Beispiel #1
0
//
// 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)
}
Beispiel #2
0
// Find the closest point on a ray
//
// **params**
// + point to project
// + origin for ray
// + direction of ray 1, assumed normalized
//
// **returns**
// + pt
func (this Ray) ClosestPoint(pt vec3.T) vec3.T {
	o2pt := vec3.Sub(&pt, &this.Origin)
	do2ptr := vec3.Dot(&o2pt, &this.Dir)
	dirScaled := this.Dir.Scaled(do2ptr)
	proj := vec3.Add(&this.Origin, &dirScaled)

	return proj
}
Beispiel #3
0
func (this Plane) IntersectLine(line Line) *vec3.T {
	lineVec := vec3.Sub(&line[1], &line[0])
	invLineVec := vec3.Sub(&line[0], &line[1])

	denom := vec3.Dot(&this.Normal, &invLineVec)
	if math.Abs(denom) < epsilon {
		// Line is parallel to plane
		return nil
	}

	numer := vec3.Dot(&this.Normal, &line[0]) + this.Offset
	t := numer / denom
	if t < 0 || t > 1 {
		return nil
	}

	lineVec.Scale(t)
	intersectPt := vec3.Add(&lineVec, &line[0])
	return &intersectPt
}
Beispiel #4
0
// Find the closest point on a segment
//
// **params**
// + point to project
// + first point of segment
// + second point of segment
// + first param of segment
// + second param of segment
//
// **returns**
// + *Object* with u and pt properties
func segmentClosestPoint(pt, segpt0, segpt1 *vec3.T, u0, u1 float64) CurvePoint {
	dif := vec3.Sub(segpt1, segpt0)
	l := dif.Length()

	if l < Epsilon {
		return CurvePoint{u0, *segpt0}
	}

	o := segpt0
	r := dif.Normalize()
	o2pt := vec3.Sub(pt, o)
	do2ptr := vec3.Dot(&o2pt, r)

	if do2ptr < 0 {
		return CurvePoint{u0, *segpt0}
	} else if do2ptr > l {
		return CurvePoint{u1, *segpt1}
	}

	return CurvePoint{
		u0 + (u1-u0)*do2ptr/l,
		vec3.Add(o, r.Scale(do2ptr)),
	}
}
Beispiel #5
0
// 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)
}
Beispiel #6
0
// Generate the control points, weights, and knots of a revolved surface
// (Corresponds to Algorithm A7.1 from Piegl & Tiller)
//
// **params**
// + center of the rotation axis
// + axis of the rotation axis
// + angle to revolve around axis
// + degree of the generatrix
// + control points of the generatrix
// + weights of the generatrix
//
// **returns**
// + an object with the following properties: controlPoints, weights, knots, degree
func RevolvedSurface(profile *verb.NurbsCurve, center *vec3.T, axis *vec3.T, theta float64) *verb.NurbsSurface {
	prof_controlPoints := profile.ControlPoints()
	prof_weights := profile.Weights()

	var narcs int
	var knotsU []float64

	switch {
	case theta <= math.Pi/2:
		{ // less than 90
			narcs = 1
			knotsU = make([]float64, 6+2*(narcs-1))
		}
	case theta <= math.Pi:
		{ // between 90 and 180
			narcs = 2
			knotsU = make([]float64, 6+2*(narcs-1))
			knotsU[3], knotsU[4] = 0.5, 0.5
		}
	case theta <= 3*math.Pi/2:
		{ // between 180 and 270
			narcs = 3
			knotsU = make([]float64, 6+2*(narcs-1))
			knotsU[3], knotsU[4] = 1/3, 1/3
			knotsU[5], knotsU[6] = 2/3, 2/3
		}
	default:
		{ // between 270 and 360
			narcs = 4
			knotsU = make([]float64, 6+2*(narcs-1))
			knotsU[3], knotsU[4] = 1/4, 1/4
			knotsU[5], knotsU[6] = 1/2, 1/2
			knotsU[7], knotsU[8] = 3/4, 3/4
		}
	}

	dtheta := theta / float64(narcs) // divide the interval into several points
	j := 3 + 2*(narcs-1)

	// initialize the start and end knots
	// keep in mind that we only return the knot vector for thes
	for i := 0; i < 3; i++ {
		knotsU[j+i] = 1
	}

	// do some initialization
	wm := math.Cos(dtheta / 2)
	sines, cosines := make([]float64, narcs+1), make([]float64, narcs+1)

	controlPoints := make([][]vec3.T, 2*narcs+1)
	for i := range controlPoints {
		controlPoints[i] = make([]vec3.T, len(prof_controlPoints))
	}

	weights := make([][]float64, 2*narcs+1)
	for i := range weights {
		weights[i] = make([]float64, len(prof_controlPoints))
	}

	// initialize the sines and cosines
	var angle float64
	for i := 1; i <= narcs; i++ {
		angle += dtheta
		cosines[i] = math.Cos(angle)
		sines[i] = math.Sin(angle)
	}

	// for each pt in the generatrix
	// i.e. for each row of the 2d knot vectors
	for j := range prof_controlPoints {
		// get the closest point of the generatrix point on the axis
		O := rayClosestPoint(prof_controlPoints[j], center, axis)
		// X is the vector from the axis to generatrix control pt
		X := vec3.Sub(&prof_controlPoints[j], &O)
		// radius at that height
		r := X.Length()
		// Y is perpendicular to X and axis, and complete the coordinate system
		Y := vec3.Cross(axis, &X)

		if r > internal.Epsilon {
			X.Scale(1 / r)
			Y.Scale(1 / r)
		}

		// the first row of controlPoints and weights is just the generatrix
		controlPoints[0][j] = prof_controlPoints[j]
		P0 := prof_controlPoints[j]
		weights[0][j] = prof_weights[j]

		// store T0 as the Y vector
		var T0 = Y
		var index int

		// proceed around the circle
		for i := 1; i <= narcs; i++ {
			// O + r * cos(theta) * X + r * sin(theta) * Y
			// rotated generatrix pt
			var P2 vec3.T
			if r == 0 {
				P2 = O
			} else {
				xCompon := X.Scaled(r * cosines[i])
				yCompon := Y.Scaled(r * sines[i])
				offset := xCompon.Add(&yCompon)
				P2 = vec3.Add(&O, offset)
			}

			controlPoints[index+2][j] = P2
			weights[index+2][j] = prof_weights[j]

			// construct the vector tangent to the rotation
			temp0 := Y.Scaled(cosines[i])
			temp1 := X.Scaled(sines[i])
			T2 := temp0.Sub(&temp1)

			// construct the next control pt
			if r == 0 {
				controlPoints[index+1][j] = O
			} else {
				T0Norm := T0.Normalized()
				T2Norm := T2.Normalized()
				inters := intersect.Rays(&P0, &T0Norm, &P2, &T2Norm)

				T0Scaled := T0.Scaled(inters.U0)
				P1 := T0Scaled.Add(&P0)

				controlPoints[index+1][j] = *P1
			}

			weights[index+1][j] = wm * prof_weights[j]

			index += 2

			if i < narcs {
				P0 = P2
				T0 = *T2
			}
		}
	}

	return verb.NewNurbsSurfaceUnchecked(2, profile.Degree(), controlPoints, weights, knotsU, profile.Knots())
}