//
// Generate the control points, weights, and knots of a cone
//
// **params**
// + normalized axis of cone
// + position of base of cone
// + height from base to tip
// + radius at the base of the cone
//
// **returns**
// + an object with the following properties: controlPoints, weights, knots, degree
//
func ConicalSurface(axis, xaxis *vec3.T, base *vec3.T, height, radius float64) *verb.NurbsSurface {
angle := 2 * math.Pi
profDegree := 1
heightCompon := axis.Scaled(height)
radiusCompon := xaxis.Scaled(radius)
profCtrlPts := []vec3.T{vec3.Add(base, &heightCompon), vec3.Add(base, &radiusCompon)}
profKnots := []float64{0, 0, 1, 1}
profWeights := []float64{1, 1}
prof := verb.NewNurbsCurveUnchecked(profDegree, profCtrlPts, profWeights, profKnots)
return RevolvedSurface(prof, base, axis, angle)
}
// 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
}
func Homogenized(pt vec3.T, w float64) HomoPoint {
return HomoPoint{pt.Scaled(w), w}
}
func (this *NurbsCurve) ClosestParam(p vec3.T) float64 {
// We want to solve:
//
// C'(u) * ( C(u) - P ) = 0 = f(u)
//
// C(u) is the curve, p is the point, * is a dot product
//
// We'll use newton's method:
//
// u* = u - f / f'
//
// We use the product rule in order to form the derivative, f':
//
// f' = C"(u) * ( C(u) - p ) + C'(u) * C'(u)
//
// What is the conversion criteria? (Piegl & Tiller suggest)
//
// |C(u) - p| < e1
//
// |C'(u)*(C(u) - P)|
// ------------------ < e2
// |C'(u)| |C(u) - P|
//
// 1) first check 2 & 3
// 2) if at least one of these is not, compute new value, otherwise halt
// 3) ensure the parameter stays within range
// * if not closed, don't allow outside of range a-b
// * if closed (e.g. circle), allow to move back to beginning
// 4) if |(u* - u)C'(u)| < e1, halt
//
min := math.MaxFloat64
var u float64
pts := this.regularSample(len(this.controlPoints) * this.degree)
for i := 0; i < len(pts)-1; i++ {
u0, u1 := pts[i].U, pts[i+1].U
p0 := pts[i].Pt
p1 := pts[i+1].Pt
proj := segmentClosestPoint(&p, &p0, &p1, u0, u1)
dv := vec3.Sub(&p, &proj.Pt)
d := dv.Length()
if d < min {
min = d
u = proj.U
}
}
maxits := 5
var i int
var e []vec3.T
eps1, eps2 := 0.0001, 0.0005
var dif vec3.T
minu, maxu := this.knots[0], this.knots[len(this.knots)-1]
firstCtrlPt := this.controlPoints[0].Dehomogenized()
lastCtrlPt := this.controlPoints[len(this.controlPoints)-1].Dehomogenized()
closed := vec3.SquareDistance(&firstCtrlPt, &lastCtrlPt) < Epsilon
cu := u
f := func(u float64) []vec3.T {
return this.Derivatives(u, 2)
}
n := func(u float64, e []vec3.T, d vec3.T) float64 {
// C'(u) * ( C(u) - P ) = 0 = f(u)
f := vec3.Dot(&e[1], &d)
// f' = C"(u) * ( C(u) - p ) + C'(u) * C'(u)
s0 := vec3.Dot(&e[2], &d)
s1 := vec3.Dot(&e[1], &e[1])
df := s0 + s1
return u - f/df
}
for i < maxits {
e = f(cu)
dif = vec3.Sub(&e[0], &p)
// |C(u) - p| < e1
c1v := dif.Length()
// C'(u) * (C(u) - P)
// ------------------ < e2
// |C'(u)| |C(u) - P|
c2n := vec3.Dot(&e[1], &dif)
c2d := e[1].Length() * c1v
c2v := c2n / c2d
c1 := c1v < eps1
c2 := math.Abs(c2v) < eps2
// if both tolerances are met
if c1 && c2 {
return cu
}
ct := n(cu, e, dif)
// are we outside of the bounds of the curve?
if ct < minu {
if closed {
ct = maxu - (ct - minu)
} else {
ct = minu
}
} else if ct > maxu {
if closed {
ct = minu + (ct - maxu)
} else {
ct = maxu
}
}
// will our next step force us out of the curve?
c3vv := e[1].Scaled(ct - cu)
c3v := c3vv.Length()
if c3v < eps1 {
return cu
}
cu = ct
i++
}
return cu
}
// Generate the control points, weights, and knots of an elliptical arc
//
// **params**
// + the center
// + the scaled x axis
// + the scaled y axis
// + start angle of the ellipse arc, between 0 and 2pi, where 0 points at the xaxis
// + end angle of the arc, between 0 and 2pi, greater than the start angle
//
// **returns**
// + a NurbsCurveData object representing a NURBS curve
func EllipseArc(center *vec3.T, xaxis, yaxis *vec3.T, startAngle, endAngle float64) *verb.NurbsCurve {
xradius, yradius := xaxis.Length(), yaxis.Length()
xaxisNorm, yaxisNorm := xaxis.Normalized(), yaxis.Normalized()
// if the end angle is less than the start angle, do a circle
if endAngle < startAngle {
endAngle = 2.0*math.Pi + startAngle
}
theta := endAngle - startAngle
// how many arcs?
var numArcs int
if theta <= math.Pi/2 {
numArcs = 1
} else {
if theta <= math.Pi {
numArcs = 2
} else if theta <= 3*math.Pi/2 {
numArcs = 3
} else {
numArcs = 4
}
}
dtheta := theta / float64(numArcs)
w1 := math.Cos(dtheta / 2)
xCompon := xaxisNorm.Scaled(xradius * math.Cos(startAngle))
yCompon := yaxisNorm.Scaled(yradius * math.Sin(startAngle))
P0 := vec3.Add(&xCompon, &yCompon)
temp0 := yaxisNorm.Scaled(math.Cos(startAngle))
temp1 := xaxisNorm.Scaled(math.Sin(startAngle))
T0 := vec3.Sub(&temp0, &temp1)
controlPoints := make([]vec3.T, 2*numArcs+1)
knots := make([]float64, 2*numArcs+3)
index := 0
angle := startAngle
weights := make([]float64, numArcs*2)
controlPoints[0] = P0
weights[0] = 1.0
for i := 1; i <= numArcs; i++ {
angle += dtheta
xCompon = xaxisNorm.Scaled(xradius * math.Cos(angle))
yCompon = yaxisNorm.Scaled(yradius * math.Sin(angle))
offset := vec3.Add(&xCompon, &yCompon)
P2 := vec3.Add(center, &offset)
weights[index+2] = 1
controlPoints[index+2] = P2
temp0 := yaxisNorm.Scaled(math.Cos(angle))
temp1 := xaxisNorm.Scaled(math.Sin(angle))
T2 := vec3.Sub(&temp0, &temp1)
T0Norm := T0.Normalized()
T2Norm := T2.Normalized()
inters := intersect.Rays(&P0, &T0Norm, &P2, &T2Norm)
T0Scaled := T0.Scaled(inters.U0)
P1 := vec3.Add(&P0, &T0Scaled)
weights[index+1] = w1
controlPoints[index+1] = P1
index += 2
if i < numArcs {
P0 = P2
T0 = T2
}
}
j := 2*numArcs + 1
for i := 0; i < 3; i++ {
knots[i] = 0.0
knots[i+j] = 1.0
}
switch numArcs {
case 2:
knots[3] = 0.5
knots[4] = 0.5
case 3:
knots[3] = 1 / 3
knots[4] = 1 / 3
knots[5] = 2 / 3
knots[6] = 2 / 3
case 4:
knots[3] = 0.25
knots[4] = 0.25
knots[5] = 0.5
knots[6] = 0.5
knots[7] = 0.75
knots[8] = 0.75
}
return verb.NewNurbsCurveUnchecked(2, controlPoints, weights, knots)
}
// Generate the control points, weights, and knots of an arbitrary arc
// (Corresponds to Algorithm A7.1 from Piegl & Tiller)
//
// **params**
// + the center of the arc
// + the xaxis of the arc
// + orthogonal yaxis of the arc
// + radius of the arc
// + start angle of the arc, between 0 and 2pi
// + end angle of the arc, between 0 and 2pi, greater than the start angle
//
// **returns**
// + a NurbsCurveData object representing a NURBS curve
func Arc(center *vec3.T, xaxis, yaxis *vec3.T, radius float64, startAngle, endAngle float64) *verb.NurbsCurve {
xaxisScaled, yaxisScaled := xaxis.Scaled(radius), yaxis.Scaled(radius)
return EllipseArc(center, &xaxisScaled, &yaxisScaled, startAngle, endAngle)
}
//
// Generate the control points, weights, and knots of a sphere
//
// **params**
// + the center of the sphere
// + normalized axis of sphere
// + vector perpendicular to axis of sphere, starting the rotation of the sphere
// + radius of the sphere
//
// **returns**
// + an object with the following properties: controlPoints, weights, knotsU, knotsV, degreeU, degreeV
//
func SphericalSurface(center *vec3.T, axis, xaxis *vec3.T, radius float64) *verb.NurbsSurface {
invAxis := axis.Inverted()
arc := Arc(center, &invAxis, xaxis, radius, 0, math.Pi)
return RevolvedSurface(arc, center, axis, 2*math.Pi)
}