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/****************************************************************************
* VCGLib o o *
* Visual and Computer Graphics Library o o *
* _ O _ *
* Copyright(C) 2004 \/)\/ *
* Visual Computing Lab /\/| *
* ISTI - Italian National Research Council | *
* \ *
* All rights reserved. *
* *
* This program is free software; you can redistribute it and/or modify *
* it under the terms of the GNU General Public License as published by *
* the Free Software Foundation; either version 2 of the License, or *
* (at your option) any later version. *
* *
* This program is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
* GNU General Public License (http://www.gnu.org/licenses/gpl.txt) *
* for more details. *
* *
****************************************************************************/
/****************************************************************************
History
$Log: not supported by cvs2svn $
Revision 1.8 2008/05/14 10:03:29 ganovelli
Point3f->Coordtype
Revision 1.7 2008/04/23 16:37:15 onnis
VertexCurvature method added.
Revision 1.6 2008/04/04 10:26:12 cignoni
Cleaned up names, now Kg() gives back Gaussian Curvature (k1*k2), while Kh() gives back Mean Curvature 1/2(k1+k2)
Revision 1.5 2008/03/25 11:00:56 ganovelli
fixed bugs sign of principal direction and mean curvature value
Revision 1.4 2008/03/17 11:29:59 ganovelli
taubin and desbrun estimates added (-> see vcg/simplex/vertex/component.h [component_ocf.h|component_occ.h ]
Revision 1.3 2006/02/27 18:02:57 ponchio
Area -> doublearea/2
added some typename
Revision 1.2 2005/10/25 09:17:41 spinelli
correct IsBorder
Revision 1.1 2005/02/22 16:40:29 ganovelli
created. This version writes the gaussian curvature on the Q() member of
the vertex
****************************************************************************/
#ifndef VCGLIB_UPDATE_CURVATURE_
#define VCGLIB_UPDATE_CURVATURE_
#include <vcg/space/index/grid_static_ptr.h>
#include <vcg/math/base.h>
#include <vcg/math/matrix.h>
#include <vcg/simplex/face/topology.h>
#include <vcg/simplex/face/pos.h>
#include <vcg/simplex/face/jumping_pos.h>
#include <vcg/container/simple_temporary_data.h>
#include <vcg/complex/trimesh/update/normal.h>
#include <vcg/complex/trimesh/point_sampling.h>
#include <vcg/complex/trimesh/append.h>
#include <vcg/complex/intersection.h>
#include <vcg/complex/trimesh/inertia.h>
#include <vcg/math/matrix33.h>
namespace vcg {
namespace tri {
/// \ingroup trimesh
/// \headerfile curvature.h vcg/complex/trimesh/update/curvature.h
/// \brief Management, updating and computation of per-vertex and per-face normals.
/**
This class is used to compute or update the normals that can be stored in the vertex or face component of a mesh.
*/
template <class MeshType>
class UpdateCurvature
{
public:
typedef typename MeshType::FaceType FaceType;
typedef typename MeshType::FacePointer FacePointer;
typedef typename MeshType::FaceIterator FaceIterator;
typedef typename MeshType::VertexIterator VertexIterator;
typedef typename MeshType::VertContainer VertContainer;
typedef typename MeshType::VertexType VertexType;
typedef typename MeshType::VertexPointer VertexPointer;
typedef vcg::face::VFIterator<FaceType> VFIteratorType;
typedef typename MeshType::CoordType CoordType;
typedef typename CoordType::ScalarType ScalarType;
private:
typedef struct AdjVertex {
VertexType * vert;
float doubleArea;
bool isBorder;
};
public:
/// \brief Compute principal direction and magniuto of curvature.
/*
Compute principal direction and magniuto of curvature as describe in the paper:
@InProceedings{bb33922,
author = "G. Taubin",
title = "Estimating the Tensor of Curvature of a Surface from a
Polyhedral Approximation",
booktitle = "International Conference on Computer Vision",
year = "1995",
pages = "902--907",
URL = "http://dx.doi.org/10.1109/ICCV.1995.466840",
bibsource = "http://www.visionbib.com/bibliography/describe440.html#TT32253",
*/
static void PrincipalDirections(MeshType &m) {
assert(m.HasVFTopology());
vcg::tri::UpdateNormals<MeshType>::PerVertexNormalized(m);
VertexIterator vi;
for (vi =m.vert.begin(); vi !=m.vert.end(); ++vi) {
if ( ! (*vi).IsD() && (*vi).VFp() != NULL) {
VertexType * central_vertex = &(*vi);
std::vector<float> weights;
std::vector<AdjVertex> vertices;
vcg::face::JumpingPos<FaceType> pos((*vi).VFp(), central_vertex);
// firstV is the first vertex of the 1ring neighboorhood
VertexType* firstV = pos.VFlip();
VertexType* tempV;
float totalDoubleAreaSize = 0.0f;
// compute the area of each triangle around the central vertex as well as their total area
do
{
// this bring the pos to the next triangle counterclock-wise
pos.FlipF();
pos.FlipE();
// tempV takes the next vertex in the 1ring neighborhood
tempV = pos.VFlip();
assert(tempV!=central_vertex);
AdjVertex v;
v.isBorder = pos.IsBorder();
v.vert = tempV;
v.doubleArea = vcg::DoubleArea(*pos.F());
totalDoubleAreaSize += v.doubleArea;
vertices.push_back(v);
}
while(tempV != firstV);
// compute the weights for the formula computing matrix M
for (int i = 0; i < vertices.size(); ++i) {
if (vertices[i].isBorder) {
weights.push_back(vertices[i].doubleArea / totalDoubleAreaSize);
} else {
weights.push_back(0.5f * (vertices[i].doubleArea + vertices[(i-1)%vertices.size()].doubleArea) / totalDoubleAreaSize);
}
assert(weights.back() < 1.0f);
}
// compute I-NN^t to be used for computing the T_i's
Matrix33<ScalarType> Tp;
for (int i = 0; i < 3; ++i)
Tp[i][i] = 1.0f - powf(central_vertex->cN()[i],2);
Tp[0][1] = Tp[1][0] = -1.0f * (central_vertex->N()[0] * central_vertex->cN()[1]);
Tp[1][2] = Tp[2][1] = -1.0f * (central_vertex->cN()[1] * central_vertex->cN()[2]);
Tp[0][2] = Tp[2][0] = -1.0f * (central_vertex->cN()[0] * central_vertex->cN()[2]);
// for all neighbors vi compute the directional curvatures k_i and the T_i
// compute M by summing all w_i k_i T_i T_i^t
Matrix33<ScalarType> tempMatrix;
Matrix33<ScalarType> M;
M.SetZero();
for (int i = 0; i < vertices.size(); ++i) {
CoordType edge = (central_vertex->cP() - vertices[i].vert->cP());
float curvature = (2.0f * (central_vertex->cN().dot(edge)) ) / edge.SquaredNorm();
CoordType T = (Tp*edge).normalized();
tempMatrix.ExternalProduct(T,T);
M += tempMatrix * weights[i] * curvature ;
}
// compute vector W for the Householder matrix
CoordType W;
CoordType e1(1.0f,0.0f,0.0f);
if ((e1 - central_vertex->cN()).SquaredNorm() > (e1 + central_vertex->cN()).SquaredNorm())
W = e1 - central_vertex->cN();
else
W = e1 + central_vertex->cN();
W.Normalize();
// compute the Householder matrix I - 2WW^t
Matrix33<ScalarType> Q;
Q.SetIdentity();
tempMatrix.ExternalProduct(W,W);
Q -= tempMatrix * 2.0f;
// compute matrix Q^t M Q
Matrix33<ScalarType> QtMQ = (Q.transpose() * M * Q);
CoordType bl = Q.GetColumn(0);
CoordType T1 = Q.GetColumn(1);
CoordType T2 = Q.GetColumn(2);
// find sin and cos for the Givens rotation
float s,c;
// Gabriel Taubin hint and Valentino Fiorin impementation
float qt21 = QtMQ[2][1];
float qt12 = QtMQ[1][2];
float alpha = QtMQ[1][1]-QtMQ[2][2];
float beta = QtMQ[2][1];
float h[2];
float delta = sqrtf(4.0f*powf(alpha, 2) +16.0f*powf(beta, 2));
h[0] = (2.0f*alpha + delta) / (2.0f*beta);
h[1] = (2.0f*alpha - delta) / (2.0f*beta);
float t[2];
float best_c, best_s;
float min_error = std::numeric_limits<ScalarType>::infinity();
for (int i=0; i<2; i++)
{
delta = sqrtf(powf(h[i], 2) + 4.0f);
t[0] = (h[i]+delta) / 2.0f;
t[1] = (h[i]-delta) / 2.0f;
for (int j=0; j<2; j++)
{
float squared_t = powf(t[j], 2);
float denominator = 1.0f + squared_t;
s = (2.0f*t[j]) / denominator;
c = (1-squared_t) / denominator;
float approximation = c*s*alpha + (powf(c, 2) - powf(s, 2))*beta;
float angle_similarity = fabs(acosf(c)/asinf(s));
float error = fabs(1.0f-angle_similarity)+fabs(approximation);
if (error<min_error)
{
min_error = error;
best_c = c;
best_s = s;
}
}
}
c = best_c;
s = best_s;
vcg::ndim::MatrixMNf minor2x2 (2,2);
vcg::ndim::MatrixMNf S (2,2);
// diagonalize M
minor2x2[0][0] = QtMQ[1][1];
minor2x2[0][1] = QtMQ[1][2];
minor2x2[1][0] = QtMQ[2][1];
minor2x2[1][1] = QtMQ[2][2];
S[0][0] = S[1][1] = c;
S[0][1] = s;
S[1][0] = -1.0f * s;
vcg::ndim::MatrixMNf StMS(S.transpose() * minor2x2 * S);
// compute curvatures and curvature directions
float Principal_Curvature1 = (3.0f * StMS[0][0]) - StMS[1][1];
float Principal_Curvature2 = (3.0f * StMS[1][1]) - StMS[0][0];
CoordType Principal_Direction1 = T1 * c - T2 * s;
CoordType Principal_Direction2 = T1 * s + T2 * c;
(*vi).PD1() = Principal_Direction1;
(*vi).PD2() = Principal_Direction2;
(*vi).K1() = Principal_Curvature1;
(*vi).K2() = Principal_Curvature2;
}
}
}
class AreaData
{
public:
float A;
};
/** Curvature meseaure as described in the paper:
Robust principal curvatures on Multiple Scales, Yong-Liang Yang, Yu-Kun Lai, Shi-Min Hu Helmut Pottmann
SGP 2004
If pointVSfaceInt==true the covariance is computed by montecarlo sampling on the mesh (faster)
If pointVSfaceInt==false the covariance is computed by (analytic)integration over the surface (slower)
*/
typedef vcg::GridStaticPtr <FaceType, ScalarType > MeshGridType;
typedef vcg::GridStaticPtr <VertexType, ScalarType > PointsGridType;
static void PrincipalDirectionsPCA(MeshType &m, ScalarType r, bool pointVSfaceInt = true) {
std::vector<VertexType*> closests;
std::vector<ScalarType> distances;
std::vector<CoordType> points;
VertexIterator vi;
ScalarType area;
MeshType tmpM;
typename std::vector<CoordType>::iterator ii;
vcg::tri::TrivialSampler<MeshType> vs;
MeshGridType mGrid;
PointsGridType pGrid;
// Fill the grid used
if(pointVSfaceInt){
area = Stat<MeshType>::ComputeMeshArea(m);
vcg::tri::SurfaceSampling<MeshType,vcg::tri::TrivialSampler<MeshType> >::Montecarlo(m,vs,1000 * area / (2*M_PI*r*r ));
vi = vcg::tri::Allocator<MeshType>::AddVertices(tmpM,m.vert.size());
for(int y = 0; y < m.vert.size(); ++y,++vi) (*vi).P() = m.vert[y].P();
pGrid.Set(tmpM.vert.begin(),tmpM.vert.end());
} else{ mGrid.Set(m.face.begin(),m.face.end()); }
for(vi = m.vert.begin(); vi != m.vert.end(); ++vi){
vcg::Matrix33<ScalarType> A,eigenvectors;
vcg::Point3<ScalarType> bp,eigenvalues;
int nrot;
// sample the neighborhood
if(pointVSfaceInt)
{
vcg::tri::GetInSphereVertex<
MeshType,
PointsGridType,std::vector<VertexType*>,
std::vector<ScalarType>,
std::vector<CoordType> >(tmpM,pGrid, (*vi).cP(),r ,closests,distances,points);
A.Covariance(points,bp);
A*=area*area/1000;
}
else{
IntersectionBallMesh<MeshType,ScalarType>( m ,vcg::Sphere3<ScalarType>((*vi).cP(),r),tmpM );
vcg::Point3<ScalarType> _bary;
vcg::tri::Inertia<MeshType>::Covariance(tmpM,_bary,A);
}
Jacobi(A, eigenvalues , eigenvectors, nrot);
// get the estimate of curvatures from eigenvalues and eigenvectors
// find the 2 most tangent eigenvectors (by finding the one closest to the normal)
int best = 0; ScalarType bestv = fabs( (*vi).cN().dot(eigenvectors.GetColumn(0).normalized()) );
for(int i = 1 ; i < 3; ++i){
ScalarType prod = fabs((*vi).cN().dot(eigenvectors.GetColumn(i).normalized()));
if( prod > bestv){bestv = prod; best = i;}
}
(*vi).PD1() = eigenvectors.GetColumn( (best+1)%3).normalized();
(*vi).PD2() = eigenvectors.GetColumn( (best+2)%3).normalized();
// project them to the plane identified by the normal
vcg::Matrix33<ScalarType> rot;
ScalarType angle = acos((*vi).PD1().dot((*vi).N()));
rot.SetRotateRad( - (M_PI*0.5 - angle),(*vi).PD1()^(*vi).N());
(*vi).PD1() = rot*(*vi).PD1();
angle = acos((*vi).PD2().dot((*vi).N()));
rot.SetRotateRad( - (M_PI*0.5 - angle),(*vi).PD2()^(*vi).N());
(*vi).PD2() = rot*(*vi).PD2();
// copmutes the curvature values
const ScalarType r5 = r*r*r*r*r;
const ScalarType r6 = r*r5;
(*vi).K1() = (2.0/5.0) * (4.0*M_PI*r5 + 15*eigenvalues[(best+2)%3]-45.0*eigenvalues[(best+1)%3])/(M_PI*r6);
(*vi).K2() = (2.0/5.0) * (4.0*M_PI*r5 + 15*eigenvalues[(best+1)%3]-45.0*eigenvalues[(best+2)%3])/(M_PI*r6);
if((*vi).K1() < (*vi).K2()) { std::swap((*vi).K1(),(*vi).K2());
std::swap((*vi).PD1(),(*vi).PD2());
}
}
}
/// \brief Computes the discrete gaussian curvature.
/** For further details, please, refer to: \n
- <em> Discrete Differential-Geometry Operators for Triangulated 2-Manifolds Mark Meyer,
Mathieu Desbrun, Peter Schroder, Alan H. Barr VisMath '02, Berlin </em>
*/
static void MeanAndGaussian(MeshType & m)
{
assert(HasFFAdjacency(m));
float area0, area1, area2, angle0, angle1, angle2;
FaceIterator fi;
VertexIterator vi;
typename MeshType::CoordType e01v ,e12v ,e20v;
SimpleTempData<VertContainer, AreaData> TDAreaPtr(m.vert);
SimpleTempData<VertContainer, typename MeshType::CoordType> TDContr(m.vert);
vcg::tri::UpdateNormals<MeshType>::PerVertexNormalized(m);
//Compute AreaMix in H (vale anche per K)
for(vi=m.vert.begin(); vi!=m.vert.end(); ++vi) if(!(*vi).IsD())
{
(TDAreaPtr)[*vi].A = 0.0;
(TDContr)[*vi] =typename MeshType::CoordType(0.0,0.0,0.0);
(*vi).Kh() = 0.0;
(*vi).Kg() = (float)(2.0 * M_PI);
}
for(fi=m.face.begin();fi!=m.face.end();++fi) if( !(*fi).IsD())
{
// angles
angle0 = math::Abs(Angle( (*fi).P(1)-(*fi).P(0),(*fi).P(2)-(*fi).P(0) ));
angle1 = math::Abs(Angle( (*fi).P(0)-(*fi).P(1),(*fi).P(2)-(*fi).P(1) ));
angle2 = M_PI-(angle0+angle1);
if((angle0 < M_PI/2) && (angle1 < M_PI/2) && (angle2 < M_PI/2)) // triangolo non ottuso
{
float e01 = SquaredDistance( (*fi).V(1)->cP() , (*fi).V(0)->cP() );
float e12 = SquaredDistance( (*fi).V(2)->cP() , (*fi).V(1)->cP() );
float e20 = SquaredDistance( (*fi).V(0)->cP() , (*fi).V(2)->cP() );
area0 = ( e20*(1.0/tan(angle1)) + e01*(1.0/tan(angle2)) ) / 8.0;
area1 = ( e01*(1.0/tan(angle2)) + e12*(1.0/tan(angle0)) ) / 8.0;
area2 = ( e12*(1.0/tan(angle0)) + e20*(1.0/tan(angle1)) ) / 8.0;
(TDAreaPtr)[(*fi).V(0)].A += area0;
(TDAreaPtr)[(*fi).V(1)].A += area1;
(TDAreaPtr)[(*fi).V(2)].A += area2;
}
else // obtuse
{
if(angle0 >= M_PI/2)
{
(TDAreaPtr)[(*fi).V(0)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 4.0;
(TDAreaPtr)[(*fi).V(1)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
(TDAreaPtr)[(*fi).V(2)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
}
else if(angle1 >= M_PI/2)
{
(TDAreaPtr)[(*fi).V(0)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
(TDAreaPtr)[(*fi).V(1)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 4.0;
(TDAreaPtr)[(*fi).V(2)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
}
else
{
(TDAreaPtr)[(*fi).V(0)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
(TDAreaPtr)[(*fi).V(1)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
(TDAreaPtr)[(*fi).V(2)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 4.0;
}
}
}
for(fi=m.face.begin();fi!=m.face.end();++fi) if( !(*fi).IsD() )
{
angle0 = math::Abs(Angle( (*fi).P(1)-(*fi).P(0),(*fi).P(2)-(*fi).P(0) ));
angle1 = math::Abs(Angle( (*fi).P(0)-(*fi).P(1),(*fi).P(2)-(*fi).P(1) ));
angle2 = M_PI-(angle0+angle1);
// Skip degenerate triangles.
if(angle0==0 || angle1==0 || angle1==0) continue;
e01v = ( (*fi).V(1)->cP() - (*fi).V(0)->cP() ) ;
e12v = ( (*fi).V(2)->cP() - (*fi).V(1)->cP() ) ;
e20v = ( (*fi).V(0)->cP() - (*fi).V(2)->cP() ) ;
TDContr[(*fi).V(0)] += ( e20v * (1.0/tan(angle1)) - e01v * (1.0/tan(angle2)) ) / 4.0;
TDContr[(*fi).V(1)] += ( e01v * (1.0/tan(angle2)) - e12v * (1.0/tan(angle0)) ) / 4.0;
TDContr[(*fi).V(2)] += ( e12v * (1.0/tan(angle0)) - e20v * (1.0/tan(angle1)) ) / 4.0;
(*fi).V(0)->Kg() -= angle0;
(*fi).V(1)->Kg() -= angle1;
(*fi).V(2)->Kg() -= angle2;
for(int i=0;i<3;i++)
{
if(vcg::face::IsBorder((*fi), i))
{
CoordType e1,e2;
vcg::face::Pos<FaceType> hp(&*fi, i, (*fi).V(i));
vcg::face::Pos<FaceType> hp1=hp;
hp1.FlipV();
e1=hp1.v->cP() - hp.v->cP();
hp1.FlipV();
hp1.NextB();
e2=hp1.v->cP() - hp.v->cP();
(*fi).V(i)->Kg() -= math::Abs(Angle(e1,e2));
}
}
}
for(vi=m.vert.begin(); vi!=m.vert.end(); ++vi) if(!(*vi).IsD() /*&& !(*vi).IsB()*/)
{
if((TDAreaPtr)[*vi].A<=std::numeric_limits<ScalarType>::epsilon())
{
(*vi).Kh() = 0;
(*vi).Kg() = 0;
}
else
{
(*vi).Kh() = (((TDContr)[*vi].dot((*vi).cN())>0)?1.0:-1.0)*((TDContr)[*vi] / (TDAreaPtr) [*vi].A).Norm();
(*vi).Kg() /= (TDAreaPtr)[*vi].A;
}
}
}
/// \brief Update the mean and the gaussian curvature of a vertex.
/**
The function uses the VF adiacency to walk around the vertex.
\return It will return the voronoi area around the vertex. If (norm == true) the mean and the gaussian curvature are normalized.
Based on the paper <a href="http://www2.in.tu-clausthal.de/~hormann/papers/Dyn.2001.OTU.pdf"> <em> "Optimizing 3d triangulations using discrete curvature analysis" </em> </a>
*/
static float VertexCurvature(VertexPointer v, bool norm = true)
{
// VFAdjacency required!
assert(FaceType::HasVFAdjacency());
assert(VertexType::HasVFAdjacency());
VFIteratorType vfi(v);
float A = 0;
v->Kh() = 0;
v->Kg() = 2 * M_PI;
while (!vfi.End()) {
if (!vfi.F()->IsD()) {
FacePointer f = vfi.F();
int i = vfi.I();
VertexPointer v0 = f->V0(i), v1 = f->V1(i), v2 = f->V2(i);
float ang0 = math::Abs(Angle(v1->P() - v0->P(), v2->P() - v0->P() ));
float ang1 = math::Abs(Angle(v0->P() - v1->P(), v2->P() - v1->P() ));
float ang2 = M_PI - ang0 - ang1;
float s01 = SquaredDistance(v1->P(), v0->P());
float s02 = SquaredDistance(v2->P(), v0->P());
// voronoi cell of current vertex
if (ang0 >= M_PI/2)
A += (0.5f * DoubleArea(*f) - (s01 * tan(ang1) + s02 * tan(ang2)) / 8.0 );
else if (ang1 >= M_PI/2)
A += (s01 * tan(ang0)) / 8.0;
else if (ang2 >= M_PI/2)
A += (s02 * tan(ang0)) / 8.0;
else // non obctuse triangle
A += ((s02 / tan(ang1)) + (s01 / tan(ang2))) / 8.0;
// gaussian curvature update
v->Kg() -= ang0;
// mean curvature update
ang1 = math::Abs(Angle(f->N(), v1->N()));
ang2 = math::Abs(Angle(f->N(), v2->N()));
v->Kh() += ( (math::Sqrt(s01) / 2.0) * ang1 +
(math::Sqrt(s02) / 2.0) * ang2 );
}
++vfi;
}
v->Kh() /= 4.0f;
if(norm) {
if(A <= std::numeric_limits<float>::epsilon()) {
v->Kh() = 0;
v->Kg() = 0;
}
else {
v->Kh() /= A;
v->Kg() /= A;
}
}
return A;
}
static void VertexCurvature(MeshType & m){
for(VertexIterator vi = m.vert.begin(); vi != m.vert.end(); ++vi)
VertexCurvature(&*vi,false);
}
/*
Compute principal curvature directions and value with normal cycle:
@inproceedings{CohMor03,
author = {Cohen-Steiner, David and Morvan, Jean-Marie },
booktitle = {SCG '03: Proceedings of the nineteenth annual symposium on Computational geometry},
title - {Restricted delaunay triangulations and normal cycle}
year = {2003}
}
*/
static void PrincipalDirectionsNormalCycles(MeshType & m){
assert(VertexType::HasVFAdjacency());
assert(FaceType::HasFFAdjacency());
assert(FaceType::HasFaceNormal());
typename MeshType::VertexIterator vi;
for(vi = m.vert.begin(); vi != m.vert.end(); ++vi)
if(!((*vi).IsD())){
vcg::Matrix33<ScalarType> m33;m33.SetZero();
face::JumpingPos<typename MeshType::FaceType> p((*vi).VFp(),&(*vi));
p.FlipE();
typename MeshType::VertexType * firstv = p.VFlip();
assert(p.F()->V(p.VInd())==&(*vi));
do{
if( p.F() != p.FFlip()){
Point3<ScalarType> normalized_edge = p.F()->V(p.F()->Next(p.VInd()))->cP() - (*vi).P();
ScalarType edge_length = normalized_edge.Norm();
normalized_edge/=edge_length;
Point3<ScalarType> n1 = p.F()->cN();n1.Normalize();
Point3<ScalarType> n2 = p.FFlip()->cN();n2.Normalize();
ScalarType n1n2 = (n1 ^ n2).dot(normalized_edge);
n1n2 = math::Max<ScalarType >(math::Min<ScalarType> ( 1.0,n1n2),-1.0);
ScalarType beta = math::Asin(n1n2);
m33[0][0] += beta*edge_length*normalized_edge[0]*normalized_edge[0];
m33[0][1] += beta*edge_length*normalized_edge[1]*normalized_edge[0];
m33[1][1] += beta*edge_length*normalized_edge[1]*normalized_edge[1];
m33[0][2] += beta*edge_length*normalized_edge[2]*normalized_edge[0];
m33[1][2] += beta*edge_length*normalized_edge[2]*normalized_edge[1];
m33[2][2] += beta*edge_length*normalized_edge[2]*normalized_edge[2];
}
p.NextFE();
}while(firstv != p.VFlip());
if(m33.Determinant()==0.0){ // degenerate case
(*vi).K1() = (*vi).K2() = 0.0; continue;}
m33[1][0] = m33[0][1];
m33[2][0] = m33[0][2];
m33[2][1] = m33[1][2];
Point3<ScalarType> lambda;
Matrix33<ScalarType> vect;
int n_rot;
Jacobi(m33,lambda, vect,n_rot);
vect.transposeInPlace();
ScalarType normal = std::numeric_limits<ScalarType>::min();
int normI = 0;
for(int i = 0; i < 3; ++i)
if( fabs((*vi).N().Normalize().dot(vect.GetRow(i))) > normal )
{
normal= fabs((*vi).N().Normalize().dot(vect.GetRow(i)));
normI = i;
}
int maxI = (normI+2)%3;
int minI = (normI+1)%3;
if(fabs(lambda[maxI]) < fabs(lambda[minI])) std::swap(maxI,minI);
(*vi).PD1() = *(Point3<ScalarType>*)(& vect[maxI][0]);
(*vi).PD2() = *(Point3<ScalarType>*)(& vect[minI][0]);
(*vi).K1() = lambda[maxI];
(*vi).K2() = lambda[minI];
}
}
};
}
}
#endif
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