bool segment_intersection_above(const P2 a0, const P2 a1, const P2 b0, const P2 b1, const P2 c) {
return perturbed_predicate<SegmentIntersectionAbove>(a0,a1,b0,b1,c)
^ segment_directions_oriented(a0,a1,b0,b1)
^ rightwards(a0,a1);
}
// 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]));
}
}
bool segment_intersections_ordered(const P2 a0, const P2 a1, P2 b0, P2 b1, P2 c0, P2 c1) {
return perturbed_predicate<SegmentIntersectionsOrdered>(a0,a1,b0,b1,c0,c1)
^ segment_directions_oriented(a0,a1,b0,b1)
^ segment_directions_oriented(a0,a1,c0,c1);
}
