// 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 inline FaceId bsp_search(RawArray<const Node> bsp, RawField<const Perturbed2,VertexId> X, const VertexId v) {
if (!bsp.size())
return FaceId(0);
int node = 0;
do {
const Node& n = bsp[node];
node = n.children[triangle_oriented(X,n.test.x,n.test.y,v)];
} while (node >= 0);
return FaceId(~node);
}
GEODE_ALWAYS_INLINE static inline bool triangle_oriented(const RawField<const Perturbed2,VertexId> X, VertexId v0, VertexId v1, VertexId v2) {
const auto x0 = X[v0],
x1 = X[v1],
x2 = X[v2];
// If we have a nonsentinel triangle, use a normal orientation test
if (maxabs(x0.value().x,x1.value().x,x2.value().x)!=bound)
return triangle_oriented(x0,x1,x2);
// Fall back to sentinel case analysis
return triangle_oriented_sentinels(x0,x1,x2);
}
GEODE_COLD static void assert_delaunay(const char* prefix,
const TriangleTopology& mesh, RawField<const Perturbed2,VertexId> X,
const Hashtable<Vector<VertexId,2>>& constrained=Tuple<>(),
const bool oriented_only=false,
const bool check_boundary=true) {
// Verify that all faces are correctly oriented
for (const auto f : mesh.faces()) {
const auto v = mesh.vertices(f);
GEODE_ASSERT(triangle_oriented(X,v.x,v.y,v.z));
}
if (oriented_only)
return;
// Verify that all internal edges are Delaunay
if (!constrained.size()) {
for (const auto e : mesh.interior_halfedges())
if (!mesh.is_boundary(mesh.reverse(e)) && mesh.src(e)<mesh.dst(e))
if (!is_delaunay(mesh,X,e))
throw RuntimeError(format("%snon delaunay edge: e%d, v%d v%d",prefix,e.id,mesh.src(e).id,mesh.dst(e).id));
} else {
for (const auto v : constrained)
if (!mesh.halfedge(v.x,v.y).valid())
throw RuntimeError(format("%smissing constraint edge: v%d v%d",prefix,v.x.id,v.y.id));
for (const auto e : mesh.interior_halfedges()) {
const auto s = mesh.src(e), d = mesh.dst(e);
if (!mesh.is_boundary(mesh.reverse(e)) && s<d && !constrained.contains(vec(s,d)))
if (!is_delaunay(mesh,X,e))
throw RuntimeError(format("%snon delaunay edge: e%d, v%d v%d",prefix,e.id,mesh.src(e).id,mesh.dst(e).id));
}
}
// Verify that all boundary vertices are convex
if (check_boundary)
for (const auto e : mesh.boundary_edges()) {
const auto v0 = mesh.src(mesh.prev(e)),
v1 = mesh.src(e),
v2 = mesh.dst(e);
GEODE_ASSERT(triangle_oriented(X,v2,v1,v0));
}
}
// Test whether an edge containing sentinels is Delaunay
GEODE_COLD GEODE_PURE static inline bool is_delaunay_sentinels(Perturbed2 x0, Perturbed2 x1, Perturbed2 x2, Perturbed2 x3) {
// Unfortunately, the sentinels need to be at infinity for purposes of Delaunay testing, and our SOS predicates
// don't support infinities. Therefore, we need some case analysis. First, we move all the sentinels to the end,
// sorted in decreasing order of index. Different sentinels will be placed at different orders of infinity,
// with earlier indices further away, so our order will place larger infinities last.
bool parity = 0;
COMPARATOR(0,1)
COMPARATOR(2,3)
COMPARATOR(0,2)
COMPARATOR(1,3)
COMPARATOR(1,2)
if (!is_sentinel(x2)) // One infinity: A finite circle contains infinity iff it is inside out, so we reduce to an orientation test
return parity^triangle_oriented(x0,x1,x2);
else if (!is_sentinel(x1)) // Two infinities: also an orientation test, but with the last point at infinity
return parity^segment_to_direction_oriented(x0,x1,x2);
else // Three infinities: the finite point no longer matters.
return parity^directions_oriented(x1,x2);
}
static void predicate_tests() {
IntervalScope scope;
typedef Vector<double,2> TV2;
typedef Vector<Quantized,2> QV2;
// Compare triangle_oriented and incircle against approximate floating point versions
struct F {
static inline double triangle_oriented(const TV2 p0, const TV2 p1, const TV2 p2) {
return edet(p1-p0,p2-p0);
};
static inline double incircle(const TV2 p0, const TV2 p1, const TV2 p2, const TV2 p3) {
const auto d0 = p0-p3, d1 = p1-p3, d2 = p2-p3;
return edet(ROW(d0),ROW(d1),ROW(d2));
}
};
const auto random = new_<Random>(9817241);
for (int step=0;step<100;step++) {
#define MAKE(i) \
const auto p##i = tuple(i,QV2(random->uniform<Vector<ExactInt,2>>(-exact::bound,exact::bound))); \
const TV2 x##i(p##i.y);
MAKE(0) MAKE(1) MAKE(2) MAKE(3)
GEODE_ASSERT(triangle_oriented(p0,p1,p2)==(F::triangle_oriented(x0,x1,x2)>0));
GEODE_ASSERT(incircle(p0,p1,p2,p3)==(F::incircle(x0,x1,x2,x3)>0));
}
// Test behavior for large numbers, using the scale invariance and antisymmetry of incircle.
for (const int i : range(exact::log_bound)) {
const auto bound = ExactInt(1)<<i;
const auto p0 = tuple(0,QV2(-bound,-bound)), // Four points on a circle of radius sqrt(2)*bound
p1 = tuple(1,QV2( bound,-bound)),
p2 = tuple(2,QV2( bound, bound)),
p3 = tuple(3,QV2(-bound, bound));
GEODE_ASSERT(!incircle(p0,p1,p2,p3));
GEODE_ASSERT( incircle(p0,p1,p3,p2));
}
}
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);
}
bool segments_intersect(const P2 a0, const P2 a1, const P2 b0, const P2 b1) {
return triangle_oriented(a0,a1,b0)!=triangle_oriented(a0,a1,b1)
&& triangle_oriented(b0,b1,a0)!=triangle_oriented(b0,b1,a1);
}
