Matrix3<Real> ImplicitSurface<Real>::GetHessian (const Vector3<Real>& rkP)
const
{
Real fFXX = FXX(rkP);
Real fFXY = FXY(rkP);
Real fFXZ = FXZ(rkP);
Real fFYY = FYY(rkP);
Real fFYZ = FYZ(rkP);
Real fFZZ = FZZ(rkP);
return Matrix3<Real>(fFXX,fFXY,fFXZ,fFXY,fFYY,fFYZ,fFXZ,fFYZ,fFZZ);
}
Matrix3<Real> ImplicitSurface<Real>::GetHessian (const Vector3<Real>& pos)
const
{
Real fxx = FXX(pos);
Real fxy = FXY(pos);
Real fxz = FXZ(pos);
Real fyy = FYY(pos);
Real fyz = FYZ(pos);
Real fzz = FZZ(pos);
return Matrix3<Real>(fxx, fxy, fxz, fxy, fyy, fyz, fxz, fyz, fzz);
}
bool ImplicitSurface<Real>::ComputePrincipalCurvatureInfo (
const Vector3<Real>& rkP, Real& rfCurv0, Real& rfCurv1,
Vector3<Real>& rkDir0, Vector3<Real>& rkDir1)
{
// Principal curvatures and directions for implicitly defined surfaces
// F(x,y,z) = 0.
//
// DF = (Fx,Fy,Fz), L = Length(DF)
//
// D^2 F = +- -+
// | Fxx Fxy Fxz |
// | Fxy Fyy Fyz |
// | Fxz Fyz Fzz |
// +- -+
//
// adj(D^2 F) = +- -+
// | Fyy*Fzz-Fyz*Fyz Fyz*Fxz-Fxy*Fzz Fxy*Fyz-Fxz*Fyy |
// | Fyz*Fxz-Fxy*Fzz Fxx*Fzz-Fxz*Fxz Fxy*Fxz-Fxx*Fyz |
// | Fxy*Fyz-Fxz*Fyy Fxy*Fxz-Fxx*Fyz Fxx*Fyy-Fxy*Fxy |
// +- -+
//
// Gaussian curvature = [DF^t adj(D^2 F) DF]/L^4
//
// Mean curvature = 0.5*[trace(D^2 F)/L - (DF^t D^2 F DF)/L^3]
// first derivatives
Real fFX = FX(rkP);
Real fFY = FY(rkP);
Real fFZ = FZ(rkP);
Real fL = Math<Real>::Sqrt(fFX*fFX + fFY*fFY + fFZ*fFZ);
if (fL <= Math<Real>::ZERO_TOLERANCE)
{
return false;
}
Real fFXFX = fFX*fFX;
Real fFXFY = fFX*fFY;
Real fFXFZ = fFX*fFZ;
Real fFYFY = fFY*fFY;
Real fFYFZ = fFY*fFZ;
Real fFZFZ = fFZ*fFZ;
Real fInvL = ((Real)1.0)/fL;
Real fInvL2 = fInvL*fInvL;
Real fInvL3 = fInvL*fInvL2;
Real fInvL4 = fInvL2*fInvL2;
// second derivatives
Real fFXX = FXX(rkP);
Real fFXY = FXY(rkP);
Real fFXZ = FXZ(rkP);
Real fFYY = FYY(rkP);
Real fFYZ = FYZ(rkP);
Real fFZZ = FZZ(rkP);
// mean curvature
Real fMCurv = ((Real)0.5)*fInvL3*(fFXX*(fFYFY+fFZFZ) + fFYY*(fFXFX+fFZFZ)
+ fFZZ*(fFXFX+fFYFY)
- ((Real)2.0)*(fFXY*fFXFY+fFXZ*fFXFZ+fFYZ*fFYFZ));
// Gaussian curvature
Real fGCurv = fInvL4*(fFXFX*(fFYY*fFZZ-fFYZ*fFYZ)
+ fFYFY*(fFXX*fFZZ-fFXZ*fFXZ) + fFZFZ*(fFXX*fFYY-fFXY*fFXY)
+ ((Real)2.0)*(fFXFY*(fFXZ*fFYZ-fFXY*fFZZ)
+ fFXFZ*(fFXY*fFYZ-fFXZ*fFYY)
+ fFYFZ*(fFXY*fFXZ-fFXX*fFYZ)));
// solve for principal curvatures
Real fDiscr = Math<Real>::Sqrt(Math<Real>::FAbs(fMCurv*fMCurv-fGCurv));
rfCurv0 = fMCurv - fDiscr;
rfCurv1 = fMCurv + fDiscr;
Real fM00 = ((-(Real)1.0 + fFXFX*fInvL2)*fFXX)*fInvL + (fFXFY*fFXY)*fInvL3
+ (fFXFZ*fFXZ)*fInvL3;
Real fM01 = ((-(Real)1.0 + fFXFX*fInvL2)*fFXY)*fInvL + (fFXFY*fFYY)*fInvL3
+ (fFXFZ*fFYZ)*fInvL3;
Real fM02 = ((-(Real)1.0 + fFXFX*fInvL2)*fFXZ)*fInvL + (fFXFY*fFYZ)*fInvL3
+ (fFXFZ*fFZZ)*fInvL3;
Real fM10 = (fFXFY*fFXX)*fInvL3 + ((-(Real)1.0 + fFYFY*fInvL2)*fFXY)*fInvL
+ (fFYFZ*fFXZ)*fInvL3;
Real fM11 = (fFXFY*fFXY)*fInvL3 + ((-(Real)1.0 + fFYFY*fInvL2)*fFYY)*fInvL
+ (fFYFZ*fFYZ)*fInvL3;
Real fM12 = (fFXFY*fFXZ)*fInvL3 + ((-(Real)1.0 + fFYFY*fInvL2)*fFYZ)*fInvL
+ (fFYFZ*fFZZ)*fInvL3;
Real fM20 = (fFXFZ*fFXX)*fInvL3 + (fFYFZ*fFXY)*fInvL3 + ((-(Real)1.0
+ fFZFZ*fInvL2)*fFXZ)*fInvL;
Real fM21 = (fFXFZ*fFXY)*fInvL3 + (fFYFZ*fFYY)*fInvL3 + ((-(Real)1.0
+ fFZFZ*fInvL2)*fFYZ)*fInvL;
Real fM22 = (fFXFZ*fFXZ)*fInvL3 + (fFYFZ*fFYZ)*fInvL3 + ((-(Real)1.0
+ fFZFZ*fInvL2)*fFZZ)*fInvL;
// solve for principal directions
Real fTmp1 = fM00 + rfCurv0;
Real fTmp2 = fM11 + rfCurv0;
Real fTmp3 = fM22 + rfCurv0;
Vector3<Real> akU[3];
Real afLength[3];
akU[0].X() = fM01*fM12-fM02*fTmp2;
akU[0].Y() = fM02*fM10-fM12*fTmp1;
akU[0].Z() = fTmp1*fTmp2-fM01*fM10;
afLength[0] = akU[0].Length();
akU[1].X() = fM01*fTmp3-fM02*fM21;
akU[1].Y() = fM02*fM20-fTmp1*fTmp3;
akU[1].Z() = fTmp1*fM21-fM01*fM20;
afLength[1] = akU[1].Length();
akU[2].X() = fTmp2*fTmp3-fM12*fM21;
akU[2].Y() = fM12*fM20-fM10*fTmp3;
akU[2].Z() = fM10*fM21-fM20*fTmp2;
afLength[2] = akU[2].Length();
int iMaxIndex = 0;
Real fMax = afLength[0];
if (afLength[1] > fMax)
{
iMaxIndex = 1;
fMax = afLength[1];
}
if (afLength[2] > fMax)
{
iMaxIndex = 2;
}
Real fInvLength = ((Real)1.0)/afLength[iMaxIndex];
akU[iMaxIndex] *= fInvLength;
rkDir1 = akU[iMaxIndex];
rkDir0 = rkDir1.UnitCross(Vector3<Real>(fFX,fFY,fFZ));
return true;
}
bool ImplicitSurface<Real>::ComputePrincipalCurvatureInfo (
const Vector3<Real>& pos, Real& curv0, Real& curv1, Vector3<Real>& dir0,
Vector3<Real>& dir1)
{
// Principal curvatures and directions for implicitly defined surfaces
// F(x,y,z) = 0.
//
// DF = (Fx,Fy,Fz), L = Length(DF)
//
// D^2 F = +- -+
// | Fxx Fxy Fxz |
// | Fxy Fyy Fyz |
// | Fxz Fyz Fzz |
// +- -+
//
// adj(D^2 F) = +- -+
// | Fyy*Fzz-Fyz*Fyz Fyz*Fxz-Fxy*Fzz Fxy*Fyz-Fxz*Fyy |
// | Fyz*Fxz-Fxy*Fzz Fxx*Fzz-Fxz*Fxz Fxy*Fxz-Fxx*Fyz |
// | Fxy*Fyz-Fxz*Fyy Fxy*Fxz-Fxx*Fyz Fxx*Fyy-Fxy*Fxy |
// +- -+
//
// Gaussian curvature = [DF^t adj(D^2 F) DF]/L^4
//
// Mean curvature = 0.5*[trace(D^2 F)/L - (DF^t D^2 F DF)/L^3]
// first derivatives
Real fx = FX(pos);
Real fy = FY(pos);
Real fz = FZ(pos);
Real fLength = Math<Real>::Sqrt(fx*fx + fy*fy + fz*fz);
if (fLength <= Math<Real>::ZERO_TOLERANCE)
{
return false;
}
Real fxfx = fx*fx;
Real fxfy = fx*fy;
Real fxfz = fx*fz;
Real fyfy = fy*fy;
Real fyfz = fy*fz;
Real fzfz = fz*fz;
Real invLength = ((Real)1)/fLength;
Real invLength2 = invLength*invLength;
Real invLength3 = invLength*invLength2;
Real invLength4 = invLength2*invLength2;
// second derivatives
Real fxx = FXX(pos);
Real fxy = FXY(pos);
Real fxz = FXZ(pos);
Real fyy = FYY(pos);
Real fyz = FYZ(pos);
Real fzz = FZZ(pos);
// mean curvature
Real meanCurv = ((Real)0.5)*invLength3*(fxx*(fyfy + fzfz) +
fyy*(fxfx + fzfz) + fzz*(fxfx + fyfy) - ((Real)2)*(fxy*fxfy + fxz*fxfz +
fyz*fyfz));
// Gaussian curvature
Real gaussCurv = invLength4*(fxfx*(fyy*fzz - fyz*fyz)
+ fyfy*(fxx*fzz - fxz*fxz) + fzfz*(fxx*fyy - fxy*fxy)
+ ((Real)2)*(fxfy*(fxz*fyz - fxy*fzz) + fxfz*(fxy*fyz - fxz*fyy)
+ fyfz*(fxy*fxz - fxx*fyz)));
// solve for principal curvatures
Real discr = Math<Real>::Sqrt(Math<Real>::FAbs(meanCurv*meanCurv-gaussCurv));
curv0 = meanCurv - discr;
curv1 = meanCurv + discr;
Real m00 = ((-(Real)1 + fxfx*invLength2)*fxx)*invLength +
(fxfy*fxy)*invLength3 + (fxfz*fxz)*invLength3;
Real m01 = ((-(Real)1 + fxfx*invLength2)*fxy)*invLength +
(fxfy*fyy)*invLength3 + (fxfz*fyz)*invLength3;
Real m02 = ((-(Real)1 + fxfx*invLength2)*fxz)*invLength +
(fxfy*fyz)*invLength3 + (fxfz*fzz)*invLength3;
Real m10 = (fxfy*fxx)*invLength3 +
((-(Real)1 + fyfy*invLength2)*fxy)*invLength + (fyfz*fxz)*invLength3;
Real m11 = (fxfy*fxy)*invLength3 +
((-(Real)1 + fyfy*invLength2)*fyy)*invLength + (fyfz*fyz)*invLength3;
Real m12 = (fxfy*fxz)*invLength3 +
((-(Real)1 + fyfy*invLength2)*fyz)*invLength + (fyfz*fzz)*invLength3;
Real m20 = (fxfz*fxx)*invLength3 + (fyfz*fxy)*invLength3 +
((-(Real)1 + fzfz*invLength2)*fxz)*invLength;
Real m21 = (fxfz*fxy)*invLength3 + (fyfz*fyy)*invLength3 +
((-(Real)1 + fzfz*invLength2)*fyz)*invLength;
Real m22 = (fxfz*fxz)*invLength3 + (fyfz*fyz)*invLength3 +
((-(Real)1 + fzfz*invLength2)*fzz)*invLength;
// solve for principal directions
Real tmp1 = m00 + curv0;
Real tmp2 = m11 + curv0;
Real tmp3 = m22 + curv0;
Vector3<Real> U[3];
Real lengths[3];
U[0].X() = m01*m12-m02*tmp2;
U[0].Y() = m02*m10-m12*tmp1;
U[0].Z() = tmp1*tmp2-m01*m10;
lengths[0] = U[0].Length();
U[1].X() = m01*tmp3-m02*m21;
U[1].Y() = m02*m20-tmp1*tmp3;
U[1].Z() = tmp1*m21-m01*m20;
lengths[1] = U[1].Length();
U[2].X() = tmp2*tmp3-m12*m21;
U[2].Y() = m12*m20-m10*tmp3;
U[2].Z() = m10*m21-m20*tmp2;
lengths[2] = U[2].Length();
int maxIndex = 0;
Real maxValue = lengths[0];
if (lengths[1] > maxValue)
{
maxIndex = 1;
maxValue = lengths[1];
}
if (lengths[2] > maxValue)
{
maxIndex = 2;
}
invLength = ((Real)1)/lengths[maxIndex];
U[maxIndex] *= invLength;
dir1 = U[maxIndex];
dir0 = dir1.UnitCross(Vector3<Real>(fx, fy, fz));
return true;
}