//----------------------------------------------------------------------------- // Routines for ray-casting: intersecting a line segment or an infinite line // with a surface or shell. Ray-casting against a shell is used for point-in- // shell testing, and the intersection of edge line segments against surfaces // is used to get rough surface-curve intersections, which are later refined // numerically. // // Copyright 2008-2013 Jonathan Westhues. //----------------------------------------------------------------------------- #include "solvespace.h" // Dot product tolerance for perpendicular; this is on the direction cosine, // so it's about 0.001 degrees. const double SShell::DOTP_TOL = 1e-5; extern int FLAG; double SSurface::DepartureFromCoplanar(void) { int i, j; int ia, ja, ib, jb, ic, jc; double best; // Grab three points to define a plane; first choose (0, 0) arbitrarily. ia = ja = 0; // Then the point farthest from pt a. best = VERY_NEGATIVE; for(i = 0; i <= degm; i++) { for(j = 0; j <= degn; j++) { if(i == ia && j == ja) continue; double dist = (ctrl[i][j]).Minus(ctrl[ia][ja]).Magnitude(); if(dist > best) { best = dist; ib = i; jb = j; } } } // Then biggest magnitude of ab cross ac. best = VERY_NEGATIVE; for(i = 0; i <= degm; i++) { for(j = 0; j <= degn; j++) { if(i == ia && j == ja) continue; if(i == ib && j == jb) continue; double mag = ((ctrl[ia][ja].Minus(ctrl[ib][jb]))).Cross( (ctrl[ia][ja].Minus(ctrl[i ][j ]))).Magnitude(); if(mag > best) { best = mag; ic = i; jc = j; } } } Vector n = ((ctrl[ia][ja].Minus(ctrl[ib][jb]))).Cross( (ctrl[ia][ja].Minus(ctrl[ic][jc]))); n = n.WithMagnitude(1); double d = (ctrl[ia][ja]).Dot(n); // Finally, calculate the deviation from each point to the plane. double farthest = VERY_NEGATIVE; for(i = 0; i <= degm; i++) { for(j = 0; j <= degn; j++) { double dist = fabs(n.Dot(ctrl[i][j]) - d); if(dist > farthest) { farthest = dist; } } } return farthest; } void SSurface::WeightControlPoints(void) { int i, j; for(i = 0; i <= degm; i++) { for(j = 0; j <= degn; j++) { ctrl[i][j] = (ctrl[i][j]).ScaledBy(weight[i][j]); } } } void SSurface::UnWeightControlPoints(void) { int i, j; for(i = 0; i <= degm; i++) { for(j = 0; j <= degn; j++) { ctrl[i][j] = (ctrl[i][j]).ScaledBy(1.0/weight[i][j]); } } } void SSurface::CopyRowOrCol(bool row, int this_ij, SSurface *src, int src_ij) { if(row) { int j; for(j = 0; j <= degn; j++) { ctrl [this_ij][j] = src->ctrl [src_ij][j]; weight[this_ij][j] = src->weight[src_ij][j]; } } else { int i; for(i = 0; i <= degm; i++) { ctrl [i][this_ij] = src->ctrl [i][src_ij]; weight[i][this_ij] = src->weight[i][src_ij]; } } } void SSurface::BlendRowOrCol(bool row, int this_ij, SSurface *a, int a_ij, SSurface *b, int b_ij) { if(row) { int j; for(j = 0; j <= degn; j++) { Vector c = (a->ctrl [a_ij][j]).Plus(b->ctrl [b_ij][j]); double w = (a->weight[a_ij][j] + b->weight[b_ij][j]); ctrl [this_ij][j] = c.ScaledBy(0.5); weight[this_ij][j] = w / 2; } } else { int i; for(i = 0; i <= degm; i++) { Vector c = (a->ctrl [i][a_ij]).Plus(b->ctrl [i][b_ij]); double w = (a->weight[i][a_ij] + b->weight[i][b_ij]); ctrl [i][this_ij] = c.ScaledBy(0.5); weight[i][this_ij] = w / 2; } } } void SSurface::SplitInHalf(bool byU, SSurface *sa, SSurface *sb) { sa->degm = sb->degm = degm; sa->degn = sb->degn = degn; // by de Casteljau's algorithm in a projective space; so we must work // on points (w*x, w*y, w*z, w) WeightControlPoints(); switch(byU ? degm : degn) { case 1: sa->CopyRowOrCol (byU, 0, this, 0); sb->CopyRowOrCol (byU, 1, this, 1); sa->BlendRowOrCol(byU, 1, this, 0, this, 1); sb->BlendRowOrCol(byU, 0, this, 0, this, 1); break; case 2: sa->CopyRowOrCol (byU, 0, this, 0); sb->CopyRowOrCol (byU, 2, this, 2); sa->BlendRowOrCol(byU, 1, this, 0, this, 1); sb->BlendRowOrCol(byU, 1, this, 1, this, 2); sa->BlendRowOrCol(byU, 2, sa, 1, sb, 1); sb->BlendRowOrCol(byU, 0, sa, 1, sb, 1); break; case 3: { SSurface st; st.degm = degm; st.degn = degn; sa->CopyRowOrCol (byU, 0, this, 0); sb->CopyRowOrCol (byU, 3, this, 3); sa->BlendRowOrCol(byU, 1, this, 0, this, 1); sb->BlendRowOrCol(byU, 2, this, 2, this, 3); st. BlendRowOrCol(byU, 0, this, 1, this, 2); // scratch var sa->BlendRowOrCol(byU, 2, sa, 1, &st, 0); sb->BlendRowOrCol(byU, 1, sb, 2, &st, 0); sa->BlendRowOrCol(byU, 3, sa, 2, sb, 1); sb->BlendRowOrCol(byU, 0, sa, 2, sb, 1); break; } default: oops(); } sa->UnWeightControlPoints(); sb->UnWeightControlPoints(); UnWeightControlPoints(); } //----------------------------------------------------------------------------- // Find all points where the indicated finite (if segment) or infinite (if not // segment) line intersects our surface. Report them in uv space in the list. // We first do a bounding box check; if the line doesn't intersect, then we're // done. If it does, then we check how small our surface is. If it's big, // then we subdivide into quarters and recurse. If it's small, then we refine // by Newton's method and record the point. //----------------------------------------------------------------------------- void SSurface::AllPointsIntersectingUntrimmed(Vector a, Vector b, int *cnt, int *level, List *l, bool segment, SSurface *sorig) { // Test if the line intersects our axis-aligned bounding box; if no, then // no possibility of an intersection if(LineEntirelyOutsideBbox(a, b, segment)) return; if(*cnt > 2000) { dbp("!!! too many subdivisions (level=%d)!", *level); dbp("degm = %d degn = %d", degm, degn); return; } (*cnt)++; // If we might intersect, and the surface is small, then switch to Newton // iterations. if(DepartureFromCoplanar() < 0.2*SS.ChordTolMm()) { Vector p = (ctrl[0 ][0 ]).Plus( ctrl[0 ][degn]).Plus( ctrl[degm][0 ]).Plus( ctrl[degm][degn]).ScaledBy(0.25); Inter inter; sorig->ClosestPointTo(p, &(inter.p.x), &(inter.p.y), false); if(sorig->PointIntersectingLine(a, b, &(inter.p.x), &(inter.p.y))) { Vector p = sorig->PointAt(inter.p.x, inter.p.y); // Debug check, verify that the point lies in both surfaces // (which it ought to, since the surfaces should be coincident) double u, v; ClosestPointTo(p, &u, &v); l->Add(&inter); } else { // Might not converge if line is almost tangent to surface... } return; } // But the surface is big, so split it, alternating by u and v SSurface surf0, surf1; SplitInHalf((*level & 1) == 0, &surf0, &surf1); int nextLevel = (*level) + 1; (*level) = nextLevel; surf0.AllPointsIntersectingUntrimmed(a, b, cnt, level, l, segment, sorig); (*level) = nextLevel; surf1.AllPointsIntersectingUntrimmed(a, b, cnt, level, l, segment, sorig); } //----------------------------------------------------------------------------- // Find all points where a line through a and b intersects our surface, and // add them to the list. If seg is true then report only intersections that // lie within the finite line segment (not including the endpoints); otherwise // we work along the infinite line. And we report either just intersections // inside the trim curve, or any intersection with u, v in [0, 1]. And we // either disregard or report tangent points. //----------------------------------------------------------------------------- void SSurface::AllPointsIntersecting(Vector a, Vector b, List *l, bool seg, bool trimmed, bool inclTangent) { if(LineEntirelyOutsideBbox(a, b, seg)) return; Vector ba = b.Minus(a); double bam = ba.Magnitude(); List inters; ZERO(&inters); // All the intersections between the line and the surface; either special // cases that we can quickly solve in closed form, or general numerical. Vector center, axis, start, finish; double radius; if(degm == 1 && degn == 1) { // Against a plane, easy. Vector n = NormalAt(0, 0).WithMagnitude(1); double d = n.Dot(PointAt(0, 0)); // Trim to line segment now if requested, don't generate points that // would just get discarded later. if(!seg || (n.Dot(a) > d + LENGTH_EPS && n.Dot(b) < d - LENGTH_EPS) || (n.Dot(b) > d + LENGTH_EPS && n.Dot(a) < d - LENGTH_EPS)) { Vector p = Vector::AtIntersectionOfPlaneAndLine(n, d, a, b, NULL); Inter inter; ClosestPointTo(p, &(inter.p.x), &(inter.p.y)); inters.Add(&inter); } } else if(IsCylinder(&axis, ¢er, &radius, &start, &finish)) { // This one can be solved in closed form too. Vector ab = b.Minus(a); if(axis.Cross(ab).Magnitude() < LENGTH_EPS) { // edge is parallel to axis of cylinder, no intersection points return; } // A coordinate system centered at the center of the circle, with // the edge under test horizontal Vector u, v, n = axis.WithMagnitude(1); u = (ab.Minus(n.ScaledBy(ab.Dot(n)))).WithMagnitude(1); v = n.Cross(u); Point2d ap = (a.Minus(center)).DotInToCsys(u, v, n).ProjectXy(), bp = (b.Minus(center)).DotInToCsys(u, v, n).ProjectXy(), sp = (start. Minus(center)).DotInToCsys(u, v, n).ProjectXy(), fp = (finish.Minus(center)).DotInToCsys(u, v, n).ProjectXy(); double thetas = atan2(sp.y, sp.x), thetaf = atan2(fp.y, fp.x); Point2d ip[2]; int ip_n = 0; if(fabs(fabs(ap.y) - radius) < LENGTH_EPS) { // tangent if(inclTangent) { ip[0] = Point2d::From(0, ap.y); ip_n = 1; } } else if(fabs(ap.y) < radius) { // two intersections double xint = sqrt(radius*radius - ap.y*ap.y); ip[0] = Point2d::From(-xint, ap.y); ip[1] = Point2d::From( xint, ap.y); ip_n = 2; } int i; for(i = 0; i < ip_n; i++) { double t = (ip[i].Minus(ap)).DivPivoting(bp.Minus(ap)); // This is a point on the circle; but is it on the arc? Point2d pp = ap.Plus((bp.Minus(ap)).ScaledBy(t)); double theta = atan2(pp.y, pp.x); double dp = WRAP_SYMMETRIC(theta - thetas, 2*PI), df = WRAP_SYMMETRIC(thetaf - thetas, 2*PI); double tol = LENGTH_EPS/radius; if((df > 0 && ((dp < -tol) || (dp > df + tol))) || (df < 0 && ((dp > tol) || (dp < df - tol)))) { continue; } Vector p = a.Plus((b.Minus(a)).ScaledBy(t)); Inter inter; ClosestPointTo(p, &(inter.p.x), &(inter.p.y)); inters.Add(&inter); } } else { // General numerical solution by subdivision, fallback int cnt = 0, level = 0; AllPointsIntersectingUntrimmed(a, b, &cnt, &level, &inters, seg, this); } // Remove duplicate intersection points inters.ClearTags(); int i, j; for(i = 0; i < inters.n; i++) { for(j = i + 1; j < inters.n; j++) { if(inters.elem[i].p.Equals(inters.elem[j].p)) { inters.elem[j].tag = 1; } } } inters.RemoveTagged(); for(i = 0; i < inters.n; i++) { Point2d puv = inters.elem[i].p; // Make sure the point lies within the finite line segment Vector pxyz = PointAt(puv.x, puv.y); double t = (pxyz.Minus(a)).DivPivoting(ba); if(seg && (t > 1 - LENGTH_EPS/bam || t < LENGTH_EPS/bam)) { continue; } // And that it lies inside our trim region Point2d dummy = { 0, 0 }; int c = bsp->ClassifyPoint(puv, dummy, this); if(trimmed && c == SBspUv::OUTSIDE) { continue; } // It does, so generate the intersection SInter si; si.p = pxyz; si.surfNormal = NormalAt(puv.x, puv.y); si.pinter = puv; si.srf = this; si.onEdge = (c != SBspUv::INSIDE); l->Add(&si); } inters.Clear(); } void SShell::AllPointsIntersecting(Vector a, Vector b, List *il, bool seg, bool trimmed, bool inclTangent) { SSurface *ss; for(ss = surface.First(); ss; ss = surface.NextAfter(ss)) { ss->AllPointsIntersecting(a, b, il, seg, trimmed, inclTangent); } } int SShell::ClassifyRegion(Vector edge_n, Vector inter_surf_n, Vector edge_surf_n) { double dot = inter_surf_n.DirectionCosineWith(edge_n); if(fabs(dot) < DOTP_TOL) { // The edge's surface and the edge-on-face surface // are coincident. Test the edge's surface normal // to see if it's with same or opposite normals. if(inter_surf_n.Dot(edge_surf_n) > 0) { return COINC_SAME; } else { return COINC_OPP; } } else if(dot > 0) { return OUTSIDE; } else { return INSIDE; } } //----------------------------------------------------------------------------- // Does the given point lie on our shell? There are many cases; inside and // outside are obvious, but then there's all the edge-on-edge and edge-on-face // possibilities. // // To calculate, we intersect a ray through p with our shell, and classify // using the closest intersection point. If the ray hits a surface on edge, // then just reattempt in a different random direction. //----------------------------------------------------------------------------- bool SShell::ClassifyEdge(int *indir, int *outdir, Vector ea, Vector eb, Vector p, Vector edge_n_in, Vector edge_n_out, Vector surf_n) { List l; ZERO(&l); srand(0); // First, check for edge-on-edge int edge_inters = 0; Vector inter_surf_n[2], inter_edge_n[2]; SSurface *srf; for(srf = surface.First(); srf; srf = surface.NextAfter(srf)) { if(srf->LineEntirelyOutsideBbox(ea, eb, true)) continue; SEdgeList *sel = &(srf->edges); SEdge *se; for(se = sel->l.First(); se; se = sel->l.NextAfter(se)) { if((ea.Equals(se->a) && eb.Equals(se->b)) || (eb.Equals(se->a) && ea.Equals(se->b)) || p.OnLineSegment(se->a, se->b)) { if(edge_inters < 2) { // Edge-on-edge case Point2d pm; srf->ClosestPointTo(p, &pm, false); // A vector normal to the surface, at the intersection point inter_surf_n[edge_inters] = srf->NormalAt(pm); // A vector normal to the intersecting edge (but within the // intersecting surface) at the intersection point, pointing // out. inter_edge_n[edge_inters] = (inter_surf_n[edge_inters]).Cross((se->b).Minus((se->a))); } edge_inters++; } } } if(edge_inters == 2) { // TODO, make this use the appropriate curved normals double dotp[2]; for(int i = 0; i < 2; i++) { dotp[i] = edge_n_out.DirectionCosineWith(inter_surf_n[i]); } if(fabs(dotp[1]) < DOTP_TOL) { SWAP(double, dotp[0], dotp[1]); SWAP(Vector, inter_surf_n[0], inter_surf_n[1]); SWAP(Vector, inter_edge_n[0], inter_edge_n[1]); } int coinc = (surf_n.Dot(inter_surf_n[0])) > 0 ? COINC_SAME : COINC_OPP; if(fabs(dotp[0]) < DOTP_TOL && fabs(dotp[1]) < DOTP_TOL) { // This is actually an edge on face case, just that the face // is split into two pieces joining at our edge. *indir = coinc; *outdir = coinc; } else if(fabs(dotp[0]) < DOTP_TOL && dotp[1] > DOTP_TOL) { if(edge_n_out.Dot(inter_edge_n[0]) > 0) { *indir = coinc; *outdir = OUTSIDE; } else { *indir = INSIDE; *outdir = coinc; } } else if(fabs(dotp[0]) < DOTP_TOL && dotp[1] < -DOTP_TOL) { if(edge_n_out.Dot(inter_edge_n[0]) > 0) { *indir = coinc; *outdir = INSIDE; } else { *indir = OUTSIDE; *outdir = coinc; } } else if(dotp[0] > DOTP_TOL && dotp[1] > DOTP_TOL) { *indir = INSIDE; *outdir = OUTSIDE; } else if(dotp[0] < -DOTP_TOL && dotp[1] < -DOTP_TOL) { *indir = OUTSIDE; *outdir = INSIDE; } else { // Edge is tangent to the shell at shell's edge, so can't be // a boundary of the surface. return false; } return true; } if(edge_inters != 0) dbp("bad, edge_inters=%d", edge_inters); // Next, check for edge-on-surface. The ray-casting for edge-inside-shell // would catch this too, but test separately, for speed (since many edges // are on surface) and for numerical stability, so we don't pick up // the additional error from the line intersection. for(srf = surface.First(); srf; srf = surface.NextAfter(srf)) { if(srf->LineEntirelyOutsideBbox(ea, eb, true)) continue; Point2d puv; srf->ClosestPointTo(p, &(puv.x), &(puv.y), false); Vector pp = srf->PointAt(puv); if((pp.Minus(p)).Magnitude() > LENGTH_EPS) continue; Point2d dummy = { 0, 0 }; int c = srf->bsp->ClassifyPoint(puv, dummy, srf); if(c == SBspUv::OUTSIDE) continue; // Edge-on-face (unless edge-on-edge above superceded) Point2d pin, pout; srf->ClosestPointTo(p.Plus(edge_n_in), &pin, false); srf->ClosestPointTo(p.Plus(edge_n_out), &pout, false); Vector surf_n_in = srf->NormalAt(pin), surf_n_out = srf->NormalAt(pout); *indir = ClassifyRegion(edge_n_in, surf_n_in, surf_n); *outdir = ClassifyRegion(edge_n_out, surf_n_out, surf_n); return true; } // Edge is not on face or on edge; so it's either inside or outside // the shell, and we'll determine which by raycasting. int cnt = 0; for(;;) { // Cast a ray in a random direction (two-sided so that we test if // the point lies on a surface, but use only one side for in/out // testing) Vector ray = Vector::From(Random(1), Random(1), Random(1)); AllPointsIntersecting( p.Minus(ray), p.Plus(ray), &l, false, true, false); // no intersections means it's outside *indir = OUTSIDE; *outdir = OUTSIDE; double dmin = VERY_POSITIVE; bool onEdge = false; edge_inters = 0; SInter *si; for(si = l.First(); si; si = l.NextAfter(si)) { double t = ((si->p).Minus(p)).DivPivoting(ray); if(t*ray.Magnitude() < -LENGTH_EPS) { // wrong side, doesn't count continue; } double d = ((si->p).Minus(p)).Magnitude(); // We actually should never hit this case; it should have been // handled above. if(d < LENGTH_EPS && si->onEdge) { edge_inters++; } if(d < dmin) { dmin = d; // Edge does not lie on surface; either strictly inside // or strictly outside if((si->surfNormal).Dot(ray) > 0) { *indir = INSIDE; *outdir = INSIDE; } else { *indir = OUTSIDE; *outdir = OUTSIDE; } onEdge = si->onEdge; } } l.Clear(); // If the point being tested lies exactly on an edge of the shell, // then our ray always lies on edge, and that's okay. Otherwise // try again in a different random direction. if(!onEdge) break; if(cnt++ > 5) { dbp("can't find a ray that doesn't hit on edge!"); dbp("on edge = %d, edge_inters = %d", onEdge, edge_inters); SS.nakedEdges.AddEdge(ea, eb); break; } } return true; }