static gdouble region_area (GtsVertex * v, GtsFace * f)
{
/* cf. Section 3.3 of [Meyer et al 2002] */
if (gts_triangle_area (GTS_TRIANGLE (f)) == 0.0) return (0.0);
if (triangle_obtuse (v, f)) {
if (angle_obtuse (v, f))
return (gts_triangle_area (GTS_TRIANGLE (f))/2.0);
else
return (gts_triangle_area (GTS_TRIANGLE (f))/4.0);
} else {
GtsEdge * e = gts_triangle_edge_opposite (GTS_TRIANGLE (f), v);
return ((cotan (GTS_SEGMENT (e)->v1, v, GTS_SEGMENT (e)->v2)*
gts_point_distance2 (GTS_POINT (v),
GTS_POINT (GTS_SEGMENT (e)->v2)) +
cotan (GTS_SEGMENT (e)->v2, v, GTS_SEGMENT (e)->v1)*
gts_point_distance2 (GTS_POINT (v),
GTS_POINT (GTS_SEGMENT (e)->v1)))
/8.0);
}
}
/*! gts_vertex_principal_directions:
* @v: a #WVertex.
* @s: a #GtsSurface.
* @Kh: mean curvature normal (a #Vec3r).
* @Kg: Gaussian curvature (a real).
* @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(WVertex *v, Vec3r Kh, real Kg, Vec3r &e1, Vec3r &e2)
{
Vec3r N;
real normKh;
Vec3r basis1, basis2, d, eig;
real ve2, vdotN;
real aterm_da, bterm_da, cterm_da, const_da;
real aterm_db, bterm_db, cterm_db, const_db;
real a, b, c;
real K1, K2;
real *weights, *kappas, *d1s, *d2s;
int edge_count;
real err_e1, err_e2;
int e;
WVertex::incoming_edge_iterator itE;
/* compute unit normal */
normKh = Kh.norm();
if (normKh > 0.0) {
Kh.normalize();
}
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;
for (itE = v->incoming_edges_begin(); itE != v->incoming_edges_end(); itE++)
N = Vec3r(N + (*itE)->GetaFace()->GetNormal());
real normN = N.norm();
if (normN <= 0.0)
return;
N.normalize();
}
/* 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 */
basis2 = (N ^ basis1);
basis2.normalize();
/* make basis1 orthogonal to N and basis2 */
basis1 = (N ^ basis2);
basis1.normalize();
aterm_da = bterm_da = cterm_da = const_da = 0.0;
aterm_db = bterm_db = cterm_db = const_db = 0.0;
int nb_edges = v->GetEdges().size();
weights = (real *)malloc(sizeof(real) * nb_edges);
kappas = (real *)malloc(sizeof(real) * nb_edges);
d1s = (real *)malloc(sizeof(real) * nb_edges);
d2s = (real *)malloc(sizeof(real) * nb_edges);
edge_count = 0;
for (itE = v->incoming_edges_begin(); itE != v->incoming_edges_end(); itE++) {
WOEdge *e;
WFace *f1, *f2;
real weight, kappa, d1, d2;
Vec3r vec_edge;
if (!*itE)
continue;
e = *itE;
/* 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 = e->GetaFace();
f2 = e->GetbFace();
/* 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 */
vec_edge = Vec3r(-1 * e->GetVec());
ve2 = vec_edge.squareNorm();
vdotN = 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. */
weight = 0.0;
if (!triangle_obtuse(v, f1)) {
weight += ve2 * cotan(f1->GetNextOEdge(e->twin())->GetbVertex(), e->GetaVertex(), e->GetbVertex()) / 8.0;
}
else {
if (angle_obtuse(v, f1)) {
weight += ve2 * f1->getArea() / 4.0;
}
else {
weight += ve2 * f1->getArea() / 8.0;
}
}
if (!triangle_obtuse(v, f2)) {
weight += ve2 * cotan (f2->GetNextOEdge(e)->GetbVertex(), e->GetaVertex(), e->GetbVertex()) / 8.0;
}
else {
if (angle_obtuse(v, f2)) {
weight += ve2 * f1->getArea() / 4.0;
}
else {
weight += ve2 * f1->getArea() / 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];
d.normalize();
/* not explicit in the paper, but necessary. Move d to 2D basis. */
d1 = d * basis1;
d2 = 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);
}
/* 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++) {
real weight, kappa, d1, d2;
real temp1, temp2;
real 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;
}
free (weights);
free (kappas);
free (d1s);
free (d2s);
/* rotate eig by a right angle if that would decrease the error */
if (err_e2 < err_e1) {
real 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];
e1.normalize();
/* make N,e1,e2 a right handed coordinate sytem */
e2 = N ^ e1;
e2.normalize();
}
/**
* 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);
}
