//
// 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
//.........这里部分代码省略.........

// 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
//.........这里部分代码省略.........

// 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)
}