// Compute the derivatives at a point on a NURBS surface
//
// **params**
// + NurbsSurfaceData object representing the surface
// + number of derivatives to evaluate
// + u parameter at which to evaluate the derivatives
// + v parameter at which to evaluate the derivatives
//
// **returns**
// + a point represented by an array of length (dim)
func (this *NurbsSurface) Derivatives(uv UV, numDerivs int) [][]vec3.T {
ders := this.nonRationalDerivatives(uv, numDerivs)
wders := Weight2d(ders)
skl := make([][]vec3.T, numDerivs+1)
for k := 0; k <= numDerivs; k++ {
for l := 0; l <= numDerivs-k; l++ {
v := ders[k][l].Vec3
for j := 1; j <= l; j++ {
scaled := skl[k][l-j].Scaled(binomial(l, j) * wders[0][j])
v.Sub(&scaled)
}
for i := 1; i <= k; i++ {
scaled := skl[k-i][l].Scaled(binomial(k, i) * wders[i][0])
v.Sub(&scaled)
var v2 vec3.T
for j := 1; j <= l; j++ {
scaled := skl[k-i][l-j].Scaled(binomial(l, j) * wders[i][j])
v2.Add(&scaled)
}
scaled = v2.Scaled(binomial(k, i))
v.Sub(&scaled)
}
v.Scale(1 / wders[0][0])
skl[k][l] = v
}
}
return skl
}
//
// Compute a point in a non-uniform, non-rational B spline volume
//
// **params**
// + VolumeData
// + u parameter at which to evaluate the volume point
// + v parameter at which to evaluate the volume point
// + w parameter at which to evaluate the volume point
//
// **returns**
// + a point represented by an array of length (dim)
func (this *volume) PointGivenNML(n, m, l int, uvw UVW) vec3.T {
if !areValidRelations(this.DegreeU, len(this.ControlPoints), len(this.KnotsU)) ||
!areValidRelations(this.DegreeV, len(this.ControlPoints[0]), len(this.KnotsV)) ||
!areValidRelations(this.DegreeW, len(this.ControlPoints[0][0]), len(this.KnotsW)) {
panic("Invalid relations between control points and knot vector")
}
controlPoints := this.ControlPoints
degreeU, degreeV, degreeW := this.DegreeU, this.DegreeV, this.DegreeW
knotsU, knotsV, knotsW := this.KnotsU, this.KnotsV, this.KnotsW
knotSpanIndexU := knotsU.SpanGivenN(n, degreeU, uvw[0])
knotSpanIndexV := knotsV.SpanGivenN(m, degreeV, uvw[1])
knotSpanIndexW := knotsW.SpanGivenN(l, degreeW, uvw[2])
uBasisVals := BasisFunctionsGivenKnotSpanIndex(knotSpanIndexU, uvw[0], degreeU, knotsU)
vBasisVals := BasisFunctionsGivenKnotSpanIndex(knotSpanIndexV, uvw[0], degreeV, knotsV)
wBasisVals := BasisFunctionsGivenKnotSpanIndex(knotSpanIndexV, uvw[0], degreeW, knotsW)
uind := knotSpanIndexU - degreeU
var position, temp, temp2 vec3.T
for i := 0; i <= degreeW; i++ {
temp2 = vec3.Zero
wind := knotSpanIndexW - degreeW + i
for j := 0; j <= degreeV; j++ {
temp = vec3.Zero
vind := knotSpanIndexV - degreeV + j
for k := 0; k <= degreeU; k++ {
scaled := controlPoints[uind+k][vind][wind].Scaled(uBasisVals[k])
temp.Add(&scaled)
}
// add weighted contribution of u isoline
scaled := temp.Scaled(vBasisVals[j])
temp2.Add(&scaled)
}
// add weighted contribution from uv isosurfaces
scaled := temp2.Scaled(wBasisVals[i])
position.Add(&scaled)
}
return position
}