// 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 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())
}