static gboolean points_are_folded (GtsPoint * A, GtsPoint * B, GtsPoint * C, GtsPoint * D, gdouble max) { GtsVector AB, AC, AD; GtsVector n1, n2; gdouble nn1, nn2, n1n2; gts_vector_init (AB, A, B); gts_vector_init (AC, A, C); gts_vector_init (AD, A, D); gts_vector_cross (n1, AB, AC); gts_vector_cross (n2, AD, AB); nn1 = gts_vector_scalar (n1, n1); nn2 = gts_vector_scalar (n2, n2); if (nn1 >= AREA_RATIO_MAX2*nn2 || nn2 >= AREA_RATIO_MAX2*nn1) return TRUE; n1n2 = gts_vector_scalar (n1, n2); if (n1n2 > 0.) return FALSE; if (n1n2*n1n2/(nn1*nn2) > max) return TRUE; return FALSE; }
void gts_triangulate_convex_polygon(GtsSurface *s, GtsEdgePool *pool, GCList *p) { guint poly_length = g_clist_length(p); while (poly_length > 2) { while (poly_length > 3) { GtsVector e1, e2; GtsPoint *v1 = GTS_POINT(p->prev->data); GtsPoint *v2 = GTS_POINT(p->data); GtsPoint *v3 = GTS_POINT(p->next->data); GtsPoint *v4 = GTS_POINT(p->next->next->data); if (v1 == v3 || v2 == v4) { g_debug("kill degenerated triangle"); GCList *p1 = p; GCList *p2 = p->next; p=g_clist_delete_link(p,p1); p=g_clist_delete_link(p,p2); poly_length -= 2; continue; } gts_vector_init(e1, v1, v3); gts_vector_init(e2, v2, v4); if (gts_vector_scalar(e1, e1) < gts_vector_scalar(e2, e2)) { add_face_from_polygon_corner(s, pool, GTS_VERTEX(v1), GTS_VERTEX(v2), GTS_VERTEX(v3)); p = g_clist_delete_link(p, p); } else { add_face_from_polygon_corner(s, pool, GTS_VERTEX(v2), GTS_VERTEX(v3), GTS_VERTEX(v4)); p = g_clist_delete_link(p, p->next); } --poly_length; } if (g_clist_length(p) > 2) { add_face_from_polygon_corner(s, pool, GTS_VERTEX(p->prev->data), GTS_VERTEX(p->data), GTS_VERTEX(p->next->data)); p = g_clist_delete_link(p, p); --poly_length; } } g_clist_free(p); }
/** * gts_bb_tree_triangle_distance: * @tree: a bounding box tree. * @t: a #GtsTriangle. * @distance: a #GtsBBoxDistFunc. * @delta: spatial scale of the sampling to be used. * @range: a #GtsRange to be filled with the results. * * Given a triangle @t, points are sampled regularly on its surface * using @delta as increment. The distance from each of these points * to the closest object of @tree is computed using @distance and the * gts_bb_tree_point_distance() function. The fields of @range are * filled with the number of points sampled, the minimum, average and * maximum value and the standard deviation. */ void gts_bb_tree_triangle_distance (GNode * tree, GtsTriangle * t, GtsBBoxDistFunc distance, gdouble delta, GtsRange * range) { GtsPoint * p1, * p2, * p3, * p; GtsVector p1p2, p1p3; gdouble l1, t1, dt1; guint i, n1; g_return_if_fail (tree != NULL); g_return_if_fail (t != NULL); g_return_if_fail (distance != NULL); g_return_if_fail (delta > 0.); g_return_if_fail (range != NULL); gts_triangle_vertices (t, (GtsVertex **) &p1, (GtsVertex **) &p2, (GtsVertex **) &p3); gts_vector_init (p1p2, p1, p2); gts_vector_init (p1p3, p1, p3); gts_range_init (range); p = GTS_POINT (gts_object_new (GTS_OBJECT_CLASS (gts_point_class ()))); l1 = sqrt (gts_vector_scalar (p1p2, p1p2)); n1 = l1/delta + 1; dt1 = 1.0/(gdouble) n1; t1 = 0.0; for (i = 0; i <= n1; i++, t1 += dt1) { gdouble t2 = 1. - t1; gdouble x = t2*p1p3[0]; gdouble y = t2*p1p3[1]; gdouble z = t2*p1p3[2]; gdouble l2 = sqrt (x*x + y*y + z*z); guint j, n2 = (guint) (l2/delta + 1); gdouble dt2 = t2/(gdouble) n2; x = t2*p1->x + t1*p2->x; y = t2*p1->y + t1*p2->y; z = t2*p1->z + t1*p2->z; t2 = 0.0; for (j = 0; j <= n2; j++, t2 += dt2) { p->x = x + t2*p1p3[0]; p->y = y + t2*p1p3[1]; p->z = z + t2*p1p3[2]; gts_range_add_value (range, gts_bb_tree_point_distance (tree, p, distance, NULL)); } } gts_object_destroy (GTS_OBJECT (p)); gts_range_update (range); }
static gdouble angle_from_cotan (GtsVertex * vo, GtsVertex * v1, GtsVertex * v2) { /* cf. Appendix B and the caption of Table 1 from [Meyer et al 2002] */ GtsVector u, v; gdouble udotv, denom; gts_vector_init (u, GTS_POINT (vo), GTS_POINT (v1)); gts_vector_init (v, GTS_POINT (vo), GTS_POINT (v2)); udotv = gts_vector_scalar (u, v); denom = sqrt (gts_vector_scalar (u,u)*gts_vector_scalar (v,v) - udotv*udotv); /* Note: I assume this is what they mean by using atan2 (). -Ray Jones */ /* tan = denom/udotv = y/x (see man page for atan2) */ return (fabs (atan2 (denom, udotv))); }
static gboolean angle_obtuse (GtsVertex * v, GtsFace * f) { GtsEdge * e = gts_triangle_edge_opposite (GTS_TRIANGLE (f), v); GtsVector vec1, vec2; gts_vector_init (vec1, GTS_POINT (v), GTS_POINT (GTS_SEGMENT (e)->v1)); gts_vector_init (vec2, GTS_POINT (v), GTS_POINT (GTS_SEGMENT (e)->v2)); return (gts_vector_scalar (vec1, vec2) < 0.0); }
static gdouble cotan (GtsVertex * vo, GtsVertex * v1, GtsVertex * v2) { /* cf. Appendix B of [Meyer et al 2002] */ GtsVector u, v; gdouble udotv, denom; gts_vector_init (u, GTS_POINT (vo), GTS_POINT (v1)); gts_vector_init (v, GTS_POINT (vo), GTS_POINT (v2)); udotv = gts_vector_scalar (u, v); denom = sqrt (gts_vector_scalar (u,u)*gts_vector_scalar (v,v) - udotv*udotv); /* denom can be zero if u==v. Returning 0 is acceptable, based on * the callers of this function below. */ if (denom == 0.0) return (0.0); return (udotv/denom); }
static gboolean is_convex(GtsPoint *p0, GtsPoint *p1, GtsPoint *p2, GtsVector orientation) { g_assert(GTS_IS_POINT(p0)); g_assert(GTS_IS_POINT(p1)); g_assert(GTS_IS_POINT(p2)); GtsVector a; GtsVector b; GtsVector c; gts_vector_init(a, p1, p0); gts_vector_init(b, p1, p2); gts_vector_cross(c, a, b); return gts_vector_scalar(c, orientation) >= 0; }
/** * gts_bb_tree_segment_distance: * @tree: a bounding box tree. * @s: a #GtsSegment. * @distance: a #GtsBBoxDistFunc. * @delta: spatial scale of the sampling to be used. * @range: a #GtsRange to be filled with the results. * * Given a segment @s, points are sampled regularly on its length * using @delta as increment. The distance from each of these points * to the closest object of @tree is computed using @distance and the * gts_bb_tree_point_distance() function. The fields of @range are * filled with the number of points sampled, the minimum, average and * maximum value and the standard deviation. */ void gts_bb_tree_segment_distance (GNode * tree, GtsSegment * s, gdouble (*distance) (GtsPoint *, gpointer), gdouble delta, GtsRange * range) { GtsPoint * p1, * p2, * p; GtsVector p1p2; gdouble l, t, dt; guint i, n; g_return_if_fail (tree != NULL); g_return_if_fail (s != NULL); g_return_if_fail (distance != NULL); g_return_if_fail (delta > 0.); g_return_if_fail (range != NULL); p1 = GTS_POINT (s->v1); p2 = GTS_POINT (s->v2); gts_vector_init (p1p2, p1, p2); gts_range_init (range); p = GTS_POINT (gts_object_new (GTS_OBJECT_CLASS (gts_point_class()))); l = sqrt (gts_vector_scalar (p1p2, p1p2)); n = (guint) (l/delta + 1); dt = 1.0/(gdouble) n; t = 0.0; for (i = 0; i <= n; i++, t += dt) { p->x = p1->x + t*p1p2[0]; p->y = p1->y + t*p1p2[1]; p->z = p1->z + t*p1p2[2]; gts_range_add_value (range, gts_bb_tree_point_distance (tree, p, distance, NULL)); } gts_object_destroy (GTS_OBJECT (p)); gts_range_update (range); }
/** * gts_vertex_principal_directions: * @v: a #GtsVertex. * @s: a #GtsSurface. * @Kh: mean curvature normal (a #GtsVector). * @Kg: Gaussian curvature (a gdouble). * @e1: first principal curvature direction (direction of largest curvature). * @e2: second principal curvature direction. * * Computes the principal curvature directions at a point given @Kh * and @Kg, the mean curvature normal and Gaussian curvatures at that * point, computed with gts_vertex_mean_curvature_normal() and * gts_vertex_gaussian_curvature(), respectively. * * Note that this computation is very approximate and tends to be * unstable. Smoothing of the surface or the principal directions may * be necessary to achieve reasonable results. */ void gts_vertex_principal_directions (GtsVertex * v, GtsSurface * s, GtsVector Kh, gdouble Kg, GtsVector e1, GtsVector e2) { GtsVector N; gdouble normKh; GSList * i, * j; GtsVector basis1, basis2, d, eig; gdouble ve2, vdotN; gdouble aterm_da, bterm_da, cterm_da, const_da; gdouble aterm_db, bterm_db, cterm_db, const_db; gdouble a, b, c; gdouble K1, K2; gdouble *weights, *kappas, *d1s, *d2s; gint edge_count; gdouble err_e1, err_e2; int e; /* compute unit normal */ normKh = sqrt (gts_vector_scalar (Kh, Kh)); if (normKh > 0.0) { N[0] = Kh[0] / normKh; N[1] = Kh[1] / normKh; N[2] = Kh[2] / normKh; } else { /* This vertex is a point of zero mean curvature (flat or saddle * point). Compute a normal by averaging the adjacent triangles */ N[0] = N[1] = N[2] = 0.0; i = gts_vertex_faces (v, s, NULL); while (i) { gdouble x, y, z; gts_triangle_normal (GTS_TRIANGLE ((GtsFace *) i->data), &x, &y, &z); N[0] += x; N[1] += y; N[2] += z; i = i->next; } g_return_if_fail (gts_vector_norm (N) > 0.0); gts_vector_normalize (N); } /* construct a basis from N: */ /* set basis1 to any component not the largest of N */ basis1[0] = basis1[1] = basis1[2] = 0.0; if (fabs (N[0]) > fabs (N[1])) basis1[1] = 1.0; else basis1[0] = 1.0; /* make basis2 orthogonal to N */ gts_vector_cross (basis2, N, basis1); gts_vector_normalize (basis2); /* make basis1 orthogonal to N and basis2 */ gts_vector_cross (basis1, N, basis2); gts_vector_normalize (basis1); aterm_da = bterm_da = cterm_da = const_da = 0.0; aterm_db = bterm_db = cterm_db = const_db = 0.0; weights = g_malloc (sizeof (gdouble)*g_slist_length (v->segments)); kappas = g_malloc (sizeof (gdouble)*g_slist_length (v->segments)); d1s = g_malloc (sizeof (gdouble)*g_slist_length (v->segments)); d2s = g_malloc (sizeof (gdouble)*g_slist_length (v->segments)); edge_count = 0; i = v->segments; while (i) { GtsEdge * e; GtsFace * f1, * f2; gdouble weight, kappa, d1, d2; GtsVector vec_edge; if (! GTS_IS_EDGE (i->data)) { i = i->next; continue; } e = i->data; /* since this vertex passed the tests in * gts_vertex_mean_curvature_normal(), this should be true. */ g_assert (gts_edge_face_number (e, s) == 2); /* identify the two triangles bordering e in s */ f1 = f2 = NULL; j = e->triangles; while (j) { if ((! GTS_IS_FACE (j->data)) || (! gts_face_has_parent_surface (GTS_FACE (j->data), s))) { j = j->next; continue; } if (f1 == NULL) f1 = GTS_FACE (j->data); else { f2 = GTS_FACE (j->data); break; } j = j->next; } g_assert (f2 != NULL); /* We are solving for the values of the curvature tensor * B = [ a b ; b c ]. * The computations here are from section 5 of [Meyer et al 2002]. * * The first step is to calculate the linear equations governing * the values of (a,b,c). These can be computed by setting the * derivatives of the error E to zero (section 5.3). * * Since a + c = norm(Kh), we only compute the linear equations * for dE/da and dE/db. (NB: [Meyer et al 2002] has the * equation a + b = norm(Kh), but I'm almost positive this is * incorrect.) * * Note that the w_ij (defined in section 5.2) are all scaled by * (1/8*A_mixed). We drop this uniform scale factor because the * solution of the linear equations doesn't rely on it. * * The terms of the linear equations are xterm_dy with x in * {a,b,c} and y in {a,b}. There are also const_dy terms that are * the constant factors in the equations. */ /* find the vector from v along edge e */ gts_vector_init (vec_edge, GTS_POINT (v), GTS_POINT ((GTS_SEGMENT (e)->v1 == v) ? GTS_SEGMENT (e)->v2 : GTS_SEGMENT (e)->v1)); ve2 = gts_vector_scalar (vec_edge, vec_edge); vdotN = gts_vector_scalar (vec_edge, N); /* section 5.2 - There is a typo in the computation of kappa. The * edges should be x_j-x_i. */ kappa = 2.0 * vdotN / ve2; /* section 5.2 */ /* I don't like performing a minimization where some of the * weights can be negative (as can be the case if f1 or f2 are * obtuse). To ensure all-positive weights, we check for * obtuseness and use values similar to those in region_area(). */ weight = 0.0; if (! triangle_obtuse(v, f1)) { weight += ve2 * cotan (gts_triangle_vertex_opposite (GTS_TRIANGLE (f1), e), GTS_SEGMENT (e)->v1, GTS_SEGMENT (e)->v2) / 8.0; } else { if (angle_obtuse (v, f1)) { weight += ve2 * gts_triangle_area (GTS_TRIANGLE (f1)) / 4.0; } else { weight += ve2 * gts_triangle_area (GTS_TRIANGLE (f1)) / 8.0; } } if (! triangle_obtuse(v, f2)) { weight += ve2 * cotan (gts_triangle_vertex_opposite (GTS_TRIANGLE (f2), e), GTS_SEGMENT (e)->v1, GTS_SEGMENT (e)->v2) / 8.0; } else { if (angle_obtuse (v, f2)) { weight += ve2 * gts_triangle_area (GTS_TRIANGLE (f2)) / 4.0; } else { weight += ve2 * gts_triangle_area (GTS_TRIANGLE (f2)) / 8.0; } } /* projection of edge perpendicular to N (section 5.3) */ d[0] = vec_edge[0] - vdotN * N[0]; d[1] = vec_edge[1] - vdotN * N[1]; d[2] = vec_edge[2] - vdotN * N[2]; gts_vector_normalize (d); /* not explicit in the paper, but necessary. Move d to 2D basis. */ d1 = gts_vector_scalar (d, basis1); d2 = gts_vector_scalar (d, basis2); /* store off the curvature, direction of edge, and weights for later use */ weights[edge_count] = weight; kappas[edge_count] = kappa; d1s[edge_count] = d1; d2s[edge_count] = d2; edge_count++; /* Finally, update the linear equations */ aterm_da += weight * d1 * d1 * d1 * d1; bterm_da += weight * d1 * d1 * 2 * d1 * d2; cterm_da += weight * d1 * d1 * d2 * d2; const_da += weight * d1 * d1 * (- kappa); aterm_db += weight * d1 * d2 * d1 * d1; bterm_db += weight * d1 * d2 * 2 * d1 * d2; cterm_db += weight * d1 * d2 * d2 * d2; const_db += weight * d1 * d2 * (- kappa); i = i->next; } /* now use the identity (Section 5.3) a + c = |Kh| = 2 * kappa_h */ aterm_da -= cterm_da; const_da += cterm_da * normKh; aterm_db -= cterm_db; const_db += cterm_db * normKh; /* check for solvability of the linear system */ if (((aterm_da * bterm_db - aterm_db * bterm_da) != 0.0) && ((const_da != 0.0) || (const_db != 0.0))) { linsolve (aterm_da, bterm_da, -const_da, aterm_db, bterm_db, -const_db, &a, &b); c = normKh - a; eigenvector (a, b, c, eig); } else { /* region of v is planar */ eig[0] = 1.0; eig[1] = 0.0; } /* Although the eigenvectors of B are good estimates of the * principal directions, it seems that which one is attached to * which curvature direction is a bit arbitrary. This may be a bug * in my implementation, or just a side-effect of the inaccuracy of * B due to the discrete nature of the sampling. * * To overcome this behavior, we'll evaluate which assignment best * matches the given eigenvectors by comparing the curvature * estimates computed above and the curvatures calculated from the * discrete differential operators. */ gts_vertex_principal_curvatures (0.5 * normKh, Kg, &K1, &K2); err_e1 = err_e2 = 0.0; /* loop through the values previously saved */ for (e = 0; e < edge_count; e++) { gdouble weight, kappa, d1, d2; gdouble temp1, temp2; gdouble delta; weight = weights[e]; kappa = kappas[e]; d1 = d1s[e]; d2 = d2s[e]; temp1 = fabs (eig[0] * d1 + eig[1] * d2); temp1 = temp1 * temp1; temp2 = fabs (eig[1] * d1 - eig[0] * d2); temp2 = temp2 * temp2; /* err_e1 is for K1 associated with e1 */ delta = K1 * temp1 + K2 * temp2 - kappa; err_e1 += weight * delta * delta; /* err_e2 is for K1 associated with e2 */ delta = K2 * temp1 + K1 * temp2 - kappa; err_e2 += weight * delta * delta; } g_free (weights); g_free (kappas); g_free (d1s); g_free (d2s); /* rotate eig by a right angle if that would decrease the error */ if (err_e2 < err_e1) { gdouble temp = eig[0]; eig[0] = eig[1]; eig[1] = -temp; } e1[0] = eig[0] * basis1[0] + eig[1] * basis2[0]; e1[1] = eig[0] * basis1[1] + eig[1] * basis2[1]; e1[2] = eig[0] * basis1[2] + eig[1] * basis2[2]; gts_vector_normalize (e1); /* make N,e1,e2 a right handed coordinate sytem */ gts_vector_cross (e2, N, e1); gts_vector_normalize (e2); }