/****************************************************************************
* VCGLib                                                            o o     *
* Visual and Computer Graphics Library                            o     o   *
*                                                                _   O  _   *
* Copyright(C) 2004                                                \/)\/    *
* Visual Computing Lab                                            /\/|      *
* ISTI - Italian National Research Council                           |      *
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* (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.                                                         *
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/****************************************************************************
  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/algorithms/update/normal.h>
#include <vcg/complex/algorithms/point_sampling.h>
#include <vcg/complex/append.h>
#include <vcg/complex/algorithms/intersection.h>
#include <vcg/complex/algorithms/inertia.h>
#include <vcg/math/matrix33.h>


namespace vcg {
namespace tri {

/// \ingroup trimesh

/// \headerfile curvature.h vcg/complex/algorithms/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:
                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 (size_t 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 (size_t 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 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,vcg::CallBackPos * cb = NULL) {
			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(size_t 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()); }
				int jj = 0;
				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());
					if (cb) 
					{
						(*cb)(int(100.0f * (float)jj / (float)m.vn),"Vertices Analysis");	
						++jj;
					}													}
			}


		}
 /// \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 = std::max(std::min( ScalarType(1.0),n1n2),ScalarType(-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
