/*------------------------------------------------------------------------------*/
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;
}
}
