void ParametricSurface<Real>::GetFrame (Real fU, Real fV,
Vector3<Real>& kPosition, Vector3<Real>& kTangent0,
Vector3<Real>& kTangent1, Vector3<Real>& kNormal) const
{
kPosition = GetPosition(fU,fV);
kTangent0 = GetDerivativeU(fU,fV);
kTangent1 = GetDerivativeV(fU,fV);
kTangent0.Normalize();
kTangent1.Normalize();
kNormal = kTangent0.UnitCross(kTangent1);
}
void ParametricSurface<Real>::GetFrame (Real u, Real v,
Vector3<Real>& position, Vector3<Real>& tangent0,
Vector3<Real>& tangent1, Vector3<Real>& normal) const
{
position = P(u, v);
tangent0 = PU(u, v);
tangent1 = PV(u, v);
tangent0.Normalize(); // T0
tangent1.Normalize(); // temporary T1 just to compute N
normal = tangent0.UnitCross(tangent1); // N
// The normalized first derivatives are not necessarily orthogonal.
// Recompute T1 so that {T0,T1,N} is an orthonormal set.
tangent1 = normal.Cross(tangent0);
}
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;
}
void ParametricSurface<Real>::ComputePrincipalCurvatureInfo (Real u, Real v,
Real& curv0, Real& curv1, Vector3<Real>& dir0,
Vector3<Real>& dir1)
{
// Tangents: T0 = (x_u,y_u,z_u), T1 = (x_v,y_v,z_v)
// Normal: N = Cross(T0,T1)/Length(Cross(T0,T1))
// Metric Tensor: G = +- -+
// | Dot(T0,T0) Dot(T0,T1) |
// | Dot(T1,T0) Dot(T1,T1) |
// +- -+
//
// Curvature Tensor: B = +- -+
// | -Dot(N,T0_u) -Dot(N,T0_v) |
// | -Dot(N,T1_u) -Dot(N,T1_v) |
// +- -+
//
// Principal curvatures k are the generalized eigenvalues of
//
// Bw = kGw
//
// If k is a curvature and w=(a,b) is the corresponding solution to
// Bw = kGw, then the principal direction as a 3D vector is d = a*U+b*V.
//
// Let k1 and k2 be the principal curvatures. The mean curvature
// is (k1+k2)/2 and the Gaussian curvature is k1*k2.
// Compute derivatives.
Vector3<Real> derU = PU(u,v);
Vector3<Real> derV = PV(u,v);
Vector3<Real> derUU = PUU(u,v);
Vector3<Real> derUV = PUV(u,v);
Vector3<Real> derVV = PVV(u,v);
// Compute the metric tensor.
Matrix2<Real> metricTensor;
metricTensor[0][0] = derU.Dot(derU);
metricTensor[0][1] = derU.Dot(derV);
metricTensor[1][0] = metricTensor[0][1];
metricTensor[1][1] = derV.Dot(derV);
// Compute the curvature tensor.
Vector3<Real> normal = derU.UnitCross(derV);
Matrix2<Real> curvatureTensor;
curvatureTensor[0][0] = -normal.Dot(derUU);
curvatureTensor[0][1] = -normal.Dot(derUV);
curvatureTensor[1][0] = curvatureTensor[0][1];
curvatureTensor[1][1] = -normal.Dot(derVV);
// Characteristic polynomial is 0 = det(B-kG) = c2*k^2+c1*k+c0.
Real c0 = curvatureTensor.Determinant();
Real c1 = ((Real)2)*curvatureTensor[0][1]* metricTensor[0][1] -
curvatureTensor[0][0]*metricTensor[1][1] -
curvatureTensor[1][1]*metricTensor[0][0];
Real c2 = metricTensor.Determinant();
// Principal curvatures are roots of characteristic polynomial.
Real temp = Math<Real>::Sqrt(Math<Real>::FAbs(c1*c1 - ((Real)4)*c0*c2));
Real mult = ((Real)0.5)/c2;
curv0 = -mult*(c1+temp);
curv1 = mult*(-c1+temp);
// Principal directions are solutions to (B-kG)w = 0,
// w1 = (b12-k1*g12,-(b11-k1*g11)) OR (b22-k1*g22,-(b12-k1*g12)).
Real a0 = curvatureTensor[0][1] - curv0*metricTensor[0][1];
Real a1 = curv0*metricTensor[0][0] - curvatureTensor[0][0];
Real length = Math<Real>::Sqrt(a0*a0 + a1*a1);
if (length >= Math<Real>::ZERO_TOLERANCE)
{
dir0 = a0*derU + a1*derV;
}
else
{
a0 = curvatureTensor[1][1] - curv0*metricTensor[1][1];
a1 = curv0*metricTensor[0][1] - curvatureTensor[0][1];
length = Math<Real>::Sqrt(a0*a0 + a1*a1);
if (length >= Math<Real>::ZERO_TOLERANCE)
{
dir0 = a0*derU + a1*derV;
}
else
{
// Umbilic (surface is locally sphere, any direction principal).
dir0 = derU;
}
}
dir0.Normalize();
// Second tangent is cross product of first tangent and normal.
dir1 = dir0.Cross(normal);
}
void ParametricSurface<Real>::ComputePrincipalCurvatureInfo (Real fU, Real fV,
Real& rfCurv0, Real& rfCurv1, Vector3<Real>& rkDir0,
Vector3<Real>& rkDir1)
{
// Tangents: U = (x_s,y_s,z_s), V = (x_t,y_t,z_t)
// Normal: N = Cross(U,V)/Length(Cross(U,V))
// Metric Tensor: G = +- -+
// | Dot(U,U) Dot(U,V) |
// | Dot(V,U) Dot(V,V) |
// +- -+
//
// Curvature Tensor: B = +- -+
// | -Dot(N,U_s) -Dot(N,U_t) |
// | -Dot(N,V_s) -Dot(N,V_t) |
// +- -+
//
// Principal curvatures k are the generalized eigenvalues of
//
// Bw = kGw
//
// If k is a curvature and w=(a,b) is the corresponding solution to
// Bw=kGw, then the principal direction as a 3D vector is d = a*U+b*V.
//
// Let k1 and k2 be the principal curvatures. The mean curvature
// is (k1+k2)/2 and the Gaussian curvature is k1*k2.
// derivatives
Vector3<Real> kDerU = GetDerivativeU(fU,fV);
Vector3<Real> kDerV = GetDerivativeV(fU,fV);
Vector3<Real> kDerUU = GetDerivativeUU(fU,fV);
Vector3<Real> kDerUV = GetDerivativeUV(fU,fV);
Vector3<Real> kDerVV = GetDerivativeVV(fU,fV);
// metric tensor
Matrix3<Real> kMetricTensor;
kMetricTensor[0][0] = kDerU.Dot(kDerU);
kMetricTensor[0][1] = kDerU.Dot(kDerV);
kMetricTensor[1][0] = kMetricTensor[1][0];
kMetricTensor[1][1] = kDerV.Dot(kDerV);
// curvature tensor
Vector3<Real> kNormal = kDerU.UnitCross(kDerV);
Matrix3<Real> kCurvatureTensor;
kCurvatureTensor[0][0] = -kNormal.Dot(kDerUU);
kCurvatureTensor[0][1] = -kNormal.Dot(kDerUV);
kCurvatureTensor[1][0] = kCurvatureTensor[0][1];
kCurvatureTensor[1][1] = -kNormal.Dot(kDerVV);
// characteristic polynomial is 0 = det(B-kG) = c2*k^2+c1*k+c0
Real fC0 = kCurvatureTensor.Determinant();
Real fC1 = ((Real)2.0)*kCurvatureTensor[0][1]* kMetricTensor[0][1] -
kCurvatureTensor[0][0]*kMetricTensor[1][1] -
kCurvatureTensor[1][1]*kMetricTensor[0][0];
Real fC2 = kMetricTensor.Determinant();
// principal curvatures are roots of characteristic polynomial
Real fTemp = Math<Real>::Sqrt(Math<Real>::FAbs(fC1*fC1 -
((Real)4.0)*fC0*fC2));
rfCurv0 = -((Real)0.5)*(fC1+fTemp);
rfCurv1 = ((Real)0.5)*(-fC1+fTemp);
// principal directions are solutions to (B-kG)w = 0
// w1 = (b12-k1*g12,-(b11-k1*g11)) OR (b22-k1*g22,-(b12-k1*g12))
Real fA0 = kCurvatureTensor[0][1] - rfCurv0*kMetricTensor[0][1];
Real fA1 = rfCurv0*kMetricTensor[0][0] - kCurvatureTensor[0][0];
Real fLength = Math<Real>::Sqrt(fA0*fA0+fA1*fA1);
if ( fLength >= Math<Real>::EPSILON )
{
rkDir0 = fA0*kDerU + fA1*kDerV;
}
else
{
fA0 = kCurvatureTensor[1][1] - rfCurv0*kMetricTensor[1][1];
fA1 = rfCurv0*kMetricTensor[0][1] - kCurvatureTensor[0][1];
fLength = Math<Real>::Sqrt(fA0*fA0+fA1*fA1);
if ( fLength >= Math<Real>::EPSILON )
{
rkDir0 = fA0*kDerU + fA1*kDerV;
}
else
{
// umbilic (surface is locally sphere, any direction principal)
rkDir0 = kDerU;
}
}
rkDir0.Normalize();
// second tangent is cross product of first tangent and normal
rkDir1 = rkDir0.Cross(kNormal);
}