GW_Float GW_Mesh::GetPerimeter( T_VertexList& boundary, GW_Bool bCyclic )
{
GW_Float rPerimeter = 0;
GW_Vertex* pPrev = NULL;
GW_Vertex* pVert = NULL;
for( IT_VertexList it = boundary.begin(); it!=boundary.end(); ++it )
{
pVert = *it;
if( pPrev!=NULL )
{
rPerimeter += (pPrev->GetPosition()-pVert->GetPosition()).Norm();
}
pPrev = pVert;
}
if( boundary.size()>1 && bCyclic )
{
rPerimeter += (boundary.front()->GetPosition()-pVert->GetPosition()).Norm();
}
return rPerimeter;
}
/*------------------------------------------------------------------------------*/
void GW_TriangularInterpolation_Cubic::ComputeLocalGradient( GW_GeodesicVertex& Vert )
{
/* compute the total angle */
GW_Vector3D PrevEdge;
GW_Float rTotalAngle = 0;
for( GW_VertexIterator it=Vert.BeginVertexIterator(); it!=Vert.EndVertexIterator(); ++it )
{
GW_Vertex* pVert = *it;
GW_ASSERT( pVert!=NULL );
if( it==Vert.BeginVertexIterator() )
{
PrevEdge = pVert->GetPosition() - Vert.GetPosition();
PrevEdge.Normalize();
}
else
{
GW_Vector3D NextEdge = pVert->GetPosition() - Vert.GetPosition();
NextEdge.Normalize();
rTotalAngle += acos( NextEdge*PrevEdge );
PrevEdge = NextEdge;
}
}
/* matrix and RHS for least square minimusation */
GW_Float M[2][2] = {{0,0},{0,0}};
GW_Float b[2] = {0,0};
GW_Float rCurAngle = 0;
PrevEdge.SetZero();
for( GW_VertexIterator it=Vert.BeginVertexIterator(); it!=Vert.EndVertexIterator(); ++it )
{
GW_GeodesicVertex* pVert = (GW_GeodesicVertex*) *it;
GW_ASSERT( pVert!=NULL );
GW_Vector3D Edge = pVert->GetPosition() - Vert.GetPosition();
GW_Float a = Edge.Norm();
Edge /= a;
if( it!=Vert.BeginVertexIterator() )
{
/* update the angle */
rCurAngle += acos( Edge*PrevEdge );
}
/* directional gradient estimation */
GW_Float delta = (pVert->GetDistance() - Vert.GetDistance())/a;
/* coordonate of the edge on (u,v) [flatened coords] */
GW_Float eu = a*cos( rCurAngle/rTotalAngle );
GW_Float ev = a*sin( rCurAngle/rTotalAngle );
/* update the matrix */
M[0][0] += eu*eu;
M[0][1] += eu*ev;
M[1][0] += eu*ev;
M[1][1] += ev*ev;
b[0] += delta*eu;
b[1] += delta*ev;
PrevEdge = Edge;
}
/* invert the system */
GW_Float det = M[0][0]*M[1][1] - M[0][1]*M[1][0];
GW_ASSERT( det!=0 );
GW_Float gu = 1/det * ( M[1][1]*b[0] - M[0][1]*b[1] );
GW_Float gv = 1/det * (-M[1][0]*b[0] + M[0][0]*b[1] );
/* set the gradient in local coords for each surrounding face */
rCurAngle = 0;
for( GW_FaceIterator it = Vert.BeginFaceIterator(); it!=Vert.EndFaceIterator(); ++it )
{
GW_GeodesicFace* pFace = (GW_GeodesicFace*) *it;
GW_ASSERT( pFace!=NULL );
GW_Vertex* pVert1 = it.GetLeftVertex(); GW_ASSERT( pVert1!=NULL );
GW_Vertex* pVert2 = it.GetRightVertex(); GW_ASSERT( pVert1!=NULL );
GW_Vector3D e1 = pVert1->GetPosition() - Vert.GetPosition();
GW_Vector3D e2 = pVert2->GetPosition() - Vert.GetPosition();
GW_Float a1 = e1.Norm();
GW_Float a2 = e2.Norm();
e1 /= a1;
e2 /= a2;
GW_Float rInnerAngle = acos( e1*e2 );
/* flattened position of the two vertex */
GW_Float p1[2], p2[2];
p1[0] = cos( rCurAngle );
p1[1] = sin( rCurAngle );
p2[0] = cos( rCurAngle+rInnerAngle );
p2[1] = sin( rCurAngle+rInnerAngle );
/* we have grad = gu*u + gv*v
we are searching for grad = g1*p1 + g2*p2, so:
gu = g1*<p1,u> + g2*<p2,u>
gv = g1*<p1,v> + g2*<p2,v>
i.e.
|p1[0] p2[0]| |g1| |gu|
|p1[1] p2[1]|*|g2| = |gv|
*/
det = p1[0]*p2[1]-p1[1]*p2[0];
GW_ASSERT( det!=0 );
GW_Float g1 = 1/det * ( p2[1]*gu - p2[0]*gv );
GW_Float g2 = 1/det * (-p1[1]*gu + p1[0]*gv );
/* now compute the gradient in world coords */
GW_Vector3D LocGrad = e1*g1 + e2*g2;
GW_TriangularInterpolation_ABC* pInterp = pFace->GetTriangularInterpolation();
if( pInterp==NULL )
{
pInterp = new GW_TriangularInterpolation_Cubic;
pFace->SetTriangularInterpolation( *pInterp );
}
GW_ASSERT( pInterp->GetType()==kCubicTriangulationInterpolation );
((GW_TriangularInterpolation_Cubic*) pInterp)->SetLocalGradient( LocGrad, *pFace, Vert );
rCurAngle += rInnerAngle;
}
}
/*------------------------------------------------------------------------------*/
void GW_Vertex::ComputeCurvatureDirections( GW_Float rArea )
{
GW_Vector3D CurEdge;
GW_Vector3D CurEdgeNormalized;
GW_Float rCurEdgeLength;
GW_Float rCotan;
GW_Vector3D TempEdge1, TempEdge2;
GW_Vertex* pTempVert = NULL;
GW_Float rDotP;
/***********************************************************************************/
/* compute the two curvature directions */
/* (v1,v2) form a basis of the tangent plante */
GW_Vector3D v1 = Normal_ ^ GW_Vector3D(0,0,1);
GW_Float rNorm = v1.Norm();
if( rNorm<GW_EPSILON )
{
/* orthogonalize using another direction */
v1 = Normal_ ^ GW_Vector3D(0,1,0);
rNorm = v1.Norm();
GW_ASSERT( rNorm>GW_EPSILON );
}
v1 /= rNorm;
GW_Vector3D v2 = Normal_ ^ v1;
/* now we must find the curvature matrix entry by minimising a mean square problem
the 3 entry of the symetric curvature matrix in (v1,v2) basis are (a,b,c), stored in vector x.
IMPORTANT : we must ensure a<c, so that eigenvalues are in correct order. */
GW_Float a = 0, b = 0, c = 0; // the vector (a,b,c) we are searching. We use a+c=2*MeanCurv so we don't take care of c.
GW_Float D[2] = {0,0}; // the right side of the equation.
GW_Float M00 = 0, M11 = 0, M01 = 0; // the positive-definite matrix entries of the mean-square problem.
GW_Float d1, d2; // decomposition of current edge on (v1,v2) basis
GW_Float w, k; // current weight and current approximated directional curvature.
/* now build the matrix by iterating around the vertex */
for( GW_VertexIterator it = this->BeginVertexIterator(); it!=this->EndVertexIterator(); ++it )
{
GW_Vertex* pVert = *it;
GW_ASSERT( pVert!=NULL );
CurEdge = pVert->GetPosition() - this->GetPosition();
rCurEdgeLength = CurEdge.Norm();
CurEdgeNormalized = CurEdge/rCurEdgeLength;
/* here we store the value of the sum of the cotan */
rCotan = 0;
/* compute projection onto v1,v2 basis */
d1 = v1*CurEdge;
d2 = v2*CurEdge;
rNorm = sqrt(d1*d1 + d2*d2);
if( rNorm>0 )
{
d1 /= rNorm;
d2 /= rNorm;
}
/* compute left angle */
pTempVert = it.GetLeftVertex();
if( pTempVert!=NULL )
{
TempEdge1 = this->GetPosition() - pTempVert->GetPosition();
TempEdge2 = pVert->GetPosition() - pTempVert->GetPosition();
TempEdge1.Normalize();
TempEdge2.Normalize();
/* we use tan(acos(x))=sqrt(1-x^2)/x */
rDotP = TempEdge1*TempEdge2;
if( rDotP!=1 && rDotP!=-1 )
rCotan += rDotP/sqrt(1-rDotP*rDotP);
}
/* compute right angle AND Gaussian contribution */
pTempVert = it.GetRightVertex();
if( pTempVert!=NULL )
{
TempEdge1 = this->GetPosition() - pTempVert->GetPosition();
TempEdge2 = pVert->GetPosition() - pTempVert->GetPosition();
TempEdge1.Normalize();
TempEdge2.Normalize();
/* we use tan(acos(x))=sqrt(1-x^2)/x */
rDotP = TempEdge1*TempEdge2;
if( rDotP!=1 && rDotP!=-1 )
rCotan += (GW_Float) rDotP/sqrt(1-rDotP*rDotP);
}
GW_CHECK_MATHSBIT();
/*compute weight */
w = 0.125/rArea*rCotan*rCurEdgeLength*rCurEdgeLength;
/* compute directional curvature */
k = -2*(CurEdge*Normal_)/(rCurEdgeLength*rCurEdgeLength);
k = k-(rMinCurv_+rMaxCurv_); // modified by the fact that we use a+c=2*MeanCurv
/* add contribution to M matrix and D vector*/
M00 += w*(d1*d1-d2*d2)*(d1*d1-d2*d2);
M11 += 4*w*d1*d1*d2*d2;
M01 += 2*w*(d1*d1-d2*d2)*d1*d2;
D[0] += w*k*( d1*d1-d2*d2);
D[1] += 2*w*k*d1*d2;
}
GW_CHECK_MATHSBIT();
/* solve the system */
GW_Float rDet = M00*M11 - M01*M01;
if( rDet!=0 )
{
/* The inverse matrix is : | M11 -M01|
1/rDet * |-M01 M00| */
a = 1/rDet * ( M11*D[0] - M01*D[1] );
b = 1/rDet * (-M01*D[0] + M00*D[1] );
}
c = (rMinCurv_+rMaxCurv_) - a;
// GW_ORDER(a,c);
/* compute the direction via Givens rotations */
GW_Float rTheta;
if( GW_ABS(c-a) < GW_EPSILON )
{
if( b==0 )
rTheta = 0;
else
rTheta = GW_HALFPI;
}
else
{
rTheta = (GW_Float) 2.0f*b/(c-a);
rTheta = (GW_Float) 0.5f*atan(rTheta);
}
GW_CHECK_MATHSBIT();
CurvDirMin_ = v1*cos(rTheta) - v2*sin(rTheta);
CurvDirMax_ = v1*sin(rTheta) + v2*cos(rTheta);
GW_Float vp1 = 0, vp2 = 0;
if( rTheta!=0 )
{
GW_Float r1 = a*cos(rTheta) - b*sin(rTheta);
GW_Float r2 = b*cos(rTheta) - c*sin(rTheta);
r1 =(GW_Float) r1/cos(rTheta);
r2 = (GW_Float) -r2/sin(rTheta);
vp1 = r1;
// GW_ASSERT( GW_ABS(r1-r2)<0.001*GW_ABS(r1) );
r1 = (GW_Float) a*sin(rTheta) + b*cos(rTheta);
r2 = (GW_Float) b*sin(rTheta) + c*cos(rTheta);
r1 = (GW_Float) r1/sin(rTheta);
r2 = (GW_Float) r2/cos(rTheta);
vp2 = r2;
// GW_ASSERT( GW_ABS(r1-r2)<0.001*GW_ABS(r1) );
}
if( vp1>vp2 )
{
GW_Vector3D vtemp = CurvDirMin_;
CurvDirMin_ = CurvDirMax_;
CurvDirMax_ = vtemp;
}
}
/*------------------------------------------------------------------------------*/
void GW_GeodesicDisplayer::DisplayPath( GW_GeodesicPath& CurPath, GW_Vector3D& Color, GW_Float rLineWidth )
{
glLineWidth( (GLfloat) rLineWidth );
glPointSize( (GLfloat) rLineWidth );
T_GeodesicPointList& PointList = CurPath.GetPointList();
glDisable( GL_LIGHTING );
glColor( Color );
glBegin( GL_LINE_STRIP );
for( IT_GeodesicPointList it = PointList.begin(); it!=PointList.end(); ++it )
{
GW_GeodesicPoint* pPoint = *it;
GW_ASSERT( pPoint->GetVertex1()!=NULL );
GW_ASSERT( pPoint->GetVertex2()!=NULL );
GW_Vector3D Pos = pPoint->GetVertex1()->GetPosition()*pPoint->GetCoord() +
pPoint->GetVertex2()->GetPosition()*(1-pPoint->GetCoord());
glColor( Color );
glVertex( Pos );
GW_GeodesicFace* pFace = pPoint->GetCurFace();
GW_ASSERT( pFace!=NULL );
T_SubPointVector& SubPointVector = pPoint->GetSubPointVector();
GW_Vector3D& v0 = pPoint->GetVertex1()->GetPosition();
GW_Vector3D& v1 = pPoint->GetVertex2()->GetPosition();
GW_Vertex* pLastVert = pFace->GetVertex( *pPoint->GetVertex1(), *pPoint->GetVertex2() );
GW_ASSERT( pLastVert!=NULL );
GW_Vector3D& v2 = pLastVert->GetPosition();
for( IT_SubPointVector it=SubPointVector.begin(); it!=SubPointVector.end(); ++it )
{
GW_Vector3D& coord = *it;
Pos = v0*coord[0] + v1*coord[1] + v2*coord[2];
glVertex( Pos );
}
}
glEnd();
#if 0
glColor3f( 0,0,0 );
glPointSize(4);
glBegin( GL_POINTS );
for( IT_GeodesicPointList it = PointList.begin(); it!=PointList.end(); ++it )
{
GW_GeodesicPoint* pPoint = *it;
GW_ASSERT( pPoint->GetVertex1()!=NULL );
GW_ASSERT( pPoint->GetVertex2()!=NULL );
GW_Vector3D Pos = pPoint->GetVertex1()->GetPosition()*pPoint->GetCoord() +
pPoint->GetVertex2()->GetPosition()*(1-pPoint->GetCoord());
glVertex( Pos );
GW_GeodesicFace* pFace = pPoint->GetCurFace();
GW_ASSERT( pFace!=NULL );
T_SubPointVector& SubPointVector = pPoint->GetSubPointVector();
GW_Vector3D& v0 = pPoint->GetVertex1()->GetPosition();
GW_Vector3D& v1 = pPoint->GetVertex2()->GetPosition();
GW_Vertex* pLastVert = pFace->GetVertex( *pPoint->GetVertex1(), *pPoint->GetVertex2() );
GW_ASSERT( pLastVert!=NULL );
GW_Vector3D& v2 = pLastVert->GetPosition();
for( IT_SubPointVector it=SubPointVector.begin(); it!=SubPointVector.end(); ++it )
{
GW_Vector3D& coord = *it;
Pos = v0*coord[0] + v1*coord[1] + v2*coord[2];
glVertex( Pos );
}
}
glEnd();
glEnable( GL_LIGHTING );
glLineWidth( 1 );
#endif
}
/*------------------------------------------------------------------------------*/
void GW_Vertex::ComputeNormalAndCurvature( GW_Float& rArea )
{
GW_Vector3D CurEdge;
GW_Vector3D CurEdgeNormalized;
GW_Float rCurEdgeLength;
GW_Float rCotan;
GW_Vector3D TempEdge1, TempEdge2;
GW_Float rTempEdge1Length, rTempEdge2Length;
GW_Float rAngle, rInnerAngle;
GW_Vertex* pTempVert = NULL;
GW_Float rDotP;
Normal_.SetZero();
rArea = 0;
GW_Float rGaussianCurv = 0;
for( GW_VertexIterator it = this->BeginVertexIterator(); it!=this->EndVertexIterator(); ++it )
{
GW_Vertex* pVert = *it;
GW_ASSERT( pVert!=NULL );
CurEdge = pVert->GetPosition() - this->GetPosition();
rCurEdgeLength = CurEdge.Norm();
CurEdgeNormalized = CurEdge/rCurEdgeLength;
/* here we store the value of the sum of the cotan */
rCotan = 0;
/* compute left angle */
pTempVert = it.GetLeftVertex();
if( pTempVert!=NULL )
{
TempEdge1 = this->GetPosition() - pTempVert->GetPosition();
TempEdge2 = pVert->GetPosition() - pTempVert->GetPosition();
TempEdge1.Normalize();
TempEdge2.Normalize();
/* we use tan(acos(x))=sqrt(1-x^2)/x */
rDotP = TempEdge1*TempEdge2;
if( rDotP!=1 && rDotP!=-1 )
rCotan += (GW_Float) rDotP/((GW_Float) sqrt(1-rDotP*rDotP));
}
/* compute right angle AND Gaussian contribution */
pTempVert = it.GetRightVertex();
if( pTempVert!=NULL )
{
TempEdge1 = this->GetPosition() - pTempVert->GetPosition();
TempEdge2 = pVert->GetPosition() - pTempVert->GetPosition();
rTempEdge1Length = TempEdge1.Norm();
rTempEdge2Length = TempEdge2.Norm();
TempEdge1 /= rTempEdge1Length;
TempEdge2 /= rTempEdge2Length;
rAngle = (GW_Float) acos( -(TempEdge1 * CurEdgeNormalized) );
/* we use tan(acos(x))=sqrt(1-x^2)/x */
rDotP = TempEdge1*TempEdge2;
rInnerAngle = (GW_Float) acos( rDotP );
if( rDotP!=1 && rDotP!=-1 )
rCotan += rDotP/((GW_Float) sqrt(1-rDotP*rDotP));
rGaussianCurv += rAngle;
/* compute the contribution to area, testing for special obtuse angle */
if( rAngle<GW_HALFPI // condition on 1st angle
&& rInnerAngle<GW_HALFPI // condition on 2nd angle
&& (GW_PI-rAngle-rInnerAngle)<GW_HALFPI ) // condition on 3rd angle
{
/* non-obtuse : 1/8*( |PR|²cot(Q)+|PQ|²cot(R) ) where P=this, Q=pVert, R=pTempVert */
rArea += ( rCurEdgeLength*rCurEdgeLength*rDotP/sqrt(1-rDotP*rDotP)
+ rTempEdge1Length*rTempEdge1Length/tan(GW_PI-rAngle-rInnerAngle) )*0.125;
}
else if( rAngle>=GW_HALFPI )
{
/* obtuse at the central vertex : 0.5*area(T) */
rArea += 0.25*rCurEdgeLength*rTempEdge1Length* ~(CurEdgeNormalized^TempEdge1);
}
else
{
/* obtuse at one of side vertex */
rArea += 0.125*rCurEdgeLength*rTempEdge1Length* ~(CurEdgeNormalized^TempEdge1);
}
}
GW_CHECK_MATHSBIT();
/* add the contribution to Normal */
Normal_ -= CurEdge*rCotan;
}
GW_CHECK_MATHSBIT();
GW_ASSERT( rArea!=0 ); // remove this !
/* the Gaussian curv */
rGaussianCurv = (GW_TWOPI - rGaussianCurv)/rArea;
/* compute Normal and mean curv */
Normal_ /= 4.0*rArea;
GW_Float rMeanCurv = Normal_.Norm();
if( GW_ABS(rMeanCurv)>GW_EPSILON )
{
GW_Vector3D Normal = Normal_/rMeanCurv;
/* see if we need to flip the normal */
this->BuildRawNormal();
if( Normal*Normal_<0 )
Normal_ = -Normal;
else
Normal_ = Normal;
}
else
{
/* we must use another method to compute normal */
this->BuildRawNormal();
}
GW_Vertex::rTotalArea_ += rArea;
/* compute the two curv values */
GW_Float rDelta = rMeanCurv*rMeanCurv - rGaussianCurv;
if( rDelta<0 )
rDelta = 0;
rDelta = sqrt(rDelta);
rMinCurv_ = rMeanCurv - rDelta;
rMaxCurv_ = rMeanCurv + rDelta;
}
