func (this *Mesh) TriangleUvFromPoint(faceIndex int, f *vec3.T) verb.UV {
tri := this.Faces[faceIndex]
p0 := this.Points[tri[0]]
p1 := this.Points[tri[1]]
p2 := this.Points[tri[2]]
uv0 := this.UVs[tri[0]]
uv1 := this.UVs[tri[1]]
uv2 := this.UVs[tri[2]]
f0 := vec3.Sub(&p0, f)
f1 := vec3.Sub(&p1, f)
f2 := vec3.Sub(&p2, f)
// calculate the areas and factors (order of parameters doesn't matter):
p1.Sub(&p0)
p2.Sub(&p0)
aVec := vec3.Cross(&p1, &p2)
a := aVec.Length()
a0Vec := vec3.Cross(&f1, &f2)
a1Vec := vec3.Cross(&f2, &f0)
a2Vec := vec3.Cross(&f0, &f1)
a0 := a0Vec.Length() / a
a1 := a1Vec.Length() / a
a2 := a2Vec.Length() / a
// find the uv corresponding to point f (uv1/uv2/uv3 are associated to p1/p2/p3):
return verb.UV{
a0*uv0[0] + a1*uv1[0] + a2*uv2[0],
a0*uv0[1] + a1*uv1[1] + a2*uv2[1],
}
}
//
// Sample a NURBS curve at 3 points, facilitating adaptive sampling
//
// **params**
// + NurbsCurveData object
// + start parameter for sampling
// + end parameter for sampling
// + whether to prefix the point with the parameter
//
// **returns**
// + an array of dim + 1 length where the first element is the param where it was sampled and the remaining the pt
//
func (this *NurbsCurve) adaptiveSampleRange(start, end, tol float64, includeU bool) []CurvePoint {
// sample curve at three pts
p1, p3 := this.Point(start), this.Point(end)
t := 0.5 + 0.2*rand.Float64()
mid := start + (end-start)*t
p2 := this.Point(mid)
// if the two end control points are coincident, the three point test will always return 0, let's split the curve
diff := vec3.Sub(&p1, &p3)
diff2 := vec3.Sub(&p1, &p2)
// the first condition checks if the curve makes up a loop, if so, we will need to continue evaluation
if (vec3.Dot(&diff, &diff) < tol && vec3.Dot(&diff2, &diff2) > tol) || !threePointsAreCollinear(&p1, &p2, &p3, tol) {
// get the exact middle
exact_mid := start + (end-start)*0.5
// recurse on the two halves
left_pts := this.adaptiveSampleRange(start, exact_mid, tol, includeU)
right_pts := this.adaptiveSampleRange(exact_mid, end, tol, includeU)
// concatenate the two
leftEnd := len(left_pts) - 1
return append(left_pts[:leftEnd:leftEnd], right_pts...)
} else {
return []CurvePoint{{start, p1}, {end, p3}}
}
}
// Determine if three points form a straight line within a given tolerance for their 2 * squared area
//
// * p2
// / \
// / \
// / \
// / \
// * p1 ---- * p3
//
// The area metric is 2 * the squared norm of the cross product of two edges, requiring no square roots and no divisions
//
// **params**
// + p1
// + p2
// + p3
// + The tolerance
//
// **returns**
// + Whether the triangle passes the test
//
func threePointsAreCollinear(p1, p2, p3 *vec3.T, tol float64) bool {
// find the area of the triangle without using a square root
p2mp1 := vec3.Sub(p2, p1)
p3mp1 := vec3.Sub(p3, p1)
norm := vec3.Cross(&p2mp1, &p3mp1)
area := vec3.Dot(&norm, &norm)
return area < tol
}
func (this Line) SameSide(p0, p1 vec3.T) bool {
lineVec := vec3.Sub(&this[1], &this[0])
p0Vec := vec3.Sub(&p0, &this[0])
p1Vec := vec3.Sub(&p1, &this[0])
cp0 := vec3.Cross(&p0Vec, &lineVec)
cp1 := vec3.Cross(&p1Vec, &lineVec)
return vec3.Dot(&cp0, &cp1) >= 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
}
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
}
func distToSegment(a, b, c *vec3.T) float64 {
// check if ac is zero length
acv := vec3.Sub(c, a)
acl := acv.Length()
// subtract b from a
var bma = vec3.Sub(b, a)
if acl < Tolerance {
return bma.Length()
}
// normalize acv
acv.Normalize()
// project b - a to acv = p
p := vec3.Dot(&bma, &acv)
// multiply ac by d = acd
acv.Scale(p).Add(a)
return vec3.Distance(&acv, b)
}
func (this *adaptiveRefinementNode) ShouldDivide(options *adaptiveRefinementOptions, currentDepth int) bool {
if currentDepth < options.MinDepth {
return true
}
if currentDepth >= options.MaxDepth {
return false
}
if this.HasBadNormals() {
this.FixNormals()
// don't divide any further when encountering a degenerate normal
return false
}
normDiff01 := vec3.Sub(this.corners[0].Normal, this.corners[1].Normal)
normDiff23 := vec3.Sub(this.corners[2].Normal, this.corners[3].Normal)
this.splitVert = normDiff01.LengthSqr() > options.NormTol || normDiff23.LengthSqr() > options.NormTol
normDiff12 := vec3.Sub(this.corners[1].Normal, this.corners[2].Normal)
normDiff30 := vec3.Sub(this.corners[3].Normal, this.corners[0].Normal)
this.splitHoriz = normDiff12.LengthSqr() > options.NormTol || normDiff30.LengthSqr() > options.NormTol
if this.splitVert || this.splitHoriz {
return true
}
center := this.Center()
for _, corner := range this.corners {
diffVec := vec3.Sub(center.Normal, corner.Normal)
if diffVec.LengthSqr() > options.NormTol {
return true
}
}
return false
}
// 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)),
}
}
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 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())
}