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triangulate.cpp
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738 lines (641 loc) · 24.3 KB
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//-----------------------------------------------------------------------------
// Triangulate a surface. If the surface is curved, then we first superimpose
// a grid of quads, with spacing to achieve our chord tolerance. We then
// proceed by ear-clipping; the resulting mesh should be watertight and not
// awful numerically, but has no special properties (Delaunay, etc.).
//
// Copyright 2008-2013 Jonathan Westhues.
//-----------------------------------------------------------------------------
#include "solvespace.h"
namespace SolveSpace {
void SPolygon::UvTriangulateInto(SMesh *m, SSurface *srf) {
if(l.n <= 0) return;
//int64_t in = GetMilliseconds();
normal = Vector::From(0, 0, 1);
while(l.n > 0) {
FixContourDirections();
l.ClearTags();
// Find a top-level contour, and start with that. Then build bridges
// in order to merge all its islands into a single contour.
SContour *top;
for(top = l.First(); top; top = l.NextAfter(top)) {
if(top->timesEnclosed == 0) {
break;
}
}
if(!top) {
dbp("polygon has no top-level contours?");
return;
}
// Start with the outer contour
SContour merged = {};
top->tag = 1;
top->CopyInto(&merged);
merged.l.RemoveLast(1);
// List all of the edges, for testing whether bridges work.
SEdgeList el = {};
top->MakeEdgesInto(&el);
List<Vector> vl = {};
// And now find all of its holes. Note that we will also find any
// outer contours that lie entirely within this contour, and any
// holes for those contours. But that's okay, because we can merge
// those too.
SContour *sc;
for(sc = l.First(); sc; sc = l.NextAfter(sc)) {
if(sc->timesEnclosed != 1) continue;
if(sc->l.n < 2) continue;
// Test the midpoint of an edge. Our polygon may not be self-
// intersecting, but two contours may share a vertex; so a
// vertex could be on the edge of another polygon, in which
// case ContainsPointProjdToNormal returns indeterminate.
Vector tp = sc->AnyEdgeMidpoint();
if(top->ContainsPointProjdToNormal(normal, tp)) {
sc->tag = 2;
sc->MakeEdgesInto(&el);
sc->FindPointWithMinX();
}
}
// dbp("finished finding holes: %d ms", (int)(GetMilliseconds() - in));
for(;;) {
double xmin = 1e10;
SContour *scmin = NULL;
for(sc = l.First(); sc; sc = l.NextAfter(sc)) {
if(sc->tag != 2) continue;
if(sc->xminPt.x < xmin) {
xmin = sc->xminPt.x;
scmin = sc;
}
}
if(!scmin) break;
if(!merged.BridgeToContour(scmin, &el, &vl)) {
dbp("couldn't merge our hole");
return;
}
// dbp(" bridged to contour: %d ms", (int)(GetMilliseconds() - in));
scmin->tag = 3;
}
// dbp("finished merging holes: %d ms", (int)(GetMilliseconds() - in));
merged.UvTriangulateInto(m, srf);
// dbp("finished ear clippping: %d ms", (int)(GetMilliseconds() - in));
merged.l.Clear();
el.Clear();
vl.Clear();
// Careful, need to free the points within the contours, and not just
// the contours themselves. This was a tricky memory leak.
for(sc = l.First(); sc; sc = l.NextAfter(sc)) {
if(sc->tag) {
sc->l.Clear();
}
}
l.RemoveTagged();
}
}
bool SContour::BridgeToContour(SContour *sc,
SEdgeList *avoidEdges, List<Vector> *avoidPts)
{
int i, j;
bool withbridge = true;
// Start looking for a bridge on our new hole near its leftmost (min x)
// point.
int sco = 0;
10BC0
for(i = 0; i < (sc->l.n - 1); i++) {
if((sc->l[i].p).EqualsExactly(sc->xminPt)) {
sco = i;
}
}
// And start looking on our merged contour at whichever point is nearest
// to the leftmost point of the new segment.
int thiso = 0;
double dmin = 1e10;
for(i = 0; i < l.n-1; i++) {
Vector p = l[i].p;
double d = (p.Minus(sc->xminPt)).MagSquared();
if(d < dmin) {
dmin = d;
thiso = i;
}
}
int thisp, scp;
Vector a, b, *f;
// First check if the contours share a point; in that case we should
// merge them there, without a bridge.
for(i = 0; i < l.n; i++) {
thisp = WRAP(i+thiso, l.n-1);
a = l[thisp].p;
for(f = avoidPts->First(); f; f = avoidPts->NextAfter(f)) {
if(f->Equals(a)) break;
}
if(f) continue;
for(j = 0; j < (sc->l.n - 1); j++) {
scp = WRAP(j+sco, (sc->l.n - 1));
b = sc->l[scp].p;
if(a.Equals(b)) {
withbridge = false;
goto haveEdge;
}
}
}
// If that fails, look for a bridge that does not intersect any edges.
for(i = 0; i < l.n; i++) {
thisp = WRAP(i+thiso, l.n);
a = l[thisp].p;
for(f = avoidPts->First(); f; f = avoidPts->NextAfter(f)) {
if(f->Equals(a)) break;
}
if(f) continue;
for(j = 0; j < (sc->l.n - 1); j++) {
scp = WRAP(j+sco, (sc->l.n - 1));
b = sc->l[scp].p;
for(f = avoidPts->First(); f; f = avoidPts->NextAfter(f)) {
if(f->Equals(b)) break;
}
if(f) continue;
if(avoidEdges->AnyEdgeCrossings(a, b) > 0) {
// doesn't work, bridge crosses an existing edge
} else {
goto haveEdge;
}
}
}
// Tried all the possibilities, didn't find an edge
return false;
haveEdge:
SContour merged = {};
for(i = 0; i < l.n; i++) {
if(withbridge || (i != thisp)) {
merged.AddPoint(l[i].p);
}
if(i == thisp) {
// less than or equal; need to duplicate the join point
for(j = 0; j <= (sc->l.n - 1); j++) {
int jp = WRAP(j + scp, (sc->l.n - 1));
merged.AddPoint((sc->l[jp]).p);
}
// and likewise duplicate join point for the outer curve
if(withbridge) {
merged.AddPoint(l[i].p);
}
}
}
// and future bridges mustn't cross our bridge, and it's tricky to get
// things right if two bridges come from the same point
if(withbridge) {
avoidEdges->AddEdge(a, b);
avoidPts->Add(&a);
}
avoidPts->Add(&b);
l.Clear();
l = merged.l;
return true;
}
bool SContour::IsEmptyTriangle(int ap, int bp, int cp, double scaledEPS) const {
STriangle tr = {};
tr.a = l[ap].p;
tr.b = l[bp].p;
tr.c = l[cp].p;
// Accelerate with an axis-aligned bounding box test
Vector maxv = tr.a, minv = tr.a;
(tr.b).MakeMaxMin(&maxv, &minv);
(tr.c).MakeMaxMin(&maxv, &minv);
Vector n = Vector::From(0, 0, -1);
int i;
for(i = 0; i < l.n; i++) {
if(i == ap || i == bp || i == cp) continue;
Vector p = l[i].p;
if(p.OutsideAndNotOn(maxv, minv)) continue;
// A point on the edge of the triangle is considered to be inside,
// and therefore makes it a non-ear; but a point on the vertex is
// "outside", since that's necessary to make bridges work.
if(p.EqualsExactly(tr.a)) continue;
if(p.EqualsExactly(tr.b)) continue;
if(p.EqualsExactly(tr.c)) continue;
if(tr.ContainsPointProjd(n, p)) {
return false;
}
}
return true;
}
// Test if ray b->d passes through triangle a,b,c
static bool RayIsInside(Vector a, Vector c, Vector b, Vector d) {
// coincident edges are not considered to intersect the triangle
if (d.Equals(a)) return false;
if (d.Equals(c)) return false;
// if d and c are on opposite sides of ba, we are ok
// likewise if d and a are on opposite sides of bc
Vector ba = a.Minus(b);
Vector bc = c.Minus(b);
Vector bd = d.Minus(b);
// perpendicular to (x,y) is (x,-y) so dot that with the two points. If they
// have opposite signs their product will be negative. If bd and bc are on
// opposite sides of ba the ray does not intersect. Likewise for bd,ba and bc.
if ( (bd.x*(ba.y) + (bd.y * (-ba.x))) * ( bc.x*(ba.y) + (bc.y * (-ba.x))) < LENGTH_EPS)
return false;
if ( (bd.x*(bc.y) + (bd.y * (-bc.x))) * ( ba.x*(bc.y) + (ba.y * (-bc.x))) < LENGTH_EPS)
return false;
return true;
}
bool SContour::IsEar(int bp, double scaledEps) const {
int ap = WRAP(bp-1, l.n),
cp = WRAP(bp+1, l.n);
STriangle tr = {};
tr.a = l[ap].p;
tr.b = l[bp].p;
tr.c = l[cp].p;
if((tr.a).Equals(tr.c)) {
// This is two coincident and anti-parallel edges. Zero-area, so
// won't generate a real triangle, but we certainly can clip it.
return true;
}
Vector n = Vector::From(0, 0, -1);
if((tr.Normal()).Dot(n) < scaledEps) {
// This vertex is reflex, or between two collinear edges; either way,
// it's not an ear.
return false;
}
// Accelerate with an axis-aligned bounding box test
Vector maxv = tr.a, minv = tr.a;
(tr.b).MakeMaxMin(&maxv, &minv);
(tr.c).MakeMaxMin(&maxv, &minv);
int i;
for(i = 0; i < l.n; i++) {
if(i == ap || i == bp || i == cp) continue;
Vector p = l[i].p;
if(p.OutsideAndNotOn(maxv, minv)) continue;
// A point on the edge of the triangle is considered to be inside,
// and therefore makes it a non-ear; but a point on the vertex is
// "outside", since that's necessary to make bridges work.
if(p.EqualsExactly(tr.a)) continue;
if(p.EqualsExactly(tr.c)) continue;
// points coincident with bp have to be allowed for bridges but edges
// from that other point must not cross through our triangle.
if(p.EqualsExactly(tr.b)) {
int j = WRAP(i-1, l.n);
int k = WRAP(i+1, l.n);
Vector jp = l[j].p;
Vector kp = l[k].p;
// two consecutive bridges (A,B,C) and later (C,B,A) are not an ear
if (jp.Equals(tr.c) && kp.Equals(tr.a)) return false;
// check both edges from the point in question
if (!RayIsInside(tr.a, tr.c, p,jp) && !RayIsInside(tr.a, tr.c, p,kp))
continue;
}
if(tr.ContainsPointProjd(n, p)) {
return false;
}
}
return true;
}
void SContour::ClipEarInto(SMesh *m, int bp, double scaledEps) {
int ap = WRAP(bp-1, l.n),
cp = WRAP(bp+1, l.n);
STriangle tr = {};
tr.a = l[ap].p;
tr.b = l[bp].p;
tr.c = l[cp].p;
if(tr.Normal().MagSquared() < scaledEps*scaledEps) {
// A vertex with more than two edges will cause us to generate
// zero-area triangles, which must be culled.
} else {
m->AddTriangle(&tr);
}
// By deleting the point at bp, we may change the ear-ness of the points
// on either side.
l[ap].ear = EarType::UNKNOWN;
l[cp].ear = EarType::UNKNOWN;
l.ClearTags();
l[bp].tag = 1;
l.RemoveTagged();
}
void SContour::UvTriangulateInto(SMesh *m, SSurface *srf) {
Vector tu, tv;
srf->TangentsAt(0.5, 0.5, &tu, &tv);
double s = sqrt(tu.MagSquared() + tv.MagSquared());
// We would like to apply our tolerances in xyz; but that would be a lot
// of work, so at least scale the epsilon semi-reasonably. That's
// perfect for square planes, less perfect for anything else.
double scaledEps = LENGTH_EPS / s;
int i;
// Clean the original contour by removing any zero-length edges.
// initialize eartypes to unknown while we're going over them.
l.ClearTags();
l[0].ear = EarType::UNKNOWN;
for(i = 1; i < l.n; i++) {
l[i].ear = EarType::UNKNOWN;
if((l[i].p).Equals(l[i-1].p)) {
l[i].tag = 1;
}
}
if( (l[0].p).Equals(l[l.n-1].p) ) {
l[l.n-1].tag = 1;
}
l.RemoveTagged();
// Handle simple triangle fans all at once. This pass is optional.
if(srf->degm == 1 && srf->degn == 1) {
l.ClearTags();
int j=0;
int pstart = 0;
double elen = -1.0;
double oldspan = 0.0;
for(i = 1; i < l.n; i++) {
Vector ab = l[i].p.Minus(l[i-1].p);
// first time just measure the segment
if (elen < 0.0) {
elen = ab.Dot(ab);
oldspan = elen;
j = 1;
continue;
}
// check for consecutive segments of similar size which are also
// ears and where the group forms a convex ear
bool end = false;
double ratio = ab.Dot(ab) / elen;
if ((ratio < 0.25) || (ratio > 4.0)) end = true;
double slen = l[pstart].p.Minus(l[i].p).MagSquared();
if (slen < oldspan) end = true;
if (!IsEar(i-1, scaledEps) ) end = true;
// if ((j>0) && !IsEar(pstart, i-1, i, scaledEps)) end = true;
if ((j>0) && !IsEmptyTriangle(pstart, i-1, i, scaledEps)) end = true;
// the new segment is valid so add to the fan
if (!end) {
j++;
oldspan = slen;
}
// we need to stop at the end of polygon but may still
if (i == l.n-1) {
end = true;
}
if (end) { // triangulate the fan and tag the vertices
if (j > 3) {
Vector center = l[pstart+1].p.Plus(l[pstart+j-1].p).ScaledBy(0.5);
for (int x=0; x<j; x++) {
STriangle tr = {};
tr.a = center;
tr.b = l[pstart+x].p;
tr.c = l[pstart+x+1].p;
m->AddTriangle(&tr);
}
for (int x=1; x<j; x++) {
l[pstart+x].tag = 1;
}
STriangle tr = {};
tr.a = center;
tr.b = l[pstart+j].p;
tr.c = l[pstart].p;
m->AddTriangle(&tr);
}
pstart = i-1;
elen = ab.Dot(ab);
oldspan = elen;
j = 1;
}
}
l.RemoveTagged();
} // end optional fan creation pass
bool toggle = false;
while(l.n > 3) {
int bestEar = -1;
double bestChordTol = VERY_POSITIVE;
// Alternate the starting position so we generate strip-like
// triangulations instead of fan-like
toggle = !toggle;
int offset = toggle ? -1 : 0;
for(i = 0; i < l.n; i++) {
int ear = WRAP(i+offset, l.n);
if(l[ear].ear == EarType::UNKNOWN) {
(l[ear]).ear = IsEar(ear, scaledEps) ? EarType::EAR : EarType::NOT_EAR;
}
if(l[ear].ear == EarType::EAR) {
if(srf->degm == 1 && srf->degn == 1) {
// This is a plane; any ear is a good ear.
bestEar = ear;
break;
}
// If we are triangulating a curved surface, then try to
// clip ears that have a small chord tolerance from the
// surface.
Vector prev = l[WRAP((i+offset-1), l.n)].p,
next = l[WRAP((i+offset+1), l.n)].p;
double tol = srf->ChordToleranceForEdge(prev, next);
if(tol < bestChordTol - scaledEps) {
bestEar = ear;
bestChordTol = tol;
}
if(bestChordTol < 0.1*SS.ChordTolMm()) {
break;
}
}
}
if(bestEar < 0) {
dbp("couldn't find an ear! fail");
return;
}
ClipEarInto(m, bestEar, scaledEps);
}
ClipEarInto(m, 0, scaledEps); // add the last triangle
}
double SSurface::ChordToleranceForEdge(Vector a, Vector b) const {
Vector as = PointAt(a.x, a.y), bs = PointAt(b.x, b.y);
double worst = VERY_NEGATIVE;
int i;
for(i = 1; i <= 3; i++) {
Vector p = a. Plus((b. Minus(a )).ScaledBy(i/4.0)),
ps = as.Plus((bs.Minus(as)).ScaledBy(i/4.0));
Vector pps = PointAt(p.x, p.y);
worst = max(worst, (pps.Minus(ps)).MagSquared());
}
return sqrt(worst);
}
Vector SSurface::PointAtMaybeSwapped(double u, double v, bool swapped) const {
if(swapped) {
return PointAt(v, u);
} else {
return PointAt(u, v);
}
}
Vector SSurface::NormalAtMaybeSwapped(double u, double v, bool swapped) const {
Vector du, dv;
if(swapped) {
TangentsAt(v, u, &dv, &du);
} else {
TangentsAt(u, v, &du, &dv);
}
return du.Cross(dv).WithMagnitude(1.0);
}
void SSurface::MakeTriangulationGridInto(List<double> *l, double vs, double vf,
bool swapped, int depth) const
{
double worst = 0;
// Try piecewise linearizing four curves, at u = 0, 1/3, 2/3, 1; choose
// the worst chord tolerance of any of those.
double worst_twist = 1.0;
int i;
for(i = 0; i <= 3; i++) {
double u = i/3.0;
// This chord test should be identical to the one in SBezier::MakePwl
// to make the piecewise linear edges line up with the grid more or
// less.
Vector ps = PointAtMaybeSwapped(u, vs, swapped),
pf = PointAtMaybeSwapped(u, vf, swapped);
double vm1 = (2*vs + vf) / 3,
vm2 = (vs + 2*vf) / 3;
Vector pm1 = PointAtMaybeSwapped(u, vm1, swapped),
pm2 = PointAtMaybeSwapped(u, vm2, swapped);
// 0.999 is about 2.5 degrees of twist over the middle 1/3 V-span.
// we don't check at the ends because the derivative may not be valid there.
double twist = 1.0;
if (degm == 1) twist = NormalAtMaybeSwapped(u, vm1, swapped).Dot(
NormalAtMaybeSwapped(u, vm2, swapped) );
if (twist < worst_twist) worst_twist = twist;
worst = max(worst, pm1.DistanceToLine(ps, pf.Minus(ps)));
worst = max(worst, pm2.DistanceToLine(ps, pf.Minus(ps)));
}
double step = 1.0/SS.GetMaxSegments();
if( ((vf - vs) < step || worst < SS.ChordTolMm())
&& ((worst_twist > 0.999) || (depth > 3)) ) {
l->Add(&vf);
} else {
MakeTriangulationGridInto(l, vs, (vs+vf)/2, swapped, depth+1);
MakeTriangulationGridInto(l, (vs+vf)/2, vf, swapped, depth+1);
}
}
void SPolygon::UvGridTriangulateInto(SMesh *mesh, SSurface *srf) {
SEdgeList orig = {};
MakeEdgesInto(&orig);
SEdgeList holes = {};
normal = Vector::From(0, 0, 1);
FixContourDirections();
// Build a rectangular grid, with horizontal and vertical lines in the
// uv plane. The spacing of these lines is adaptive, so calculate that.
List<double> li, lj;
li = {};
lj = {};
double v[5] = {0.0, 0.25, 0.5, 0.75, 1.0};
li.Add(&v[0]);
srf->MakeTriangulationGridInto(&li, 0, 1, /*swapped=*/true, 0);
lj.Add(&v[0]);
srf->MakeTriangulationGridInto(&lj, 0, 1, /*swapped=*/false, 0);
// force 2nd order grid to have at least 4 segments in each direction
if ((li.n < 5) && (srf->degm>1)) { // 4 segments minimum
li.Clear();
li.Add(&v[0]);li.Add(&v[1]);li.Add(&v[2]);li.Add(&v[3]);li.Add(&v[4]);
}
if ((lj.n < 5) && (srf->degn>1)) { // 4 segments minimum
lj.Clear();
lj.Add(&v[0]);lj.Add(&v[1]);lj.Add(&v[2]);lj.Add(&v[3]);lj.Add(&v[4]);
}
if ((li.n > 3) && (lj.n > 3)) {
// Now iterate over each quad in the grid. If it's outside the polygon,
// or if it intersects the polygon, then we discard it. Otherwise we
// generate two triangles in the mesh, and cut it out of our polygon.
// Quads around the perimeter would be rejected by AnyEdgeCrossings.
std::vector<bool> bottom(lj.n, false); // did we use this quad?
Vector tu = {0,0,0}, tv = {0,0,0};
int i, j;
for(i = 1; i < (li.n-1); i++) {
bool prev_flag = false;
for(j = 1; j < (lj.n-1); j++) {
bool this_flag = true;
double us = li[i], uf = li[i+1],
vs = lj[j], vf = lj[j+1];
Vector a = Vector::From(us, vs, 0),
b = Vector::From(us, vf, 0),
c = Vector::From(uf, vf, 0),
d = Vector::From(uf, vs, 0);
// | d-----c
// | | |
// | | |
// | a-----b
// |
// +-------------> j/v axis
if( (i==(li.n-2)) || (j==(lj.n-2)) ||
orig.AnyEdgeCrossings(a, b, NULL) ||
orig.AnyEdgeCrossings(b, c, NULL) ||
orig.AnyEdgeCrossings(c, d, NULL) ||
orig.AnyEdgeCrossings(d, a, NULL))
{
this_flag = false;
}
// There's no intersections, so it doesn't matter which point
// we decide to test.
if(!this->ContainsPoint(a)) {
this_flag = false;
}
8364
if (this_flag) {
// Add the quad to our mesh
srf->TangentsAt(us,vs, &tu,&tv);
if(tu.Dot(tv) < LENGTH_EPS) {
/* Split "the other way" if angle>90
compare to LENGTH_EPS instead of zero to avoid alternating triangle
"orientations" when the tangents are orthogonal (revolve, lathe etc.)
this results in a higher quality mesh. */
STriangle tr = {};
tr.a = a;
tr.b = b;
tr.c = c;
mesh->AddTriangle(&tr);
tr.a = a;
tr.b = c;
tr.c = d;
mesh->AddTriangle(&tr);
} else{
STriangle tr = {};
tr.a = a;
tr.b = b;
tr.c = d;
mesh->AddTriangle(&tr);
tr.a = b;
tr.b = c;
tr.c = d;
mesh->AddTriangle(&tr);
}
if (!prev_flag) // add our own left edge
holes.AddEdge(d, a);
if (!bottom[j]) // add our own bottom edge
holes.AddEdge(a, b);
} else {
if (prev_flag) // add our left neighbors right edge
holes.AddEdge(a, d);
if (bottom[j]) // add our bottom neighbors top edge
holes.AddEdge(b, a);
}
prev_flag = this_flag;
bottom[j] = this_flag;
}
}
// Because no duplicate edges were created we do not need to cull them.
SPolygon hp = {};
holes.AssemblePolygon(&hp, NULL, /*keepDir=*/true);
SContour *sc;
for(sc = hp.l.First(); sc; sc = hp.l.NextAfter(sc)) {
l.Add(sc);
}
hp.l.Clear();
}
orig.Clear();
holes.Clear();
li.Clear();
lj.Clear();
UvTriangulateInto(mesh, srf);
}
void SPolygon::TriangulateInto(SMesh *m) const {
Vector n = normal;
if(n.Equals(Vector::From(0.0, 0.0, 0.0))) {
n = ComputeNormal();
}
Vector u = n.Normal(0);
Vector v = n.Normal(1);
SPolygon p = {};
this->InverseTransformInto(&p, u, v, n);
SSurface srf = SSurface::FromPlane(Vector::From(0.0, 0.0, 0.0),
Vector::From(1.0, 0.0, 0.0),
Vector::From(0.0, 1.0, 0.0));
SMesh pm = {};
p.UvTriangulateInto(&pm, &srf);
for(STriangle st : pm.l) {
st = st.Transform(u, v, n);
m->AddTriangle(&st);
}
p.Clear();
pm.Clear();
}
} // namespace SolveSpace