vertex(mesh,Xc,marked,pi,next[pi],prev[pi]);
if (marked[pi]) {
used_chew = true;
marked[pi] = false;
// Retriangulate the fans of triangles that have all three vertices marked
auto e = mesh->reverse(mesh->halfedge(pi));
auto v = mesh->src(e);
bool mv = marked[v];
marked[v] = false;
fan.clear();
do {
const auto h = mesh->prev(e);
e = mesh->reverse(mesh->next(e));
v = mesh->src(e);
const bool mv2 = marked[v];
marked[v] = false;
if (mv) {
if (mv2)
fan.append(h);
if (!mv2 || mesh->is_boundary(e)) {
chew_fan(mesh,Xc,pi,fan,random);
fan.clear();
}
}
mv = mv2;
} while (!mesh->is_boundary(e));
}
}
#undef PI
// If we ran Chew's algorithm, validate the output. I haven't tested this
// case enough to be confident of its correctness.
if (used_chew)
assert_delaunay("Failure in extreme special case. If this triggers, please email [email protected]: ",
mesh,Xc,Tuple<>(),false,false);
// Copy triangles from temporary mesh to real mesh
for (const auto f : mesh->faces()) {
const auto v = mesh->vertices(f);
parent_mesh.add_face(vec(cavity[v.x.id],cavity[v.y.id],cavity[v.z.id]));
}
}
// Delaunay retriangulate a triangle fan
static void chew_fan(MutableTriangleTopology& parent_mesh, RawField<const Perturbed2,VertexId> X,
const VertexId u, RawArray<HalfedgeId> fan, Random& random) {
chew_fan_count_ += 1;
#ifndef NDEBUG
for (const auto e : fan)
assert(parent_mesh.opposite(e)==u);
for (int i=0;i<fan.size()-1;i++)
GEODE_ASSERT(parent_mesh.src(fan[i])==parent_mesh.dst(fan[i+1]));
#endif
const int n = fan.size();
if (n < 2)
return;
chew_fan_count_ += 1024*n;
// Collect vertices
const Field<VertexId,VertexId> vertices(n+2,uninit);
vertices.flat[0] = u;
vertices.flat[1] = parent_mesh.src(fan[n-1]);
for (int i=0;i<n;i++)
vertices.flat[i+2] = parent_mesh.dst(fan[n-1-i]);
// Delete original vertices
for (const auto e : fan)
parent_mesh.erase(parent_mesh.face(e));
// Make the vertices into a doubly linked list
const Field<VertexId,VertexId> prev(n+2,uninit),
next(n+2,uninit);
prev.flat[0].id = n+1;
next.flat[n+1].id = 0;
for (int i=0;i<n+1;i++) {
prev.flat[i+1].id = i;
next.flat[i].id = i+1;
}
// Randomly shuffle the vertices, then pulling elements off the linked list in reverse order of our final shuffle.
const Array<VertexId> pi(n+2,uninit);
for (int i=0;i<n+2;i++)
pi[i].id = i;
random.shuffle(pi);
for (int i=n+1;i>=0;i--) {
const auto j = pi[i];
prev[next[j]] = prev[j];
next[prev[j]] = next[j];
}
// Make a new singleton mesh
const auto mesh = new_<MutableTriangleTopology>();
mesh->add_vertices(n+2);
small_sort(pi[0],pi[1],pi[2]);
mesh->add_face(vec(pi[0],pi[1],pi[2]));
// Insert remaining vertices
Array<HalfedgeId> work;
for (int i=3;i<n+2;i++) {
const auto j = pi[i];
const auto f = mesh->add_face(vec(j,next[j],prev[j]));
work.append(mesh->reverse(mesh->opposite(f,j)));
while (work.size()) {
auto e = work.pop();
if ( !mesh->is_boundary(e)
&& incircle(X[vertices[mesh->src(e)]],
X[vertices[mesh->dst(e)]],
X[vertices[mesh->opposite(e)]],
X[vertices[mesh->opposite(mesh->reverse(e))]])) {
work.append(mesh->reverse(mesh->next(e)));
work.append(mesh->reverse(mesh->prev(e)));
e = mesh->unsafe_flip_edge(e);
}
}
}
// Copy triangles back to parent
for (const auto f : mesh->faces()) {
const auto vs = mesh->vertices(f);
parent_mesh.add_face(vec(vertices[vs.x],vertices[vs.y],vertices[vs.z]));
}
}
// 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;
}
// positions are assumed to be at default location
Ref<MutableTriangleTopology> improve_mesh(MutableTriangleTopology const &mesh, real min_quality, real max_distance, real max_silhouette_distance, real min_normal_dot, int max_iter, real min_relevant_area, real min_quality_improvement) {
FieldId<Vector<real,3>,VertexId> posid(vertex_position_id);
Ref<MutableTriangleTopology> copy = mesh.copy();
improve_mesh_inplace(copy, copy->field(posid), ImproveOptions(min_quality, max_distance, max_silhouette_distance, min_normal_dot, max_iter, min_relevant_area, min_quality_improvement));
return copy;
}
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:;
}
}