static void laplacian_triangle_weights(LaplacianSystem *sys, int f, int i1, int i2, int i3)
{
float t1, t2, t3;
float *varea= sys->varea, *v1, *v2, *v3;
v1= sys->verts[i1];
v2= sys->verts[i2];
v3= sys->verts[i3];
/* instead of *0.5 we divided by the number of faces of the edge, it still
needs to be verified that this is indeed the correct thing to do! */
t1= cotan_weight(v1, v2, v3)/laplacian_edge_count(sys->edgehash, i2, i3);
t2= cotan_weight(v2, v3, v1)/laplacian_edge_count(sys->edgehash, i3, i1);
t3= cotan_weight(v3, v1, v2)/laplacian_edge_count(sys->edgehash, i1, i2);
nlMatrixAdd(i1, i1, (t2+t3)*varea[i1]);
nlMatrixAdd(i2, i2, (t1+t3)*varea[i2]);
nlMatrixAdd(i3, i3, (t1+t2)*varea[i3]);
nlMatrixAdd(i1, i2, -t3*varea[i1]);
nlMatrixAdd(i2, i1, -t3*varea[i2]);
nlMatrixAdd(i2, i3, -t1*varea[i2]);
nlMatrixAdd(i3, i2, -t1*varea[i3]);
nlMatrixAdd(i3, i1, -t2*varea[i3]);
nlMatrixAdd(i1, i3, -t2*varea[i1]);
if(sys->storeweights) {
sys->fweights[f][0]= t1*varea[i1];
sys->fweights[f][1]= t2*varea[i2];
sys->fweights[f][2]= t3*varea[i3];
}
}
static void laplacian_triangle_area(LaplacianSystem *sys, int i1, int i2, int i3)
{
float t1, t2, t3, len1, len2, len3, area;
float *varea= sys->varea, *v1, *v2, *v3;
int obtuse = 0;
v1= sys->verts[i1];
v2= sys->verts[i2];
v3= sys->verts[i3];
t1= cotan_weight(v1, v2, v3);
t2= cotan_weight(v2, v3, v1);
t3= cotan_weight(v3, v1, v2);
if(RAD2DEGF(angle_v3v3v3(v2, v1, v3)) > 90) obtuse= 1;
else if(RAD2DEGF(angle_v3v3v3(v1, v2, v3)) > 90) obtuse= 2;
else if(RAD2DEGF(angle_v3v3v3(v1, v3, v2)) > 90) obtuse= 3;
if (obtuse > 0) {
area= area_tri_v3(v1, v2, v3);
varea[i1] += (obtuse == 1)? area: area*0.5f;
varea[i2] += (obtuse == 2)? area: area*0.5f;
varea[i3] += (obtuse == 3)? area: area*0.5f;
}
else {
len1= len_v3v3(v2, v3);
len2= len_v3v3(v1, v3);
len3= len_v3v3(v1, v2);
t1 *= len1*len1;
t2 *= len2*len2;
t3 *= len3*len3;
varea[i1] += (t2 + t3)*0.25f;
varea[i2] += (t1 + t3)*0.25f;
varea[i3] += (t1 + t2)*0.25f;
}
}
static void init_laplacian_matrix(LaplacianSystem *sys)
{
float *v1, *v2, *v3, *v4;
float w1, w2, w3, w4;
float areaf;
int i, j;
unsigned int idv1, idv2, idv3, idv4, idv[4];
int has_4_vert;
for (i = 0; i < sys->numEdges; i++) {
idv1 = sys->medges[i].v1;
idv2 = sys->medges[i].v2;
v1 = sys->vertexCos[idv1];
v2 = sys->vertexCos[idv2];
sys->numNeEd[idv1] = sys->numNeEd[idv1] + 1;
sys->numNeEd[idv2] = sys->numNeEd[idv2] + 1;
w1 = len_v3v3(v1, v2);
if (w1 < sys->min_area) {
sys->zerola[idv1] = 1;
sys->zerola[idv2] = 1;
}
else {
w1 = 1.0f / w1;
}
sys->eweights[i] = w1;
}
for (i = 0; i < sys->numFaces; i++) {
has_4_vert = ((&sys->mfaces[i])->v4) ? 1 : 0;
idv1 = sys->mfaces[i].v1;
idv2 = sys->mfaces[i].v2;
idv3 = sys->mfaces[i].v3;
idv4 = has_4_vert ? sys->mfaces[i].v4 : 0;
sys->numNeFa[idv1] += 1;
sys->numNeFa[idv2] += 1;
sys->numNeFa[idv3] += 1;
if (has_4_vert) sys->numNeFa[idv4] += 1;
v1 = sys->vertexCos[idv1];
v2 = sys->vertexCos[idv2];
v3 = sys->vertexCos[idv3];
v4 = has_4_vert ? sys->vertexCos[idv4] : 0;
if (has_4_vert) {
areaf = area_quad_v3(v1, v2, v3, sys->vertexCos[sys->mfaces[i].v4]);
}
else {
areaf = area_tri_v3(v1, v2, v3);
}
if (fabsf(areaf) < sys->min_area) {
sys->zerola[idv1] = 1;
sys->zerola[idv2] = 1;
sys->zerola[idv3] = 1;
if (has_4_vert) sys->zerola[idv4] = 1;
}
if (has_4_vert) {
sys->ring_areas[idv1] += average_area_quad_v3(v1, v2, v3, v4);
sys->ring_areas[idv2] += average_area_quad_v3(v2, v3, v4, v1);
sys->ring_areas[idv3] += average_area_quad_v3(v3, v4, v1, v2);
sys->ring_areas[idv4] += average_area_quad_v3(v4, v1, v2, v3);
}
else {
sys->ring_areas[idv1] += areaf;
sys->ring_areas[idv2] += areaf;
sys->ring_areas[idv3] += areaf;
}
if (has_4_vert) {
idv[0] = idv1;
idv[1] = idv2;
idv[2] = idv3;
idv[3] = idv4;
for (j = 0; j < 4; j++) {
idv1 = idv[j];
idv2 = idv[(j + 1) % 4];
idv3 = idv[(j + 2) % 4];
idv4 = idv[(j + 3) % 4];
v1 = sys->vertexCos[idv1];
v2 = sys->vertexCos[idv2];
v3 = sys->vertexCos[idv3];
v4 = sys->vertexCos[idv4];
w2 = cotan_weight(v4, v1, v2) + cotan_weight(v3, v1, v2);
w3 = cotan_weight(v2, v3, v1) + cotan_weight(v4, v1, v3);
w4 = cotan_weight(v2, v4, v1) + cotan_weight(v3, v4, v1);
sys->vweights[idv1] += (w2 + w3 + w4) / 4.0f;
}
}
else {
w1 = cotan_weight(v1, v2, v3) / 2.0f;
w2 = cotan_weight(v2, v3, v1) / 2.0f;
w3 = cotan_weight(v3, v1, v2) / 2.0f;
sys->fweights[i][0] = sys->fweights[i][0] + w1;
sys->fweights[i][1] = sys->fweights[i][1] + w2;
sys->fweights[i][2] = sys->fweights[i][2] + w3;
sys->vweights[idv1] = sys->vweights[idv1] + w2 + w3;
sys->vweights[idv2] = sys->vweights[idv2] + w1 + w3;
sys->vweights[idv3] = sys->vweights[idv3] + w1 + w2;
}
}
for (i = 0; i < sys->numEdges; i++) {
idv1 = sys->medges[i].v1;
idv2 = sys->medges[i].v2;
/* if is boundary, apply scale-dependent umbrella operator only with neighboors in boundary */
if (sys->numNeEd[idv1] != sys->numNeFa[idv1] && sys->numNeEd[idv2] != sys->numNeFa[idv2]) {
sys->vlengths[idv1] += sys->eweights[i];
sys->vlengths[idv2] += sys->eweights[i];
}
}
}
static void fill_laplacian_matrix(LaplacianSystem *sys)
{
float *v1, *v2, *v3, *v4;
float w2, w3, w4;
int i, j;
int has_4_vert;
unsigned int idv1, idv2, idv3, idv4, idv[4];
for (i = 0; i < sys->numFaces; i++) {
idv1 = sys->mfaces[i].v1;
idv2 = sys->mfaces[i].v2;
idv3 = sys->mfaces[i].v3;
has_4_vert = ((&sys->mfaces[i])->v4) ? 1 : 0;
if (has_4_vert) {
idv[0] = sys->mfaces[i].v1;
idv[1] = sys->mfaces[i].v2;
idv[2] = sys->mfaces[i].v3;
idv[3] = sys->mfaces[i].v4;
for (j = 0; j < 4; j++) {
idv1 = idv[j];
idv2 = idv[(j + 1) % 4];
idv3 = idv[(j + 2) % 4];
idv4 = idv[(j + 3) % 4];
v1 = sys->vertexCos[idv1];
v2 = sys->vertexCos[idv2];
v3 = sys->vertexCos[idv3];
v4 = sys->vertexCos[idv4];
w2 = cotan_weight(v4, v1, v2) + cotan_weight(v3, v1, v2);
w3 = cotan_weight(v2, v3, v1) + cotan_weight(v4, v1, v3);
w4 = cotan_weight(v2, v4, v1) + cotan_weight(v3, v4, v1);
w2 = w2 / 4.0f;
w3 = w3 / 4.0f;
w4 = w4 / 4.0f;
if (sys->numNeEd[idv1] == sys->numNeFa[idv1] && sys->zerola[idv1] == 0) {
nlMatrixAdd(idv1, idv2, w2 * sys->vweights[idv1]);
nlMatrixAdd(idv1, idv3, w3 * sys->vweights[idv1]);
nlMatrixAdd(idv1, idv4, w4 * sys->vweights[idv1]);
}
}
}
else {
/* Is ring if number of faces == number of edges around vertice*/
if (sys->numNeEd[idv1] == sys->numNeFa[idv1] && sys->zerola[idv1] == 0) {
nlMatrixAdd(idv1, idv2, sys->fweights[i][2] * sys->vweights[idv1]);
nlMatrixAdd(idv1, idv3, sys->fweights[i][1] * sys->vweights[idv1]);
}
if (sys->numNeEd[idv2] == sys->numNeFa[idv2] && sys->zerola[idv2] == 0) {
nlMatrixAdd(idv2, idv1, sys->fweights[i][2] * sys->vweights[idv2]);
nlMatrixAdd(idv2, idv3, sys->fweights[i][0] * sys->vweights[idv2]);
}
if (sys->numNeEd[idv3] == sys->numNeFa[idv3] && sys->zerola[idv3] == 0) {
nlMatrixAdd(idv3, idv1, sys->fweights[i][1] * sys->vweights[idv3]);
nlMatrixAdd(idv3, idv2, sys->fweights[i][0] * sys->vweights[idv3]);
}
}
}
for (i = 0; i < sys->numEdges; i++) {
idv1 = sys->medges[i].v1;
idv2 = sys->medges[i].v2;
/* Is boundary */
if (sys->numNeEd[idv1] != sys->numNeFa[idv1] &&
sys->numNeEd[idv2] != sys->numNeFa[idv2] &&
sys->zerola[idv1] == 0 &&
sys->zerola[idv2] == 0)
{
nlMatrixAdd(idv1, idv2, sys->eweights[i] * sys->vlengths[idv1]);
nlMatrixAdd(idv2, idv1, sys->eweights[i] * sys->vlengths[idv2]);
}
}
}
/**
* This method computes the Laplacian Matrix and Differential Coordinates for all vertex in the mesh.
* The Linear system is LV = d
* Where L is Laplacian Matrix, V as the vertexes in Mesh, d is the differential coordinates
* The Laplacian Matrix is computes as a
* Lij = sum(Wij) (if i == j)
* Lij = Wij (if i != j)
* Wij is weight between vertex Vi and vertex Vj, we use cotangent weight
*
* The Differential Coordinate is computes as a
* di = Vi * sum(Wij) - sum(Wij * Vj)
* Where :
* di is the Differential Coordinate i
* sum (Wij) is the sum of all weights between vertex Vi and its vertexes neighbors (Vj)
* sum (Wij * Vj) is the sum of the product between vertex neighbor Vj and weight Wij for all neighborhood.
*
* This Laplacian Matrix is described in the paper:
* Desbrun M. et.al, Implicit fairing of irregular meshes using diffusion and curvature flow, SIGGRAPH '99, pag 317-324,
* New York, USA
*
* The computation of Laplace Beltrami operator on Hybrid Triangle/Quad Meshes is described in the paper:
* Pinzon A., Romero E., Shape Inflation With an Adapted Laplacian Operator For Hybrid Quad/Triangle Meshes,
* Conference on Graphics Patterns and Images, SIBGRAPI, 2013
*
* The computation of Differential Coordinates is described in the paper:
* Sorkine, O. Laplacian Surface Editing. Proceedings of the EUROGRAPHICS/ACM SIGGRAPH Symposium on Geometry Processing,
* 2004. p. 179-188.
*/
static void initLaplacianMatrix(LaplacianSystem *sys)
{
float v1[3], v2[3], v3[3], v4[3], no[3];
float w2, w3, w4;
int i, j, fi;
bool has_4_vert;
unsigned int idv1, idv2, idv3, idv4;
for (fi = 0; fi < sys->total_faces; fi++) {
const unsigned int *vidf = sys->faces[fi];
idv1 = vidf[0];
idv2 = vidf[1];
idv3 = vidf[2];
idv4 = vidf[3];
has_4_vert = vidf[3] ? 1 : 0;
if (has_4_vert) {
normal_quad_v3(no, sys->co[idv1], sys->co[idv2], sys->co[idv3], sys->co[idv4]);
add_v3_v3(sys->no[idv4], no);
i = 4;
}
else {
normal_tri_v3(no, sys->co[idv1], sys->co[idv2], sys->co[idv3]);
i = 3;
}
add_v3_v3(sys->no[idv1], no);
add_v3_v3(sys->no[idv2], no);
add_v3_v3(sys->no[idv3], no);
for (j = 0; j < i; j++) {
idv1 = vidf[j];
idv2 = vidf[(j + 1) % i];
idv3 = vidf[(j + 2) % i];
idv4 = has_4_vert ? vidf[(j + 3) % i] : 0;
copy_v3_v3(v1, sys->co[idv1]);
copy_v3_v3(v2, sys->co[idv2]);
copy_v3_v3(v3, sys->co[idv3]);
if (has_4_vert) {
copy_v3_v3(v4, sys->co[idv4]);
}
if (has_4_vert) {
w2 = (cotan_weight(v4, v1, v2) + cotan_weight(v3, v1, v2)) / 2.0f;
w3 = (cotan_weight(v2, v3, v1) + cotan_weight(v4, v1, v3)) / 2.0f;
w4 = (cotan_weight(v2, v4, v1) + cotan_weight(v3, v4, v1)) / 2.0f;
sys->delta[idv1][0] -= v4[0] * w4;
sys->delta[idv1][1] -= v4[1] * w4;
sys->delta[idv1][2] -= v4[2] * w4;
nlRightHandSideAdd(0, idv1, -v4[0] * w4);
nlRightHandSideAdd(1, idv1, -v4[1] * w4);
nlRightHandSideAdd(2, idv1, -v4[2] * w4);
nlMatrixAdd(idv1, idv4, -w4);
}
else {
w2 = cotan_weight(v3, v1, v2);
w3 = cotan_weight(v2, v3, v1);
w4 = 0.0f;
}
sys->delta[idv1][0] += v1[0] * (w2 + w3 + w4);
sys->delta[idv1][1] += v1[1] * (w2 + w3 + w4);
sys->delta[idv1][2] += v1[2] * (w2 + w3 + w4);
sys->delta[idv1][0] -= v2[0] * w2;
sys->delta[idv1][1] -= v2[1] * w2;
sys->delta[idv1][2] -= v2[2] * w2;
sys->delta[idv1][0] -= v3[0] * w3;
sys->delta[idv1][1] -= v3[1] * w3;
sys->delta[idv1][2] -= v3[2] * w3;
nlMatrixAdd(idv1, idv2, -w2);
nlMatrixAdd(idv1, idv3, -w3);
nlMatrixAdd(idv1, idv1, w2 + w3 + w4);
}
}
}
