/// <summary>
/// Computes the derivatives on a non-uniform, non-rational B-spline surface.<br/>
/// SKL is the derivative S(u,v) with respect to U K-times and V L-times.<br/>
/// <em>Corresponds to algorithm 3.6 from The NURBS Book by Piegl and Tiller.</em>
/// </summary>
/// <param name="surface">The surface.</param>
/// <param name="u">The U parameter at which to evaluate the derivatives.</param>
/// <param name="v">The V parameter at which to evaluate the derivatives.</param>
/// <param name="numDerivs">Number of derivatives to evaluate, set as default to 1.</param>
/// <returns>A tuple with 2D collection representing all derivatives(k,L) and weights(k,l): U derivatives increase by row, V by column.</returns>
internal static Tuple <Vector3[, ], double[, ]> Derivatives(NurbsSurface surface, double u, double v, int numDerivs = 1)
{
// number of basis function.
int n = surface.KnotsU.Count - surface.DegreeU - 2;
int m = surface.KnotsV.Count - surface.DegreeV - 2;
// number of derivatives.
int du = Math.Min(numDerivs, surface.DegreeU);
int dv = Math.Min(numDerivs, surface.DegreeV);
int knotSpanU = surface.KnotsU.Span(n, surface.DegreeU, u);
int knotSpanV = surface.KnotsV.Span(m, surface.DegreeV, v);
List <Vector> uDerivs = Curve.DerivativeBasisFunctionsGivenNI(knotSpanU, u, surface.DegreeU, n, surface.KnotsU);
List <Vector> vDerivs = Curve.DerivativeBasisFunctionsGivenNI(knotSpanV, v, surface.DegreeV, m, surface.KnotsV);
int dim = surface.ControlPoints[0][0].Size;
List <List <Vector> > SKLw = Vector.Zero3d(numDerivs + 1, numDerivs + 1, dim);
List <Vector> temp = Vector.Zero2d(surface.DegreeV + 1, dim);
Vector3[,] SKL = new Vector3[numDerivs + 1, numDerivs + 1];
double[,] weights = new double[numDerivs + 1, numDerivs + 1];
for (int k = 0; k < du + 1; k++)
{
for (int s = 0; s < surface.DegreeV + 1; s++)
{
temp[s] = Vector.Zero1d(dim);
for (int r = 0; r < surface.DegreeU + 1; r++)
{
Vector.AddMulMutate(temp[s], uDerivs[k][r], surface.ControlPoints[knotSpanU - surface.DegreeU + r][knotSpanV - surface.DegreeV + s]);
}
}
int dd = Math.Min(numDerivs - k, dv);
for (int l = 0; l < dd + 1; l++)
{
SKLw[k][l] = Vector.Zero1d(dim);
for (int s = 0; s < surface.DegreeV + 1; s++)
{
Vector.AddMulMutate(SKLw[k][l], vDerivs[l][s], temp[s]);
}
SKL[k, l] = new Vector3(SKLw[k][l][0], SKLw[k][l][1], SKLw[k][l][2]); // Extracting the derivatives.
weights[k, l] = SKLw[k][l][3]; // Extracting the weights.
}
}
return(new Tuple <Vector3[, ], double[, ]>(SKL, weights));
}
