/// <summary>
/// using the start, end and passed geo fit points will slide the points along their normals to achieve a geodesic
/// </summary>
/// <param name="g">the curve object to spline</param>
/// <param name="start">the starting point of this segment</param>
/// <param name="end">the ending point of this segment</param>
/// <param name="sFits">the s-positons of each geo-point</param>
/// <param name="uFits">the u-positons of each geo-point</param>
/// <returns>true if successful, false otherwise</returns>
static bool FitGeo(MouldCurve g, IFitPoint start, IFitPoint end, double[] sFits, Vect2[] uFits)
{
int NumFits = sFits.Length;
int INC = NumFits - 1;
int i;
Vect2[] uNor = new Vect2[NumFits];
Vect3 xyz = new Vect3(), dxu = new Vect3(), dxv = new Vect3();
Vect2 ut = new Vect2(), un = new Vect2();
double a11, a12, a22, det;
//calculate insurface normals at each fitpoint
uNor[0] = new Vect2();//fixed endpoint doesnt need normal
for (i = 1; i < INC; i++)
{
g.Surface.xVec(uFits[i], ref xyz, ref dxu, ref dxv);
//covariant metric tensor components
a11 = dxu.Norm;
a12 = dxu.Dot(dxv);
a22 = dxv.Norm;
det = Math.Sqrt(a11 * a22 - a12 * a12);
//tangent(secant) vector u components
ut = uFits[i + 1] - uFits[i - 1];
//contravariant normal vector components in the surface plane
un[0] = (a12 * ut[0] + a22 * ut[1]) / det;
un[1] = -(a11 * ut[0] + a12 * ut[1]) / det;
//store unit normal u components
un.Magnitude = 1;
uNor[i] = new Vect2(un);
}
uNor[INC] = new Vect2();//fixed endpoint doesnt need normal
Vect3 xPrev = new Vect3();
Vect3 ddxu = new Vect3(), ddxv = new Vect3(), dduv = new Vect3();
Vect3[] xNor = new Vect3[NumFits];
Vect3[] dxNor = new Vect3[NumFits];
Vect3[] xTan = new Vect3[NumFits];
double[] xLen = new double[NumFits];
bool Conver = false;
int nNwt; Vector x; DenseVector sNor;
for (nNwt = 0; nNwt < 50; nNwt++)
{
xNor[0] = new Vect3();
//update startpoint slide position
if (start is SlidePoint)
{
dxNor[0] = new Vect3();
(start as SlidePoint).Curve.xCvt((start as SlidePoint).SCurve, ref uFits[0], ref xPrev, ref xNor[0], ref dxNor[0]);
//uFits[0][0] = FitPoints[0][1];
//uFits[0][1] = FitPoints[0][2];
}
else
g.xVal(uFits[0], ref xPrev);
//update endpoint slide position
if (end is SlidePoint)
{
uFits[INC][0] = end[1];
uFits[INC][1] = end[2];
}
xLen[0] = 0;
//calc internal point vectors
for (i = 1; i < NumFits; i++)
{
g.Surface.xCvt(uFits[i], ref xyz, ref dxu, ref dxv, ref ddxu, ref ddxv, ref dduv);
a11 = uNor[i][0] * uNor[i][0];
a12 = uNor[i][0] * uNor[i][1] * 2;
a22 = uNor[i][1] * uNor[i][1];
// insurface normal x components
dxu.Scale(uNor[i][0]);
dxv.Scale(uNor[i][1]);
xNor[i] = dxu + dxv;
//insurface normal x derivatives
ddxu.Scale(a11);
dduv.Scale(a12);
ddxv.Scale(a22);
dxNor[i] = ddxu + dduv + ddxv;
//forward facing tangent vector
xTan[i] = xyz - xPrev;
xPrev.Set(xyz);
xLen[i] = xTan[i].Magnitude;//segment length
xLen[0] += xLen[i];//accumulate total length
xTan[i].Magnitude = 1;//unit tangent vector
}
//update endpoint slide position
if (end is SlidePoint)
{
(end as SlidePoint).Curve.xCvt((end as SlidePoint).SCurve, ref uFits[INC], ref xyz, ref xNor[INC], ref dxNor[INC]);
}
DenseMatrix A = new DenseMatrix(NumFits);
sNor = new DenseVector(NumFits);
double p0, pp, d, d0, dp, gm, g0, gp; int ix;
//slide startpoint
if (start is SlidePoint)
{
//mid point normal vector dotted with end point tangent vectors
pp = xNor[0].Dot(xTan[1]);
for (g0 = gp = 0, ix = 0; ix < 3; ix++)
{
//midpoint tangent and curavture variantion
dp = (xNor[0][ix] - pp * xTan[1][ix]) / xLen[1];
//mid and top point gradients
g0 += -dp * xNor[0][ix] + xTan[1][ix] * dxNor[0][ix];
gp += dp * xNor[1][ix];
}
//geodesic residual and gradients
A[0, 0] = g0;
A[0, 1] = gp;
sNor[0] = pp;
}
else//fixed start point
{
A[0, 0] = 1;
sNor[0] = 0;
}
for (i = 1; i < INC; i++)//internal points
{
//midpoint normal dotted with tangents
p0 = xNor[i].Dot(xTan[i]);// BLAS.dot(xNor[i], xTan[i]);
pp = xNor[i].Dot(xTan[i + 1]);//BLAS.dot(xNor[i], xTan[i + 1]);
for (gm = g0 = gp = 0, ix = 0; ix < 3; ix++)
{
//midpoint curvature vector
d = xTan[i + 1][ix] - xTan[i][ix];
//midpoint tangent and curavture variantion
d0 = (xNor[i][ix] - p0 * xTan[i][ix]) / xLen[i];
dp = (xNor[i][ix] - pp * xTan[i + 1][ix]) / xLen[i + 1];
//bottom, mid and top point gradients
gm += d0 * xNor[i - 1][ix];
g0 += (-d0 - dp) * xNor[i][ix] + d * dxNor[i][ix];
gp += dp * xNor[i + 1][ix];
}
A[i, i - 1] = gm;
A[i, i] = g0;
A[i, i + 1] = gp;
sNor[i] = -p0 + pp;
}
if (end is SlidePoint)//slide endpoint
{
p0 = xNor[i].Dot(xTan[i]);
for (gm = g0 = 0, ix = 0; ix < 3; ix++)
{
//midpoint tangent and curavture variantion
d0 = (xNor[i][ix] - p0 * xTan[i][ix]) / xLen[i];
//bottom, mid and top point gradients
gm += d0 * xNor[i - 1][ix];
g0 += -d0 * xNor[i][ix] - xTan[i][ix] * dxNor[i][ix];
}
A[i, i - 1] = gm;
A[i, i] = g0;
sNor[i] = -p0;
}
else//fixed endpoint
{
A[i, i] = 1;
sNor[i] = 0;
}
LU decomp = A.LU();
x = (Vector)decomp.Solve(sNor);
double Reduce = Math.Min(1, .05 / x.AbsoluteMaximum());
if( start is SlidePoint)
(start as SlidePoint).SCurve -= x[0] * Reduce;
if( end is SlidePoint)
(end as SlidePoint).SCurve -= x[INC] * Reduce;
for (i = 1; i < NumFits; i++)//increment uv points
{
uFits[i][0] -= x[i] * uNor[i][0] * Reduce;
uFits[i][1] -= x[i] * uNor[i][1] * Reduce;
}
if (nNwt < 5)
{
//keep initial (s)-increments within bounds
if (start is SlidePoint)
(start as SlidePoint).SCurve = Utilities.LimitRange(0, (start as SlidePoint).SCurve, 1);
if (end is SlidePoint)
(end as SlidePoint).SCurve = Utilities.LimitRange(0, (end as SlidePoint).SCurve, 1);
// keep initial (u)-increments within bounds
for (i = 1; i < NumFits - 1; i++)
{
uFits[i][0] = Utilities.LimitRange(0, uFits[i][0], 1);
uFits[i][1] = Utilities.LimitRange(-.125, uFits[i][1], 1.125);
}
}
double xmax = x.AbsoluteMaximum();
double smax = sNor.AbsoluteMaximum();
if (Conver = (x.AbsoluteMaximum() < 1e-8 && sNor.AbsoluteMaximum() < 1e-7))
break;
}
if (!Conver)
return false;
g.Length = xLen[0];//store length
//calculate unit length (s)-parameter values
sFits[0] = 0;
for (i = 1; i < NumFits; i++)
sFits[i] = sFits[i - 1] + xLen[i] / xLen[0];
//g.m_uvs = uFits;
g.ReSpline(sFits, uFits);
return true;
}
