path_intersection_util.go

package canvas

import (
	"fmt"
	"math"
	"sort"
)

// see https://github.com/signavio/svg-intersections
// see https://github.com/w8r/bezier-intersect
// see https://cs.nyu.edu/exact/doc/subdiv1.pdf

// Intersections amongst the combinations between line, quad, cube, elliptical arcs. We consider four cases: the curves do not cross nor touch (intersections is empty), the curves intersect (and cross), the curves intersect tangentially (touching), or the curves are identical (or parallel in the case of two lines). In the last case we say there are no intersections. As all curves are segments, it is considered a secant intersection when the segments touch but "intent to" cut at their ends (i.e. when position equals to 0 or 1 for either segment).

func segmentPos(start Point, d []float64, t float64) Point {
	// used for open paths in boolean
	if d[0] == LineToCmd || d[0] == CloseCmd {
		return start.Interpolate(Point{d[1], d[2]}, t)
	} else if d[0] == QuadToCmd {
		cp := Point{d[1], d[2]}
		end := Point{d[3], d[4]}
		return quadraticBezierPos(start, cp, end, t)
	} else if d[0] == CubeToCmd {
		cp1 := Point{d[1], d[2]}
		cp2 := Point{d[3], d[4]}
		end := Point{d[5], d[6]}
		return cubicBezierPos(start, cp1, cp2, end, t)
	} else if d[0] == ArcToCmd {
		rx, ry, phi := d[1], d[2], d[3]
		large, sweep := toArcFlags(d[4])
		cx, cy, theta0, theta1 := ellipseToCenter(start.X, start.Y, rx, ry, phi, large, sweep, d[5], d[6])
		return EllipsePos(rx, ry, phi, cx, cy, theta0+t*(theta1-theta0))
	}
	return Point{}
}

// returns true if p is inside q or equivalent to q, paths may not intersect
// p should not have subpaths
func (p *Path) inside(q *Path) bool {
	// if p does not fill with the EvenOdd rule, it is inside q
	p = p.Append(q)
	return !p.Filling(EvenOdd)[0]
}

type subpathIndexer []int // index from segment to subpath

func newSubpathIndexer(p *Path) subpathIndexer {
	segs := 0
	var idx subpathIndexer
	for i := 0; i < len(p.d); i += cmdLen(p.d[i]) {
		if i != 0 && p.d[i] == MoveToCmd {
			idx = append(idx, segs)
		}
		segs++
	}
	idx = append(idx, segs)
	return idx
}

func newSubpathIndexerSubpaths(ps []*Path) subpathIndexer {
	segs := 0
	idx := make(subpathIndexer, len(ps))
	for i, pi := range ps {
		segs += pi.Len()
		idx[i] = segs
	}
	return idx
}

func (idx subpathIndexer) in(i, seg int) bool {
	return (i == 0 || idx[i-1] <= seg) && seg < idx[i]
}

func (idx subpathIndexer) get(seg int) int {
	for i, n := range idx {
		if seg < n {
			return i
		}
	}
	panic("bug: segment doesn't exist on path")
}

type PathIntersectionNode struct {
	i            int // intersection index
	prevP, nextP *PathIntersectionNode
	prevQ, nextQ *PathIntersectionNode

	p, q *Path // path towards next node
	x    *Path // parallel/ovarlapping path at node, can be nil

	PintoQ           bool
	Tangent          bool
	Parallel         bool
	ParallelReversed bool
}

func (z PathIntersectionNode) ParallelTangent(onP, forwardP, forwardQ bool) bool {
	return z.Tangent && (onP && (forwardP && z.Parallel || !forwardP && z.prevP.Parallel) || !onP && (forwardQ && (z.Parallel && !z.ParallelReversed || z.nextQ.Parallel && z.nextQ.ParallelReversed) || !forwardQ && (z.prevQ.Parallel && !z.prevQ.ParallelReversed || z.Parallel && z.ParallelReversed)))
}

func (z PathIntersectionNode) String() string {
	var extra string
	if z.PintoQ {
		extra = " PintoQ"
	} else {
		extra = " QintoP"
	}
	if z.Tangent {
		extra += "-Tangent"
	}
	if z.Parallel {
		extra += " Parallel"
		if z.ParallelReversed {
			extra += "-Reversed"
		}
	}
	pos := z.p.StartPos()
	return fmt.Sprintf("(%v {%v,%v} P=[%v→·→%v] Q=[%v→·→%v]%v)", z.i, numEps(pos.X), numEps(pos.Y), z.prevP.i, z.nextP.i, z.prevQ.i, z.nextQ.i, extra)
}

func pathIntersectionNodes(p, q *Path, zp, zq []PathIntersection) []PathIntersectionNode {
	// create graph of nodes between intersections over both paths
	if len(zp) == 0 {
		return nil
	}

	// count number of nodes
	n := len(zp)
	for _, z := range zp {
		if z.Parallel && !z.Tangent {
			n--
		}
	}
	if n%2 != 0 {
		panic("bug: number of nodes must be even")
	}

	i, k := 0, 0
	ps, segs := cut(p, zp)
	idxZ := make([]int, len(zp)) // index zp to zs
	zs := make([]PathIntersectionNode, n)
	for _, seg := range segs {
		// loop over each subpath of p
		j := i
		for j < len(zp) && zp[j].Seg < seg {
			j++
		}

		i0, k0 := i, k
		for ; i < j; i++ {
			// loop over the intersections in a subpath of p
			idxZ[i] = k
			if zp[i].Parallel {
				i1, k1 := i+1, k
				if i+1 == j {
					i1, k1 = i0, k0
				}
				reversed := zq[i1].Parallel

				if zp[i].Tangent {
					zs[k].Parallel = true
					zs[k].ParallelReversed = reversed
				} else {
					// add parallel part to next node, skip this intersection
					idxZ[i] = k1
					zs[k1].Parallel = true
					zs[k1].ParallelReversed = reversed
					zs[k1].x = ps[i]
					continue
				}
			}

			zs[k].i = k
			zs[k].p = ps[i]
			zs[k].PintoQ = zp[i].Into
			zs[k].Tangent = zp[i].Tangent

			if 0 < k {
				// we overwrite the first (for non-first subpath) at the end
				zs[k].prevP = &zs[k-1]
			}
			if k+1 < len(zs) {
				// we overwrite the last (for non-first subpath) at the end
				zs[k].nextP = &zs[k+1]
			}
			k++
		}
		if k0 < k {
			zs[k0].prevP = &zs[k-1]
			zs[k-1].nextP = &zs[k0]
		}
	}

	// sort zq and keep indices of sorted to original
	idxP := make([]int, len(zq)) // index zq to zp
	for i := range zq {
		idxP[i] = i
	}
	sort.Stable(pathIntersectionSort{zq, idxP})

	i = 0
	qs, segs := cut(q, zq)
	for _, seg := range segs {
		// loop over each subpath of q
		j := i
		for j < len(zq) && zq[j].Seg < seg {
			j++
		}

		i0 := i
		for ; i < j; i++ {
			// loop over the intersections in a subpath of q
			k := idxZ[idxP[i]] // index in zq => index in zp => index in zs
			if zq[i].Parallel && !zq[i].Tangent {
				continue
			}

			zs[k].q = qs[i]
			if i0 < i {
				i1 := i - 1
				if zq[i1].Parallel && !zq[i1].Tangent {
					i1--
				}
				if i0 <= i1 {
					zs[k].prevQ = &zs[idxZ[idxP[i1]]]
				}
			}
			if i+1 < j {
				zs[k].nextQ = &zs[idxZ[idxP[i+1]]]
			}
		}
		if i0 < i {
			i1 := i - 1
			if zq[i1].Parallel && !zq[i1].Tangent {
				i1--
			}
			zs[idxZ[idxP[i0]]].prevQ = &zs[idxZ[idxP[i1]]]
			zs[idxZ[idxP[i1]]].nextQ = &zs[idxZ[idxP[i0]]]
		}
	}
	return zs
}

func cut(p *Path, zs []PathIntersection) ([]*Path, subpathIndexer) {
	// zs must be sorted
	if len(zs) == 0 {
		return []*Path{p}, newSubpathIndexer(p)
	}

	j := 0   // index into zs
	k := 0   // index into ps
	seg := 0 // segment count
	var ps []*Path
	var first, cur []float64
	segs := subpathIndexer{}
	for i := 0; i < len(p.d); i += cmdLen(p.d[i]) {
		cmd := p.d[i]
		if 0 < i && cmd == MoveToCmd {
			closed := p.d[i-1] == CloseCmd
			if first != nil {
				// there were intersections in the last subpath
				if closed {
					ps = append(ps, &Path{append(cur, first[4:]...)})
					cur = nil
				} else {
					ps = append(ps[:k], append([]*Path{{first}}, ps[k:]...)...)
				}
			} else if closed {
				cur[len(cur)-1] = CloseCmd
				cur[len(cur)-4] = CloseCmd
			}
			first = nil
			k = len(ps)
			segs = append(segs, seg)
		}
		if j < len(zs) && seg == zs[j].Seg {
			// segment has an intersection, cut it up and append first part to prev intersection
			p0, p1 := cutSegment(Point{p.d[i-3], p.d[i-2]}, p.d[i:i+cmdLen(cmd)], zs[j].T)
			if !p0.Empty() {
				cur = append(cur, p0.d[4:]...)
			}

			for j+1 < len(zs) && seg == zs[j+1].Seg {
				// next cut is on the same segment, find new t after the first cut and set path
				if first == nil {
					first = cur // take aside the path to the first intersection to later append it
				} else {
					ps = append(ps, &Path{cur})
				}
				j++
				t := (zs[j].T - zs[j-1].T) / (1.0 - zs[j-1].T)
				if !p1.Empty() {
					p0, p1 = cutSegment(Point{p1.d[1], p1.d[2]}, p1.d[4:], t)
				} else {
					p0 = p1
				}
				cur = p0.d
			}
			if first == nil {
				first = cur // take aside the path to the first intersection to later append it
			} else {
				ps = append(ps, &Path{cur})
			}
			cur = p1.d
			j++
		} else {
			// segment has no intersection
			if len(cur) == 0 || cmd != CloseCmd || p.d[i+1] != cur[len(cur)-3] || p.d[i+2] != cur[len(cur)-2] {
				// don't append point-close command
				cur = append(cur, p.d[i:i+cmdLen(cmd)]...)
				if cmd == CloseCmd {
					cur[len(cur)-1] = LineToCmd
					cur[len(cur)-cmdLen(CloseCmd)] = LineToCmd
				}
			}
		}
		seg++
	}
	closed := 0 < len(p.d) && p.d[len(p.d)-1] == CloseCmd
	if first != nil {
		// there were intersections in the last subpath
		if closed {
			cur = append(cur, first[4:]...)
		} else {
			ps = append(ps[:k], append([]*Path{{first}}, ps[k:]...)...)
		}
	} else if closed {
		cur[len(cur)-1] = CloseCmd
		cur[len(cur)-4] = CloseCmd
	}
	ps = append(ps, &Path{cur})
	segs = append(segs, seg)
	return ps, segs
}

func cutSegment(start Point, d []float64, t float64) (*Path, *Path) {
	p0, p1 := &Path{}, &Path{}
	if Equal(t, 0.0) {
		p0.MoveTo(start.X, start.Y)
		p1.MoveTo(start.X, start.Y)
		p1.d = append(p1.d, d...)
		if p1.d[cmdLen(MoveToCmd)] == CloseCmd {
			p1.d[cmdLen(MoveToCmd)] = LineToCmd
			p1.d[len(p1.d)-1] = LineToCmd
		}
		return p0, p1
	} else if Equal(t, 1.0) {
		p0.MoveTo(start.X, start.Y)
		p0.d = append(p0.d, d...)
		if p0.d[cmdLen(MoveToCmd)] == CloseCmd {
			p0.d[cmdLen(MoveToCmd)] = LineToCmd
			p0.d[len(p0.d)-1] = LineToCmd
		}
		p1.MoveTo(d[len(d)-3], d[len(d)-2])
		return p0, p1
	}
	if cmd := d[0]; cmd == LineToCmd || cmd == CloseCmd {
		c := start.Interpolate(Point{d[len(d)-3], d[len(d)-2]}, t)
		p0.MoveTo(start.X, start.Y)
		p0.LineTo(c.X, c.Y)
		p1.MoveTo(c.X, c.Y)
		p1.LineTo(d[len(d)-3], d[len(d)-2])
	} else if cmd == QuadToCmd {
		r0, r1, r2, q0, q1, q2 := quadraticBezierSplit(start, Point{d[1], d[2]}, Point{d[3], d[4]}, t)
		p0.MoveTo(r0.X, r0.Y)
		p0.QuadTo(r1.X, r1.Y, r2.X, r2.Y)
		p1.MoveTo(q0.X, q0.Y)
		p1.QuadTo(q1.X, q1.Y, q2.X, q2.Y)
	} else if cmd == CubeToCmd {
		r0, r1, r2, r3, q0, q1, q2, q3 := cubicBezierSplit(start, Point{d[1], d[2]}, Point{d[3], d[4]}, Point{d[5], d[6]}, t)
		p0.MoveTo(r0.X, r0.Y)
		p0.CubeTo(r1.X, r1.Y, r2.X, r2.Y, r3.X, r3.Y)
		p1.MoveTo(q0.X, q0.Y)
		p1.CubeTo(q1.X, q1.Y, q2.X, q2.Y, q3.X, q3.Y)
	} else if cmd == ArcToCmd {
		large, sweep := toArcFlags(d[4])
		cx, cy, theta0, theta1 := ellipseToCenter(start.X, start.Y, d[1], d[2], d[3], large, sweep, d[5], d[6])
		theta := theta0 + (theta1-theta0)*t
		c, large0, large1, ok := ellipseSplit(d[1], d[2], d[3], cx, cy, theta0, theta1, theta)
		if !ok {
			// should never happen
			panic("theta not in elliptic arc range for splitting")
		}
		p0.MoveTo(start.X, start.Y)
		p0.ArcTo(d[1], d[2], d[3]*180.0/math.Pi, large0, sweep, c.X, c.Y)
		p1.MoveTo(c.X, c.Y)
		p1.ArcTo(d[1], d[2], d[3]*180.0/math.Pi, large1, sweep, d[len(d)-3], d[len(d)-2])
	}
	return p0, p1
}

// PathIntersection is an intersection of a path.
// Intersection is either tangent or secant. Tangent intersections may be Parallel. Secant intersections either go into the other path (Into is set) or the other path goes into this path (Into is not set).
// Possible types of intersections:
//   - Crossing anywhere: Tangent=false, Parallel=false
//   - Touching anywhere: Tangent=true, Parallel=false, Into is invalid
//   - Parallel onwards:  Tangent=false, Parallel=true, Into is invalid
//
// NB: Tangent may also be true for non-closing paths when touching its endpoints
type PathIntersection struct {
	Point         // coordinate of intersection
	Seg   int     // segment index
	T     float64 // position along segment [0,1]
	Dir   float64 // direction at intersection

	Into     bool // going forward, path goes to LHS of other path
	Parallel bool // going forward, paths are parallel
	Tangent  bool // intersection is tangent (touches) instead of secant (crosses)
}

func (z PathIntersection) Less(o PathIntersection) bool {
	ti := float64(z.Seg) + z.T
	tj := float64(o.Seg) + o.T
	// TODO: this generates panics
	//if Equal(ti, tj) {
	//	// Q crosses P twice at the same point, Q must be at a tangent intersections, since
	//	// all secant and parallel tangent intersections have been removed with Settle.
	//	// Choose the parallel-end first and then the parallel-start
	//	return !z.Parallel
	//}
	return z.Seg < o.Seg || ti < tj
}

func (z PathIntersection) Equals(o PathIntersection) bool {
	return z.Point.Equals(o.Point) && z.Seg == o.Seg && Equal(z.T, o.T) && angleEqual(z.Dir, o.Dir) && z.Into == o.Into && z.Parallel == o.Parallel && z.Tangent == o.Tangent
}

func (z PathIntersection) String() string {
	extra := ""
	if z.Into {
		extra += " Into"
	}
	if z.Parallel {
		extra += " Parallel"
	}
	if z.Tangent {
		extra += " Tangent"
	}
	return fmt.Sprintf("({%v,%v} seg=%d t=%v dir=%v°%v)", numEps(z.Point.X), numEps(z.Point.Y), z.Seg, numEps(z.T), numEps(angleNorm(z.Dir)*180.0/math.Pi), extra)
}

type pathIntersectionSort struct {
	zs  []PathIntersection
	idx []int
}

func (a pathIntersectionSort) Len() int {
	return len(a.zs)
}

func (a pathIntersectionSort) Swap(i, j int) {
	a.zs[i], a.zs[j] = a.zs[j], a.zs[i]
	a.idx[i], a.idx[j] = a.idx[j], a.idx[i]
}

func (a pathIntersectionSort) Less(i, j int) bool {
	return a.zs[i].Less(a.zs[j])
}

type pathIntersectionsSort struct {
	zp []PathIntersection
	zq []PathIntersection
}

func (a pathIntersectionsSort) Len() int {
	return len(a.zp)
}

func (a pathIntersectionsSort) Swap(i, j int) {
	a.zp[i], a.zp[j] = a.zp[j], a.zp[i]
	if a.zq != nil {
		a.zq[i], a.zq[j] = a.zq[j], a.zq[i]
	}
}

func (a pathIntersectionsSort) Less(i, j int) bool {
	return a.zp[i].Less(a.zp[j])
}

// pathIntersections converts segment intersections into path intersections, resolving tangency at segment endpoints, collapsing runs of parallel/overlapping segments
func pathIntersections(p, q *Path, withTangents, withParallelTangents bool) ([]PathIntersection, []PathIntersection) {
	self := q == nil

	// TODO: pass []*Path?
	var ps, qs []*Path
	ps = p.Split()
	if self {
		q = p
		qs = ps
	} else {
		qs = q.Split()
	}

	lenQs := make([]int, len(qs))
	closedQs := make([]bool, len(qs))
	pointClosedQs := make([]bool, len(qs))
	for i := range qs {
		lenQs[i] = qs[i].Len()
		closedQs[i] = qs[i].Closed()
		pointClosedQs[i] = qs[i].PointClosed()
	}

	offsetP := 0
	var zp, zq []PathIntersection
	for i := range ps {
		offsetQ := 0
		lenP := ps[i].Len()

		j := 0
		if self {
			j = i
			offsetQ = offsetP
		}
		for j < len(qs) {
			// register segment indices [1,len), or [1,len-1) when closed by zero-length close command, we add segOffset when adding to PathIntersection
			qsj := qs[j]
			if self && i == j {
				qsj = nil
			}

			zs, segsP, segsQ := intersectionPath(ps[i], qsj)
			if 0 < len(zs) {
				// omit close command with zero length
				lenP, closedP := ps[i].Len(), ps[i].Closed()
				if closedP && ps[i].PointClosed() {
					lenP--
				}
				lenQ, closedQ := lenQs[j], closedQs[j]
				if pointClosedQs[j] {
					lenQ--
				}

				// sort by intersections on P and secondary on Q
				// move degenerate intersections at the end of the path to the start
				sort.Stable(intersectionPathSort{
					zs:      zs,
					segsP:   segsP,
					segsQ:   segsQ,
					lenP:    lenP,
					lenQ:    lenQ,
					closedP: closedP,
					closedQ: closedQ,
				})

				// Remove degenerate tangent intersections at segment endpoint:
				// - Intersection at endpoints for P and Q: 4 degenerate intersections
				// - Intersection at endpoints for P or Q: 2 degenerate intersections
				// - Parallel/overlapping sections: 4 degenerate intersections
				var n int
				for i := 0; i < len(zs); i += n {
					n = 1
					z := zs[i]
					startP, startQ := Equal(z.T[0], 0.0), Equal(z.T[1], 0.0)
					endP, endQ := Equal(z.T[0], 1.0), Equal(z.T[1], 1.0)
					endpointP, endpointQ := startP || endP, startQ || endQ

					if !z.Tangent {
						// crossing intersection in the middle of both segments
						PintoQ := z.Into()
						zp = append(zp, PathIntersection{
							Point: z.Point,
							Seg:   offsetP + segsP[i],
							T:     z.T[0],
							Dir:   z.Dir[0],
							Into:  PintoQ,
						})
						zq = append(zq, PathIntersection{
							Point: z.Point,
							Seg:   offsetQ + segsQ[i],
							T:     z.T[1],
							Dir:   z.Dir[1],
							Into:  !PintoQ,
						})
					} else if !endpointP && !endpointQ || !closedP && (segsP[i] == 1 && startP || segsP[i] == lenP-1 && endP) || !closedQ && (segsQ[i] == 1 && startQ || segsQ[i] == lenQ-1 && endQ) {
						// touching intersection in the middle of both segments
						// or touching at the start/end of an open path
						if withTangents {
							zp = append(zp, PathIntersection{
								Point:   z.Point,
								Seg:     offsetP + segsP[i],
								T:       z.T[0],
								Dir:     z.Dir[0],
								Tangent: true,
							})
							zq = append(zq, PathIntersection{
								Point:   z.Point,
								Seg:     offsetQ + segsQ[i],
								T:       z.T[1],
								Dir:     z.Dir[1],
								Tangent: true,
							})
						}
					} else {
						if endpointP && endpointQ {
							n = 4
						} else if endpointP || endpointQ {
							n = 2
						}
						if len(zs) < i+n {
							// TODO: remove
							if self {
								fmt.Printf("Path: len=%d data=%v\n", p.Len(), p)
							} else {
								fmt.Printf("Path P: len=%d data=%v\n", p.Len(), p)
								fmt.Printf("Path Q: len=%d data=%v\n", q.Len(), q)
							}
							for i, z := range zs {
								fmt.Printf("Intersection %d: seg=(%d,%d) t=(%v,%v) pos=(%v,%v) dir=(%v°,%v°)", i, segsP[i], segsQ[i], numEps(z.T[0]), numEps(z.T[1]), numEps(z.X), numEps(z.Y), numEps(z.Dir[0]*180.0/math.Pi), numEps(z.Dir[1]*180.0/math.Pi))
								if z.Tangent {
									fmt.Printf(" tangent")
								}
								fmt.Printf("\n")
							}
							panic("Bug found in path intersection code, please report on GitHub at https://github.com/tdewolff/canvas/issues with the path or paths that caused this panic.")
						}

						if parallelEnding := z.Aligned() || endQ && zs[i+1].AntiAligned() || !endQ && z.AntiAligned(); parallelEnding {
							// found end of parallel as it wraps around path end, skip until start
							continue
						}

						reversed := endQ && zs[i+n-2].AntiAligned() || !endQ && zs[i+n-1].AntiAligned()
						parallelStart := zs[i+n-1].Aligned() || reversed
						if !parallelStart || self {
							// intersection at segment endpoint of one or both paths
							// (thetaP0,thetaP1) is the LHS angle range for Q
							// PintoQ is the incoming and outgoing direction of P into LHS of Q
							j := i + n - 1
							zi, zo := z, zs[j]
							angleQo := zo.Dir[1]
							angleQi := angleQo + angleNorm(zi.Dir[1]+math.Pi-angleQo)
							PinQi := angleBetweenExclusive(zi.Dir[0]+math.Pi, angleQo, angleQi)
							PinQo := angleBetweenExclusive(zo.Dir[0], angleQo, angleQi)
							if tangent := PinQi == PinQo; withTangents || !tangent {
								zp = append(zp, PathIntersection{
									Point:   zo.Point,
									Seg:     offsetP + segsP[j],
									T:       zo.T[0],
									Dir:     zo.Dir[0],
									Into:    !tangent && PinQo,
									Tangent: tangent,
								})
								zq = append(zq, PathIntersection{
									Point:   zo.Point,
									Seg:     offsetQ + segsQ[j],
									T:       zo.T[1],
									Dir:     zo.Dir[1],
									Into:    !tangent && !PinQo,
									Tangent: tangent,
								})
							}
						} else {
							// intersection is parallel
							m := 0
							for {
								// find parallel end, skipping all parallel sections in between
								z := zs[(i+n+m)%len(zs)]
								if (Equal(z.T[0], 0.0) || Equal(z.T[0], 1.0)) && (Equal(z.T[1], 0.0) || Equal(z.T[1], 1.0)) {
									m += 4
								} else {
									m += 2
								}
								endQ := Equal(z.T[1], 1.0)
								if parallelStart := zs[(i+n+m-1)%len(zs)].Aligned() || endQ && zs[(i+n+m-2)%len(zs)].AntiAligned() || !endQ && zs[(i+n+m-1)%len(zs)].AntiAligned(); !parallelStart {
									// found end of parallel run
									break
								}
							}

							j0, j1 := i, i+1
							j2, j3 := (i+n+m-2)%len(zs), (i+n+m-1)%len(zs) // may wrap path end
							z0, z1, z2, z3 := zs[j0], zs[j1], zs[j2], zs[j3]

							// dangle is the turn angle following P over the parallel segments
							angleQo := angleNorm(z1.Dir[1])
							angleQi := angleQo + angleNorm(z0.Dir[1]+math.Pi-angleQo)
							PinQi := angleBetweenExclusive(z0.Dir[0]+math.Pi, angleQo, angleQi)

							dangle := zs[(i+n)%len(zs)].Dir[0] - zs[i+n-1].Dir[0]
							angleQo = angleNorm(z3.Dir[1])
							angleQi = angleQo + angleNorm(z2.Dir[1]+math.Pi-angleQo)
							PinQo := angleBetweenExclusive(z3.Dir[0]-dangle, angleQo-dangle, angleQi-dangle)
							if tangent := PinQi == PinQo; withParallelTangents || withTangents || !tangent {
								ji, jo := i+n-1, (i+n+m-1)%len(zs)
								zi, zo := zs[ji], zs[jo]
								zp = append(zp, PathIntersection{
									Point:    zi.Point,
									Seg:      offsetP + segsP[ji],
									T:        zi.T[0],
									Dir:      zi.Dir[0],
									Into:     withParallelTangents && !PinQo,
									Parallel: true,
									Tangent:  withParallelTangents && tangent,
								}, PathIntersection{
									Point:   zo.Point,
									Seg:     offsetP + segsP[jo],
									T:       zo.T[0],
									Dir:     zo.Dir[0],
									Into:    (withParallelTangents || !tangent) && PinQo,
									Tangent: tangent,
								})
								if !reversed {
									zq = append(zq, PathIntersection{
										Point:    zi.Point,
										Seg:      offsetQ + segsQ[ji],
										T:        zi.T[1],
										Dir:      zi.Dir[1],
										Into:     withParallelTangents && PinQo,
										Parallel: true,
										Tangent:  withParallelTangents && tangent,
									}, PathIntersection{
										Point:   zo.Point,
										Seg:     offsetQ + segsQ[jo],
										T:       zo.T[1],
										Dir:     zo.Dir[1],
										Into:    (withParallelTangents || !tangent) && !PinQo,
										Tangent: tangent,
									})
								} else {
									zq = append(zq, PathIntersection{
										Point:   zi.Point,
										Seg:     offsetQ + segsQ[ji],
										T:       zi.T[1],
										Dir:     zi.Dir[1],
										Into:    (withParallelTangents || !tangent) && !PinQo,
										Tangent: tangent,
									}, PathIntersection{
										Point:    zo.Point,
										Seg:      offsetQ + segsQ[jo],
										T:        zo.T[1],
										Dir:      zo.Dir[1],
										Into:     withParallelTangents && PinQo,
										Parallel: true,
										Tangent:  withParallelTangents && tangent,
									})
								}
							}
							i += m // skip parallel mid and end (here) and start (in for)
						}
					}
				}
			}
			offsetQ += lenQs[j]
			j++
		}
		offsetP += lenP
	}

	if self {
		zp = append(zp, zq...)
		zq = append(zq, zp[:len(zq)]...)
	}
	sort.Stable(pathIntersectionsSort{zp, zq})
	return zp, zq
}

type intersectionPathSort struct {
	zs               Intersections
	segsP, segsQ     []int
	lenP, lenQ       int
	closedP, closedQ bool
}

func (a intersectionPathSort) pos(seg int, t float64, length int, closed bool) float64 {
	if Equal(t, 1.0) {
		if closed && seg == length-1 {
			seg = 0 // intersection at path's end into first segment (MoveTo)
		}
		return float64(seg) + t - 0.5*Epsilon
	}
	return float64(seg) + t
}

func (a intersectionPathSort) Len() int {
	return len(a.zs)
}

func (a intersectionPathSort) Swap(i, j int) {
	a.zs[i], a.zs[j] = a.zs[j], a.zs[i]
	a.segsP[i], a.segsP[j] = a.segsP[j], a.segsP[i]
	a.segsQ[i], a.segsQ[j] = a.segsQ[j], a.segsQ[i]
}

func (a intersectionPathSort) Less(i, j int) bool {
	// Sort primarily by P, then by Q, and then sort endpoints (wrap around path's end)
	// - P may have multiple intersection, sort by P
	// - Q may intersect P twice in the same point, when at end points this will generate 4-degenerate intersections twice. We want them to be sorted per intersection on Q (thus sort by Q)
	// - Intersections at endpoints for P and Q will generate 4-degenerate intersections, to sort in order we subtract 0.5*Epsilon when at the end (T=1). Wrap intersection at the path's end to the start
	posPi := a.pos(a.segsP[i], a.zs[i].T[0], a.lenP, a.closedP)
	posPj := a.pos(a.segsP[j], a.zs[j].T[0], a.lenP, a.closedP)
	if Equal(posPi, posPj) { // equality holds within +-Epsilon
		posQi := a.pos(a.segsQ[i], a.zs[i].T[1], a.lenQ, a.closedQ)
		posQj := a.pos(a.segsQ[j], a.zs[j].T[1], a.lenQ, a.closedQ)
		return posPi+posQi/float64(a.lenQ) < posPj+posQj/float64(a.lenQ)
	}
	return posPi < posPj
}

// intersectionPath returns all intersections along a path including the path segments associated.
// If q is nil, it returns all intersections (non-tangent) within the same path (faster).
// All intersections are sorted by path P and then by path Q. P and Q must not have subpaths.
func intersectionPath(p, q *Path) (Intersections, []int, []int) {
	var zs Intersections
	var segsP, segsQ []int

	self := q == nil
	if self {
		q = p
	}

	// TODO: uses O(N^2), try sweep line or bently-ottman to reduce to O((N+K) log N) (or better yet https://dl.acm.org/doi/10.1145/147508.147511)
	// see https://www.webcitation.org/6ahkPQIsN        Bentley-Ottmann
	segP, segQ := 1, 1
	for i := 4; i < len(p.d); {
		pn := cmdLen(p.d[i])
		p0 := Point{p.d[i-3], p.d[i-2]}
		if p.d[i] == CloseCmd && p0.Equals(Point{p.d[i+1], p.d[i+2]}) {
			// point-closed
			i += pn
			segP++
			continue
		} else if self && p.d[i] == CubeToCmd {
			// TODO: find intersections in Cube after we support non-flat paths
		}

		j := 4
		segQ = 1
		if self {
			segQ = segP + 1
			j = i + pn
		}
		for j < len(q.d) {
			qn := cmdLen(q.d[j])
			q0 := Point{q.d[j-3], q.d[j-2]}
			if q.d[j] == CloseCmd && q0.Equals(Point{q.d[j+1], q.d[j+2]}) {
				// point-closed
				j += qn
				segQ++
				continue
			}

			k := len(zs)
			zs = intersectionSegment(zs, p0, p.d[i:i+pn], q0, q.d[j:j+qn])
			if self && (i+pn == j || i == 4) {
				// remove tangent intersections for adjacent segments on the same subpath
				for k1 := len(zs) - 1; k <= k1; k1-- {
					if !zs[k1].Tangent {
						continue
					}

					// segments are joined if either j comes after i, or if i is first and j is last (or before last if last is point-closed)
					joined := i+pn == j && Equal(zs[k1].T[0], 1.0) && Equal(zs[k1].T[1], 0.0) ||
						i == 4 && Equal(zs[k1].T[0], 0.0) && Equal(zs[k1].T[1], 1.0) &&
							(q.d[j] == CloseCmd || j+qn < len(q.d) && q.d[j+qn] == CloseCmd &&
								Point{q.d[j+qn-3], q.d[j+qn-2]}.Equals(Point{q.d[j+qn+1], q.d[j+qn+2]}))
					if joined {
						zs = append(zs[:k1], zs[k1+1:]...)
					}
				}
			}
			for ; k < len(zs); k++ {
				segsP = append(segsP, segP)
				segsQ = append(segsQ, segQ)
			}

			j += qn
			segQ++
		}
		i += pn
		segP++
	}
	return zs, segsP, segsQ
}

// intersect for path segments a and b, starting at a0 and b0
func intersectionSegment(zs Intersections, a0 Point, a []float64, b0 Point, b []float64) Intersections {
	n := len(zs)
	swapCurves := false
	if a[0] == LineToCmd || a[0] == CloseCmd {
		if b[0] == LineToCmd || b[0] == CloseCmd {
			zs = intersectionLineLine(zs, a0, Point{a[1], a[2]}, b0, Point{b[1], b[2]})
		} else if b[0] == QuadToCmd {
			zs = intersectionLineQuad(zs, a0, Point{a[1], a[2]}, b0, Point{b[1], b[2]}, Point{b[3], b[4]})
		} else if b[0] == CubeToCmd {
			zs = intersectionLineCube(zs, a0, Point{a[1], a[2]}, b0, Point{b[1], b[2]}, Point{b[3], b[4]}, Point{b[5], b[6]})
		} else if b[0] == ArcToCmd {
			rx := b[1]
			ry := b[2]
			phi := b[3] * math.Pi / 180.0
			large, sweep := toArcFlags(b[4])
			cx, cy, theta0, theta1 := ellipseToCenter(b0.X, b0.Y, rx, ry, phi, large, sweep, b[5], b[6])
			zs = intersectionLineEllipse(zs, a0, Point{a[1], a[2]}, Point{cx, cy}, Point{rx, ry}, phi, theta0, theta1)
		}
	} else if a[0] == QuadToCmd {
		if b[0] == LineToCmd || b[0] == CloseCmd {
			zs = intersectionLineQuad(zs, b0, Point{b[1], b[2]}, a0, Point{a[1], a[2]}, Point{a[3], a[4]})
			swapCurves = true
		} else if b[0] == QuadToCmd {
			panic("unsupported intersection for quad-quad")
		} else if b[0] == CubeToCmd {
			panic("unsupported intersection for quad-cube")
		} else if b[0] == ArcToCmd {
			panic("unsupported intersection for quad-arc")
		}
	} else if a[0] == CubeToCmd {
		if b[0] == LineToCmd || b[0] == CloseCmd {
			zs = intersectionLineCube(zs, b0, Point{b[1], b[2]}, a0, Point{a[1], a[2]}, Point{a[3], a[4]}, Point{a[5], a[6]})
			swapCurves = true
		} else if b[0] == QuadToCmd {
			panic("unsupported intersection for cube-quad")
		} else if b[0] == CubeToCmd {
			panic("unsupported intersection for cube-cube")
		} else if b[0] == ArcToCmd {
			panic("unsupported intersection for cube-arc")
		}
	} else if a[0] == ArcToCmd {
		rx := a[1]
		ry := a[2]
		phi := a[3] * math.Pi / 180.0
		large, sweep := toArcFlags(a[4])
		cx, cy, theta0, theta1 := ellipseToCenter(a0.X, a0.Y, rx, ry, phi, large, sweep, a[5], a[6])
		if b[0] == LineToCmd || b[0] == CloseCmd {
			zs = intersectionLineEllipse(zs, b0, Point{b[1], b[2]}, Point{cx, cy}, Point{rx, ry}, phi, theta0, theta1)
			swapCurves = true
		} else if b[0] == QuadToCmd {
			panic("unsupported intersection for arc-quad")
		} else if b[0] == CubeToCmd {
			panic("unsupported intersection for arc-cube")
		} else if b[0] == ArcToCmd {
			rx2 := b[1]
			ry2 := b[2]
			phi2 := b[3] * math.Pi / 180.0
			large2, sweep2 := toArcFlags(b[4])
			cx2, cy2, theta20, theta21 := ellipseToCenter(b0.X, b0.Y, rx2, ry2, phi2, large2, sweep2, b[5], b[6])
			zs = intersectionEllipseEllipse(zs, Point{cx, cy}, Point{rx, ry}, phi, theta0, theta1, Point{cx2, cy2}, Point{rx2, ry2}, phi2, theta20, theta21)
		}
	}

	// swap A and B in the intersection found to match segments A and B of this function
	if swapCurves {
		for i := n; i < len(zs); i++ {
			zs[i].T[0], zs[i].T[1] = zs[i].T[1], zs[i].T[0]
			zs[i].Dir[0], zs[i].Dir[1] = zs[i].Dir[1], zs[i].Dir[0]
		}
	}
	return zs
}

// Intersection is an intersection between two path segments, e.g. Line x Line.
// Note that intersection is tangent also when it is one of the endpoints, in which case it may be tangent for this segment but we should double check when converting to a PathIntersection as it may or may not cross depending on the adjacent segment(s). Also, the Into value at tangent intersections at endpoints should be interpreted as if the paths were extended and the path would go into the left-hand side of the other path.
// Possible types of intersections:
//   - Crossing not at endpoint: Tangent=false, Aligned=false
//   - Touching not at endpoint: Tangent=true,  Aligned=true, Into is invalid
//   - Touching at endpoint:     Tangent=true,  may be aligned for (partly) overlapping paths
//
// NB: for quad/cube/ellipse aligned angles at the endpoint for non-overlapping curves are deviated slightly to correctly calculate the value for Into, and will thus not be aligned
type Intersection struct {
	Point              // coordinate of intersection
	T       [2]float64 // position along segment [0,1]
	Dir     [2]float64 // direction at intersection [0,2*pi)
	Tangent bool       // intersection is tangent (touches) instead of secant (crosses)
}

// Aligned is true when both paths are aligned at the intersection (angles are equal).
func (z Intersection) Aligned() bool {
	return angleEqual(z.Dir[0], z.Dir[1])
}

// AntiAligned is true when both paths are anti-aligned at the intersection (angles are opposite).
func (z Intersection) AntiAligned() bool {
	return angleEqual(z.Dir[0], z.Dir[1]+math.Pi)
}

// Into returns true if first path goes into the left-hand side of the second path,
// i.e. the second path goes to the right-hand side of the first path.
func (z Intersection) Into() bool {
	// TODO: test that direction is either aligned, or Into is true when in [pi,2*pi]
	return angleBetweenExclusive(z.Dir[1]-z.Dir[0], math.Pi, 2.0*math.Pi)
}

func (z Intersection) Equals(o Intersection) bool {
	return z.Point.Equals(o.Point) && Equal(z.T[0], o.T[0]) && Equal(z.T[1], o.T[1]) && angleEqual(z.Dir[0], o.Dir[0]) && angleEqual(z.Dir[1], o.Dir[1]) && z.Tangent == o.Tangent
}

func (z Intersection) String() string {
	tangent := ""
	if z.Tangent {
		tangent = " Tangent"
	}
	return fmt.Sprintf("({%v,%v} t={%v,%v} dir={%v°,%v°}%v)", numEps(z.Point.X), numEps(z.Point.Y), numEps(z.T[0]), numEps(z.T[1]), numEps(angleNorm(z.Dir[0])*180.0/math.Pi), numEps(angleNorm(z.Dir[1])*180.0/math.Pi), tangent)
}

type Intersections []Intersection

// Has returns true if there are secant/tangent intersections.
func (zs Intersections) Has() bool {
	return 0 < len(zs)
}

// HasSecant returns true when there are secant intersections, i.e. the curves intersect and cross (they cut).
func (zs Intersections) HasSecant() bool {
	for _, z := range zs {
		if !z.Tangent {
			return true
		}
	}
	return false
}

// HasTangent returns true when there are tangent intersections, i.e. the curves intersect but don't cross (they touch).
func (zs Intersections) HasTangent() bool {
	for _, z := range zs {
		if z.Tangent {
			return true
		}
	}
	return false
}

func (zs Intersections) add(pos Point, ta, tb, dira, dirb float64, tangent bool) Intersections {
	ta = math.Max(0.0, math.Min(1.0, ta))
	tb = math.Max(0.0, math.Min(1.0, tb))
	return append(zs, Intersection{pos, [2]float64{ta, tb}, [2]float64{dira, dirb}, tangent})
}

// https://www.geometrictools.com/GTE/Mathematics/IntrLine2Line2.h
func intersectionLineLine(zs Intersections, a0, a1, b0, b1 Point) Intersections {
	if a0.Equals(a1) || b0.Equals(b1) {
		return zs // zero-length Close
	}

	da := a1.Sub(a0)
	db := b1.Sub(b0)
	angle0 := da.Angle()
	angle1 := db.Angle()
	if angleEqual(angle0, angle1) || angleEqual(angle0, angle1+math.Pi) {
		// parallel
		if Equal(da.PerpDot(b1.Sub(a0)), 0.0) {
			// aligned, rotate to x-axis
			a := a0.Rot(-angle0, Point{}).X
			b := a1.Rot(-angle0, Point{}).X
			c := b0.Rot(-angle0, Point{}).X
			d := b1.Rot(-angle0, Point{}).X
			if Interval(a, c, d) && Interval(b, c, d) || Interval(a, d, c) && Interval(b, d, c) {
				// a-b in c-d or a-b == c-d
				zs = zs.add(a0, 0.0, (a-c)/(d-c), angle0, angle1, true)
				zs = zs.add(a1, 1.0, (b-c)/(d-c), angle0, angle1, true)
			} else if Interval(c, a, b) && Interval(d, a, b) {
				// c-d in a-b
				zs = zs.add(b0, (c-a)/(b-a), 0.0, angle0, angle1, true)
				zs = zs.add(b1, (d-a)/(b-a), 1.0, angle0, angle1, true)
			} else if Interval(a, c, d) || Interval(a, d, c) {
				// a in c-d
				zs = zs.add(a0, 0.0, (a-c)/(d-c), angle0, angle1, true)
				if a < d-Epsilon {
					zs = zs.add(b1, (d-a)/(b-a), 1.0, angle0, angle1, true)
				} else if a < c-Epsilon {
					zs = zs.add(b0, (c-a)/(b-a), 0.0, angle0, angle1, true)
				}
			} else if Interval(b, c, d) || Interval(b, d, c) {
				// b in c-d
				if c < b-Epsilon {
					zs = zs.add(b0, (c-a)/(b-a), 0.0, angle0, angle1, true)
				} else if d < b-Epsilon {
					zs = zs.add(b1, (d-a)/(b-a), 1.0, angle0, angle1, true)
				}
				zs = zs.add(a1, 1.0, (b-c)/(d-c), angle0, angle1, true)
			}
		}
		return zs
	} else if a1.Equals(b0) {
		// handle common cases with endpoints to avoid numerical issues
		zs = zs.add(a1, 1.0, 0.0, angle0, angle1, true)
		return zs
	} else if a0.Equals(b1) {
		// handle common cases with endpoints to avoid numerical issues
		zs = zs.add(a0, 0.0, 1.0, angle0, angle1, true)
		return zs
	}

	div := da.PerpDot(db)
	ta := db.PerpDot(a0.Sub(b0)) / div
	tb := da.PerpDot(a0.Sub(b0)) / div
	if Interval(ta, 0.0, 1.0) && Interval(tb, 0.0, 1.0) {
		tangent := Equal(ta, 0.0) || Equal(ta, 1.0) || Equal(tb, 0.0) || Equal(tb, 1.0)
		zs = zs.add(a0.Interpolate(a1, ta), ta, tb, da.Angle(), db.Angle(), tangent)
	}
	return zs
}

// https://www.particleincell.com/2013/cubic-line-intersection/
func intersectionLineQuad(zs Intersections, l0, l1, p0, p1, p2 Point) Intersections {
	if l0.Equals(l1) {
		return zs // zero-length Close
	}

	// write line as A.X = bias
	A := Point{l1.Y - l0.Y, l0.X - l1.X}
	bias := l0.Dot(A)

	a := A.Dot(p0.Sub(p1.Mul(2.0)).Add(p2))
	b := A.Dot(p1.Sub(p0).Mul(2.0))
	c := A.Dot(p0) - bias

	roots := []float64{}
	r0, r1 := solveQuadraticFormula(a, b, c)
	if !math.IsNaN(r0) {
		roots = append(roots, r0)
		if !math.IsNaN(r1) {
			roots = append(roots, r1)
		}
	}

	dira := l1.Sub(l0).Angle()
	horizontal := math.Abs(l1.Y-l0.Y) <= math.Abs(l1.X-l0.X)
	for _, root := range roots {
		if Interval(root, 0.0, 1.0) {
			var s float64
			pos := quadraticBezierPos(p0, p1, p2, root)
			if horizontal {
				s = (pos.X - l0.X) / (l1.X - l0.X)
			} else {
				s = (pos.Y - l0.Y) / (l1.Y - l0.Y)
			}
			if Interval(s, 0.0, 1.0) {
				deriv := quadraticBezierDeriv(p0, p1, p2, root)
				dirb := deriv.Angle()
				endpoint := Equal(root, 0.0) || Equal(root, 1.0) || Equal(s, 0.0) || Equal(s, 1.0)
				if endpoint {
					// deviate angle slightly at endpoint when aligned to properly set Into
					deriv2 := quadraticBezierDeriv2(p0, p1, p2)
					if (0.0 <= deriv.PerpDot(deriv2)) == (Equal(root, 0.0) || !Equal(root, 1.0) && Equal(s, 0.0)) {
						dirb += Epsilon * 2.0 // t=0 and CCW, or t=1 and CW
					} else {
						dirb -= Epsilon * 2.0 // t=0 and CW, or t=1 and CCW
					}
					dirb = angleNorm(dirb)
				}
				zs = zs.add(pos, s, root, dira, dirb, endpoint || Equal(A.Dot(deriv), 0.0))
			}
		}
	}
	return zs
}

// https://www.particleincell.com/2013/cubic-line-intersection/
func intersectionLineCube(zs Intersections, l0, l1, p0, p1, p2, p3 Point) Intersections {
	if l0.Equals(l1) {
		return zs // zero-length Close
	}

	// write line as A.X = bias
	A := Point{l1.Y - l0.Y, l0.X - l1.X}
	bias := l0.Dot(A)

	a := A.Dot(p3.Sub(p0).Add(p1.Mul(3.0)).Sub(p2.Mul(3.0)))
	b := A.Dot(p0.Mul(3.0).Sub(p1.Mul(6.0)).Add(p2.Mul(3.0)))
	c := A.Dot(p1.Mul(3.0).Sub(p0.Mul(3.0)))
	d := A.Dot(p0) - bias

	roots := []float64{}
	r0, r1, r2 := solveCubicFormula(a, b, c, d)
	if !math.IsNaN(r0) {
		roots = append(roots, r0)
		if !math.IsNaN(r1) {
			roots = append(roots, r1)
			if !math.IsNaN(r2) {
				roots = append(roots, r2)
			}
		}
	}

	dira := l1.Sub(l0).Angle()
	horizontal := math.Abs(l1.Y-l0.Y) <= math.Abs(l1.X-l0.X)
	for _, root := range roots {
		if Interval(root, 0.0, 1.0) {
			var s float64
			pos := cubicBezierPos(p0, p1, p2, p3, root)
			if horizontal {
				s = (pos.X - l0.X) / (l1.X - l0.X)
			} else {
				s = (pos.Y - l0.Y) / (l1.Y - l0.Y)
			}
			if Interval(s, 0.0, 1.0) {
				deriv := cubicBezierDeriv(p0, p1, p2, p3, root)
				dirb := deriv.Angle()
				tangent := Equal(A.Dot(deriv), 0.0)
				endpoint := Equal(root, 0.0) || Equal(root, 1.0) || Equal(s, 0.0) || Equal(s, 1.0)
				if endpoint {
					// deviate angle slightly at endpoint when aligned to properly set Into
					deriv2 := cubicBezierDeriv2(p0, p1, p2, p3, root)
					if (0.0 <= deriv.PerpDot(deriv2)) == (Equal(root, 0.0) || !Equal(root, 1.0) && Equal(s, 0.0)) {
						dirb += Epsilon * 2.0 // t=0 and CCW, or t=1 and CW
					} else {
						dirb -= Epsilon * 2.0 // t=0 and CW, or t=1 and CCW
					}
				} else if angleEqual(dira, dirb) || angleEqual(dira, dirb+math.Pi) {
					// directions are parallel but the paths do cross (inflection point)
					// TODO: test better
					deriv2 := cubicBezierDeriv2(p0, p1, p2, p3, root)
					if Equal(deriv2.X, 0.0) && Equal(deriv2.Y, 0.0) {
						deriv3 := cubicBezierDeriv3(p0, p1, p2, p3, root)
						if 0.0 < deriv.PerpDot(deriv3) {
							dirb += Epsilon * 2.0
						} else {
							dirb -= Epsilon * 2.0
						}
						dirb = angleNorm(dirb)
						tangent = false
					}
				}
				zs = zs.add(pos, s, root, dira, dirb, endpoint || tangent)
			}
		}
	}
	return zs
}

// handle line-arc intersections and their peculiarities regarding angles
func addLineArcIntersection(zs Intersections, pos Point, dira, dirb, t, t0, t1, angle, theta0, theta1 float64, tangent bool) Intersections {
	if theta0 <= theta1 {
		angle = theta0 - Epsilon + angleNorm(angle-theta0+Epsilon)
	} else {
		angle = theta1 - Epsilon + angleNorm(angle-theta1+Epsilon)
	}
	endpoint := Equal(t, t0) || Equal(t, t1) || Equal(angle, theta0) || Equal(angle, theta1)
	if endpoint {
		// deviate angle slightly at endpoint when aligned to properly set Into
		if (theta0 <= theta1) == (Equal(angle, theta0) || !Equal(angle, theta1) && Equal(t, t0)) {
			dirb += Epsilon * 2.0 // t=0 and CCW, or t=1 and CW
		} else {
			dirb -= Epsilon * 2.0 // t=0 and CW, or t=1 and CCW
		}
		dirb = angleNorm(dirb)
	}

	// snap segment parameters to 0.0 and 1.0 to avoid numerical issues
	var s float64
	if Equal(t, t0) {
		t = 0.0
	} else if Equal(t, t1) {
		t = 1.0
	} else {
		t = (t - t0) / (t1 - t0)
	}
	if Equal(angle, theta0) {
		s = 0.0
	} else if Equal(angle, theta1) {
		s = 1.0
	} else {
		s = (angle - theta0) / (theta1 - theta0)
	}
	return zs.add(pos, t, s, dira, dirb, endpoint || tangent)
}

// https://www.geometrictools.com/GTE/Mathematics/IntrLine2Circle2.h
func intersectionLineCircle(zs Intersections, l0, l1, center Point, radius, theta0, theta1 float64) Intersections {
	if l0.Equals(l1) {
		return zs // zero-length Close
	}

	// solve l0 + t*(l1-l0) = P + t*D = X  (line equation)
	// and |X - center| = |X - C| = R = radius  (circle equation)
	// by substitution and squaring: |P + t*D - C|^2 = R^2
	// giving: D^2 t^2 + 2D(P-C) t + (P-C)^2-R^2 = 0
	dir := l1.Sub(l0)
	diff := l0.Sub(center) // P-C
	length := dir.Length()
	D := dir.Div(length)

	// we normalise D to be of length 1, so that the roots are in [0,length]
	a := 1.0
	b := 2.0 * D.Dot(diff)
	c := diff.Dot(diff) - radius*radius

	// find solutions for t ∈ [0,1], the parameter along the line's path
	roots := []float64{}
	r0, r1 := solveQuadraticFormula(a, b, c)
	if !math.IsNaN(r0) {
		roots = append(roots, r0)
		if !math.IsNaN(r1) && !Equal(r0, r1) {
			roots = append(roots, r1)
		}
	}

	// handle common cases with endpoints to avoid numerical issues
	// snap closest root to path's start or end
	if 0 < len(roots) {
		if pos := l0.Sub(center); Equal(pos.Length(), radius) {
			if len(roots) == 1 || math.Abs(roots[0]) < math.Abs(roots[1]) {
				roots[0] = 0.0
			} else {
				roots[1] = 0.0
			}
		}
		if pos := l1.Sub(center); Equal(pos.Length(), radius) {
			if len(roots) == 1 || math.Abs(roots[0]-length) < math.Abs(roots[1]-length) {
				roots[0] = length
			} else {
				roots[1] = length
			}
		}
	}

	// add intersections
	dira := dir.Angle()
	tangent := len(roots) == 1
	for _, root := range roots {
		pos := diff.Add(dir.Mul(root / length))
		angle := math.Atan2(pos.Y*radius, pos.X*radius)
		if Interval(root, 0.0, length) && angleBetween(angle, theta0, theta1) {
			pos = center.Add(pos)
			dirb := ellipseDeriv(radius, radius, 0.0, theta0 <= theta1, angle).Angle()
			zs = addLineArcIntersection(zs, pos, dira, dirb, root, 0.0, length, angle, theta0, theta1, tangent)
		}
	}
	return zs
}

func intersectionLineEllipse(zs Intersections, l0, l1, center, radius Point, phi, theta0, theta1 float64) Intersections {
	if Equal(radius.X, radius.Y) {
		return intersectionLineCircle(zs, l0, l1, center, radius.X, theta0, theta1)
	} else if l0.Equals(l1) {
		return zs // zero-length Close
	}

	// TODO: needs more testing
	// TODO: intersection inconsistency due to numerical stability in finding tangent collisions for subsequent paht segments (line -> ellipse), or due to the endpoint of a line not touching with another arc, but the subsequent segment does touch with its starting point
	dira := l1.Sub(l0).Angle()

	// we take the ellipse center as the origin and counter-rotate by phi
	l0 = l0.Sub(center).Rot(-phi, Origin)
	l1 = l1.Sub(center).Rot(-phi, Origin)

	// line: cx + dy + e = 0
	c := l0.Y - l1.Y
	d := l1.X - l0.X
	e := l0.PerpDot(l1)

	// follow different code paths when line is mostly horizontal or vertical
	horizontal := math.Abs(c) <= math.Abs(d)

	// ellipse: x^2/a + y^2/b = 1
	a := radius.X * radius.X
	b := radius.Y * radius.Y

	// rewrite as a polynomial by substituting x or y to obtain:
	// At^2 + Bt + C = 0, with t either x (horizontal) or y (!horizontal)
	var A, B, C float64
	A = a*c*c + b*d*d
	if horizontal {
		B = 2.0 * a * c * e
		C = a*e*e - a*b*d*d
	} else {
		B = 2.0 * b * d * e
		C = b*e*e - a*b*c*c
	}

	// find solutions
	roots := []float64{}
	r0, r1 := solveQuadraticFormula(A, B, C)
	if !math.IsNaN(r0) {
		roots = append(roots, r0)
		if !math.IsNaN(r1) && !Equal(r0, r1) {
			roots = append(roots, r1)
		}
	}

	for _, root := range roots {
		// get intersection position with center as origin
		var x, y, t0, t1 float64
		if horizontal {
			x = root
			y = -e/d - c*root/d
			t0 = l0.X
			t1 = l1.X
		} else {
			x = -e/c - d*root/c
			y = root
			t0 = l0.Y
			t1 = l1.Y
		}

		tangent := Equal(root, 0.0)
		angle := math.Atan2(y*radius.X, x*radius.Y)
		if Interval(root, t0, t1) && angleBetween(angle, theta0, theta1) {
			pos := Point{x, y}.Rot(phi, Origin).Add(center)
			dirb := ellipseDeriv(radius.X, radius.Y, phi, theta0 <= theta1, angle).Angle()
			zs = addLineArcIntersection(zs, pos, dira, dirb, root, t0, t1, angle, theta0, theta1, tangent)
		}
	}
	return zs
}

func intersectionEllipseEllipse(zs Intersections, c0, r0 Point, phi0, thetaStart0, thetaEnd0 float64, c1, r1 Point, phi1, thetaStart1, thetaEnd1 float64) Intersections {
	// TODO: needs more testing
	if !Equal(r0.X, r0.Y) || !Equal(r1.X, r1.Y) {
		panic("not handled") // ellipses
	}

	arcAngle := func(theta float64, sweep bool) float64 {
		theta += math.Pi / 2.0
		if !sweep {
			theta -= math.Pi
		}
		return angleNorm(theta)
	}

	dtheta0 := thetaEnd0 - thetaStart0
	thetaStart0 = angleNorm(thetaStart0 + phi0)
	thetaEnd0 = thetaStart0 + dtheta0

	dtheta1 := thetaEnd1 - thetaStart1
	thetaStart1 = angleNorm(thetaStart1 + phi1)
	thetaEnd1 = thetaStart1 + dtheta1

	if c0.Equals(c1) && r0.Equals(r1) {
		// parallel
		tOffset1 := 0.0
		dirOffset1 := 0.0
		if (0.0 <= dtheta0) != (0.0 <= dtheta1) {
			thetaStart1, thetaEnd1 = thetaEnd1, thetaStart1 // keep order on first arc
			dirOffset1 = math.Pi
			tOffset1 = 1.0
		}

		// will add either 1 (when touching) or 2 (when overlapping) intersections
		if t := angleTime(thetaStart0, thetaStart1, thetaEnd1); Interval(t, 0.0, 1.0) {
			// ellipse0 starts within/on border of ellipse1
			dir := arcAngle(thetaStart0, 0.0 <= dtheta0)
			pos := EllipsePos(r0.X, r0.Y, 0.0, c0.X, c0.Y, thetaStart0)
			zs = zs.add(pos, 0.0, math.Abs(t-tOffset1), dir, angleNorm(dir+dirOffset1), true)
		}
		if t := angleTime(thetaStart1, thetaStart0, thetaEnd0); IntervalExclusive(t, 0.0, 1.0) {
			// ellipse1 starts within ellipse0
			dir := arcAngle(thetaStart1, 0.0 <= dtheta0)
			pos := EllipsePos(r0.X, r0.Y, 0.0, c0.X, c0.Y, thetaStart1)
			zs = zs.add(pos, t, tOffset1, dir, angleNorm(dir+dirOffset1), true)
		}
		if t := angleTime(thetaEnd1, thetaStart0, thetaEnd0); IntervalExclusive(t, 0.0, 1.0) {
			// ellipse1 ends within ellipse0
			dir := arcAngle(thetaEnd1, 0.0 <= dtheta0)
			pos := EllipsePos(r0.X, r0.Y, 0.0, c0.X, c0.Y, thetaEnd1)
			zs = zs.add(pos, t, 1.0-tOffset1, dir, angleNorm(dir+dirOffset1), true)
		}
		if t := angleTime(thetaEnd0, thetaStart1, thetaEnd1); Interval(t, 0.0, 1.0) {
			// ellipse0 ends within/on border of ellipse1
			dir := arcAngle(thetaEnd0, 0.0 <= dtheta0)
			pos := EllipsePos(r0.X, r0.Y, 0.0, c0.X, c0.Y, thetaEnd0)
			zs = zs.add(pos, 1.0, math.Abs(t-tOffset1), dir, angleNorm(dir+dirOffset1), true)
		}
		return zs
	}

	// https://math.stackexchange.com/questions/256100/how-can-i-find-the-points-at-which-two-circles-intersect
	// https://gist.github.com/jupdike/bfe5eb23d1c395d8a0a1a4ddd94882ac
	R := c0.Sub(c1).Length()
	if R < math.Abs(r0.X-r1.X) || r0.X+r1.X < R {
		return zs
	}
	R2 := R * R

	k := r0.X*r0.X - r1.X*r1.X
	a := 0.5
	b := 0.5 * k / R2
	c := 0.5 * math.Sqrt(2.0*(r0.X*r0.X+r1.X*r1.X)/R2-k*k/(R2*R2)-1.0)

	mid := c1.Sub(c0).Mul(a + b)
	dev := Point{c1.Y - c0.Y, c0.X - c1.X}.Mul(c)

	tangent := dev.Equals(Point{})
	anglea0 := mid.Add(dev).Angle()
	anglea1 := c0.Sub(c1).Add(mid).Add(dev).Angle()
	ta0 := angleTime(anglea0, thetaStart0, thetaEnd0)
	ta1 := angleTime(anglea1, thetaStart1, thetaEnd1)
	if Interval(ta0, 0.0, 1.0) && Interval(ta1, 0.0, 1.0) {
		dir0 := arcAngle(anglea0, 0.0 <= dtheta0)
		dir1 := arcAngle(anglea1, 0.0 <= dtheta1)
		endpoint := Equal(ta0, 0.0) || Equal(ta0, 1.0) || Equal(ta1, 0.0) || Equal(ta1, 1.0)
		zs = zs.add(c0.Add(mid).Add(dev), ta0, ta1, dir0, dir1, tangent || endpoint)
	}

	if !tangent {
		angleb0 := mid.Sub(dev).Angle()
		angleb1 := c0.Sub(c1).Add(mid).Sub(dev).Angle()
		tb0 := angleTime(angleb0, thetaStart0, thetaEnd0)
		tb1 := angleTime(angleb1, thetaStart1, thetaEnd1)
		if Interval(tb0, 0.0, 1.0) && Interval(tb1, 0.0, 1.0) {
			dir0 := arcAngle(angleb0, 0.0 <= dtheta0)
			dir1 := arcAngle(angleb1, 0.0 <= dtheta1)
			endpoint := Equal(tb0, 0.0) || Equal(tb0, 1.0) || Equal(tb1, 0.0) || Equal(tb1, 1.0)
			zs = zs.add(c0.Add(mid).Sub(dev), tb0, tb1, dir0, dir1, endpoint)
		}
	}
	return zs
}

// TODO: bezier-bezier intersection
// TODO: bezier-ellipse intersection

// For Bézier-Bézier interesections:
// see T.W. Sederberg, "Computer Aided Geometric Design", 2012
// see T.W. Sederberg and T. Nishita, "Curve intersection using Bézier clipping", 1990
// see T.W. Sederberg and S.R. Parry, "Comparison of three curve intersection algorithms", 1986

func intersectionRayLine(a0, a1, b0, b1 Point) (Point, bool) {
	da := a1.Sub(a0)
	db := b1.Sub(b0)
	div := da.PerpDot(db)
	if Equal(div, 0.0) {
		// parallel
		return Point{}, false
	}

	tb := da.PerpDot(a0.Sub(b0)) / div
	if Interval(tb, 0.0, 1.0) {
		return b0.Interpolate(b1, tb), true
	}
	return Point{}, false
}

// http://mathworld.wolfram.com/Circle-LineIntersection.html
func intersectionRayCircle(l0, l1, c Point, r float64) (Point, Point, bool) {
	d := l1.Sub(l0).Norm(1.0) // along line direction, anchored in l0, its length is 1
	D := l0.Sub(c).PerpDot(d)
	discriminant := r*r - D*D
	if discriminant < 0 {
		return Point{}, Point{}, false
	}
	discriminant = math.Sqrt(discriminant)

	ax := D * d.Y
	bx := d.X * discriminant
	if d.Y < 0.0 {
		bx = -bx
	}
	ay := -D * d.X
	by := math.Abs(d.Y) * discriminant
	return c.Add(Point{ax + bx, ay + by}), c.Add(Point{ax - bx, ay - by}), true
}

// https://math.stackexchange.com/questions/256100/how-can-i-find-the-points-at-which-two-circles-intersect
// https://gist.github.com/jupdike/bfe5eb23d1c395d8a0a1a4ddd94882ac
func intersectionCircleCircle(c0 Point, r0 float64, c1 Point, r1 float64) (Point, Point, bool) {
	R := c0.Sub(c1).Length()
	if R < math.Abs(r0-r1) || r0+r1 < R || c0.Equals(c1) {
		return Point{}, Point{}, false
	}
	R2 := R * R

	k := r0*r0 - r1*r1
	a := 0.5
	b := 0.5 * k / R2
	c := 0.5 * math.Sqrt(2.0*(r0*r0+r1*r1)/R2-k*k/(R2*R2)-1.0)

	i0 := c0.Add(c1).Mul(a)
	i1 := c1.Sub(c0).Mul(b)
	i2 := Point{c1.Y - c0.Y, c0.X - c1.X}.Mul(c)
	return i0.Add(i1).Add(i2), i0.Add(i1).Sub(i2), true
}