gtsocial-umbx

loop.go (66564B)

---

 // Copyright 2015 Google Inc. All rights reserved.
 //
 // Licensed under the Apache License, Version 2.0 (the "License");
 // you may not use this file except in compliance with the License.
 // You may obtain a copy of the License at
 //
 //     http://www.apache.org/licenses/LICENSE-2.0
 //
 // Unless required by applicable law or agreed to in writing, software
 // distributed under the License is distributed on an "AS IS" BASIS,
 // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 // See the License for the specific language governing permissions and
 // limitations under the License.
 
 package s2
 
 import (
 	"fmt"
 	"io"
 	"math"
 
 	"github.com/golang/geo/r1"
 	"github.com/golang/geo/r3"
 	"github.com/golang/geo/s1"
 )
 
 // Loop represents a simple spherical polygon. It consists of a sequence
 // of vertices where the first vertex is implicitly connected to the
 // last. All loops are defined to have a CCW orientation, i.e. the interior of
 // the loop is on the left side of the edges. This implies that a clockwise
 // loop enclosing a small area is interpreted to be a CCW loop enclosing a
 // very large area.
 //
 // Loops are not allowed to have any duplicate vertices (whether adjacent or
 // not).  Non-adjacent edges are not allowed to intersect, and furthermore edges
 // of length 180 degrees are not allowed (i.e., adjacent vertices cannot be
 // antipodal). Loops must have at least 3 vertices (except for the "empty" and
 // "full" loops discussed below).
 //
 // There are two special loops: the "empty" loop contains no points and the
 // "full" loop contains all points. These loops do not have any edges, but to
 // preserve the invariant that every loop can be represented as a vertex
 // chain, they are defined as having exactly one vertex each (see EmptyLoop
 // and FullLoop).
 type Loop struct {
 	vertices []Point
 
 	// originInside keeps a precomputed value whether this loop contains the origin
 	// versus computing from the set of vertices every time.
 	originInside bool
 
 	// depth is the nesting depth of this Loop if it is contained by a Polygon
 	// or other shape and is used to determine if this loop represents a hole
 	// or a filled in portion.
 	depth int
 
 	// bound is a conservative bound on all points contained by this loop.
 	// If l.ContainsPoint(P), then l.bound.ContainsPoint(P).
 	bound Rect
 
 	// Since bound is not exact, it is possible that a loop A contains
 	// another loop B whose bounds are slightly larger. subregionBound
 	// has been expanded sufficiently to account for this error, i.e.
 	// if A.Contains(B), then A.subregionBound.Contains(B.bound).
 	subregionBound Rect
 
 	// index is the spatial index for this Loop.
 	index *ShapeIndex
 }
 
 // LoopFromPoints constructs a loop from the given points.
 func LoopFromPoints(pts []Point) *Loop {
 	l := &Loop{
 		vertices: pts,
 		index:    NewShapeIndex(),
 	}
 
 	l.initOriginAndBound()
 	return l
 }
 
 // LoopFromCell constructs a loop corresponding to the given cell.
 //
 // Note that the loop and cell *do not* contain exactly the same set of
 // points, because Loop and Cell have slightly different definitions of
 // point containment. For example, a Cell vertex is contained by all
 // four neighboring Cells, but it is contained by exactly one of four
 // Loops constructed from those cells. As another example, the cell
 // coverings of cell and LoopFromCell(cell) will be different, because the
 // loop contains points on its boundary that actually belong to other cells
 // (i.e., the covering will include a layer of neighboring cells).
 func LoopFromCell(c Cell) *Loop {
 	l := &Loop{
 		vertices: []Point{
 			c.Vertex(0),
 			c.Vertex(1),
 			c.Vertex(2),
 			c.Vertex(3),
 		},
 		index: NewShapeIndex(),
 	}
 
 	l.initOriginAndBound()
 	return l
 }
 
 // These two points are used for the special Empty and Full loops.
 var (
 	emptyLoopPoint = Point{r3.Vector{X: 0, Y: 0, Z: 1}}
 	fullLoopPoint  = Point{r3.Vector{X: 0, Y: 0, Z: -1}}
 )
 
 // EmptyLoop returns a special "empty" loop.
 func EmptyLoop() *Loop {
 	return LoopFromPoints([]Point{emptyLoopPoint})
 }
 
 // FullLoop returns a special "full" loop.
 func FullLoop() *Loop {
 	return LoopFromPoints([]Point{fullLoopPoint})
 }
 
 // initOriginAndBound sets the origin containment for the given point and then calls
 // the initialization for the bounds objects and the internal index.
 func (l *Loop) initOriginAndBound() {
 	if len(l.vertices) < 3 {
 		// Check for the special "empty" and "full" loops (which have one vertex).
 		if !l.isEmptyOrFull() {
 			l.originInside = false
 			return
 		}
 
 		// This is the special empty or full loop, so the origin depends on if
 		// the vertex is in the southern hemisphere or not.
 		l.originInside = l.vertices[0].Z < 0
 	} else {
 		// Point containment testing is done by counting edge crossings starting
 		// at a fixed point on the sphere (OriginPoint). We need to know whether
 		// the reference point (OriginPoint) is inside or outside the loop before
 		// we can construct the ShapeIndex. We do this by first guessing that
 		// it is outside, and then seeing whether we get the correct containment
 		// result for vertex 1. If the result is incorrect, the origin must be
 		// inside the loop.
 		//
 		// A loop with consecutive vertices A,B,C contains vertex B if and only if
 		// the fixed vector R = B.Ortho is contained by the wedge ABC. The
 		// wedge is closed at A and open at C, i.e. the point B is inside the loop
 		// if A = R but not if C = R. This convention is required for compatibility
 		// with VertexCrossing. (Note that we can't use OriginPoint
 		// as the fixed vector because of the possibility that B == OriginPoint.)
 		l.originInside = false
 		v1Inside := OrderedCCW(Point{l.vertices[1].Ortho()}, l.vertices[0], l.vertices[2], l.vertices[1])
 		if v1Inside != l.ContainsPoint(l.vertices[1]) {
 			l.originInside = true
 		}
 	}
 
 	// We *must* call initBound before initializing the index, because
 	// initBound calls ContainsPoint which does a bounds check before using
 	// the index.
 	l.initBound()
 
 	// Create a new index and add us to it.
 	l.index = NewShapeIndex()
 	l.index.Add(l)
 }
 
 // initBound sets up the approximate bounding Rects for this loop.
 func (l *Loop) initBound() {
 	if len(l.vertices) == 0 {
 		*l = *EmptyLoop()
 		return
 	}
 	// Check for the special "empty" and "full" loops.
 	if l.isEmptyOrFull() {
 		if l.IsEmpty() {
 			l.bound = EmptyRect()
 		} else {
 			l.bound = FullRect()
 		}
 		l.subregionBound = l.bound
 		return
 	}
 
 	// The bounding rectangle of a loop is not necessarily the same as the
 	// bounding rectangle of its vertices. First, the maximal latitude may be
 	// attained along the interior of an edge. Second, the loop may wrap
 	// entirely around the sphere (e.g. a loop that defines two revolutions of a
 	// candy-cane stripe). Third, the loop may include one or both poles.
 	// Note that a small clockwise loop near the equator contains both poles.
 	bounder := NewRectBounder()
 	for i := 0; i <= len(l.vertices); i++ { // add vertex 0 twice
 		bounder.AddPoint(l.Vertex(i))
 	}
 	b := bounder.RectBound()
 
 	if l.ContainsPoint(Point{r3.Vector{0, 0, 1}}) {
 		b = Rect{r1.Interval{b.Lat.Lo, math.Pi / 2}, s1.FullInterval()}
 	}
 	// If a loop contains the south pole, then either it wraps entirely
 	// around the sphere (full longitude range), or it also contains the
 	// north pole in which case b.Lng.IsFull() due to the test above.
 	// Either way, we only need to do the south pole containment test if
 	// b.Lng.IsFull().
 	if b.Lng.IsFull() && l.ContainsPoint(Point{r3.Vector{0, 0, -1}}) {
 		b.Lat.Lo = -math.Pi / 2
 	}
 	l.bound = b
 	l.subregionBound = ExpandForSubregions(l.bound)
 }
 
 // Validate checks whether this is a valid loop.
 func (l *Loop) Validate() error {
 	if err := l.findValidationErrorNoIndex(); err != nil {
 		return err
 	}
 
 	// Check for intersections between non-adjacent edges (including at vertices)
 	// TODO(roberts): Once shapeutil gets findAnyCrossing uncomment this.
 	// return findAnyCrossing(l.index)
 
 	return nil
 }
 
 // findValidationErrorNoIndex reports whether this is not a valid loop, but
 // skips checks that would require a ShapeIndex to be built for the loop. This
 // is primarily used by Polygon to do validation so it doesn't trigger the
 // creation of unneeded ShapeIndices.
 func (l *Loop) findValidationErrorNoIndex() error {
 	// All vertices must be unit length.
 	for i, v := range l.vertices {
 		if !v.IsUnit() {
 			return fmt.Errorf("vertex %d is not unit length", i)
 		}
 	}
 
 	// Loops must have at least 3 vertices (except for empty and full).
 	if len(l.vertices) < 3 {
 		if l.isEmptyOrFull() {
 			return nil // Skip remaining tests.
 		}
 		return fmt.Errorf("non-empty, non-full loops must have at least 3 vertices")
 	}
 
 	// Loops are not allowed to have any duplicate vertices or edge crossings.
 	// We split this check into two parts. First we check that no edge is
 	// degenerate (identical endpoints). Then we check that there are no
 	// intersections between non-adjacent edges (including at vertices). The
 	// second check needs the ShapeIndex, so it does not fall within the scope
 	// of this method.
 	for i, v := range l.vertices {
 		if v == l.Vertex(i+1) {
 			return fmt.Errorf("edge %d is degenerate (duplicate vertex)", i)
 		}
 
 		// Antipodal vertices are not allowed.
 		if other := (Point{l.Vertex(i + 1).Mul(-1)}); v == other {
 			return fmt.Errorf("vertices %d and %d are antipodal", i,
 				(i+1)%len(l.vertices))
 		}
 	}
 
 	return nil
 }
 
 // Contains reports whether the region contained by this loop is a superset of the
 // region contained by the given other loop.
 func (l *Loop) Contains(o *Loop) bool {
 	// For a loop A to contain the loop B, all of the following must
 	// be true:
 	//
 	//  (1) There are no edge crossings between A and B except at vertices.
 	//
 	//  (2) At every vertex that is shared between A and B, the local edge
 	//      ordering implies that A contains B.
 	//
 	//  (3) If there are no shared vertices, then A must contain a vertex of B
 	//      and B must not contain a vertex of A. (An arbitrary vertex may be
 	//      chosen in each case.)
 	//
 	// The second part of (3) is necessary to detect the case of two loops whose
 	// union is the entire sphere, i.e. two loops that contains each other's
 	// boundaries but not each other's interiors.
 	if !l.subregionBound.Contains(o.bound) {
 		return false
 	}
 
 	// Special cases to handle either loop being empty or full.
 	if l.isEmptyOrFull() || o.isEmptyOrFull() {
 		return l.IsFull() || o.IsEmpty()
 	}
 
 	// Check whether there are any edge crossings, and also check the loop
 	// relationship at any shared vertices.
 	relation := &containsRelation{}
 	if hasCrossingRelation(l, o, relation) {
 		return false
 	}
 
 	// There are no crossings, and if there are any shared vertices then A
 	// contains B locally at each shared vertex.
 	if relation.foundSharedVertex {
 		return true
 	}
 
 	// Since there are no edge intersections or shared vertices, we just need to
 	// test condition (3) above. We can skip this test if we discovered that A
 	// contains at least one point of B while checking for edge crossings.
 	if !l.ContainsPoint(o.Vertex(0)) {
 		return false
 	}
 
 	// We still need to check whether (A union B) is the entire sphere.
 	// Normally this check is very cheap due to the bounding box precondition.
 	if (o.subregionBound.Contains(l.bound) || o.bound.Union(l.bound).IsFull()) &&
 		o.ContainsPoint(l.Vertex(0)) {
 		return false
 	}
 	return true
 }
 
 // Intersects reports whether the region contained by this loop intersects the region
 // contained by the other loop.
 func (l *Loop) Intersects(o *Loop) bool {
 	// Given two loops, A and B, A.Intersects(B) if and only if !A.Complement().Contains(B).
 	//
 	// This code is similar to Contains, but is optimized for the case
 	// where both loops enclose less than half of the sphere.
 	if !l.bound.Intersects(o.bound) {
 		return false
 	}
 
 	// Check whether there are any edge crossings, and also check the loop
 	// relationship at any shared vertices.
 	relation := &intersectsRelation{}
 	if hasCrossingRelation(l, o, relation) {
 		return true
 	}
 	if relation.foundSharedVertex {
 		return false
 	}
 
 	// Since there are no edge intersections or shared vertices, the loops
 	// intersect only if A contains B, B contains A, or the two loops contain
 	// each other's boundaries.  These checks are usually cheap because of the
 	// bounding box preconditions.  Note that neither loop is empty (because of
 	// the bounding box check above), so it is safe to access vertex(0).
 
 	// Check whether A contains B, or A and B contain each other's boundaries.
 	// (Note that A contains all the vertices of B in either case.)
 	if l.subregionBound.Contains(o.bound) || l.bound.Union(o.bound).IsFull() {
 		if l.ContainsPoint(o.Vertex(0)) {
 			return true
 		}
 	}
 	// Check whether B contains A.
 	if o.subregionBound.Contains(l.bound) {
 		if o.ContainsPoint(l.Vertex(0)) {
 			return true
 		}
 	}
 	return false
 }
 
 // Equal reports whether two loops have the same vertices in the same linear order
 // (i.e., cyclic rotations are not allowed).
 func (l *Loop) Equal(other *Loop) bool {
 	if len(l.vertices) != len(other.vertices) {
 		return false
 	}
 
 	for i, v := range l.vertices {
 		if v != other.Vertex(i) {
 			return false
 		}
 	}
 	return true
 }
 
 // BoundaryEqual reports whether the two loops have the same boundary. This is
 // true if and only if the loops have the same vertices in the same cyclic order
 // (i.e., the vertices may be cyclically rotated). The empty and full loops are
 // considered to have different boundaries.
 func (l *Loop) BoundaryEqual(o *Loop) bool {
 	if len(l.vertices) != len(o.vertices) {
 		return false
 	}
 
 	// Special case to handle empty or full loops.  Since they have the same
 	// number of vertices, if one loop is empty/full then so is the other.
 	if l.isEmptyOrFull() {
 		return l.IsEmpty() == o.IsEmpty()
 	}
 
 	// Loop through the vertices to find the first of ours that matches the
 	// starting vertex of the other loop. Use that offset to then 'align' the
 	// vertices for comparison.
 	for offset, vertex := range l.vertices {
 		if vertex == o.Vertex(0) {
 			// There is at most one starting offset since loop vertices are unique.
 			for i := 0; i < len(l.vertices); i++ {
 				if l.Vertex(i+offset) != o.Vertex(i) {
 					return false
 				}
 			}
 			return true
 		}
 	}
 	return false
 }
 
 // compareBoundary returns +1 if this loop contains the boundary of the other loop,
 // -1 if it excludes the boundary of the other, and 0 if the boundaries of the two
 // loops cross. Shared edges are handled as follows:
 //
 //   If XY is a shared edge, define Reversed(XY) to be true if XY
 //     appears in opposite directions in both loops.
 //   Then this loop contains XY if and only if Reversed(XY) == the other loop is a hole.
 //   (Intuitively, this checks whether this loop contains a vanishingly small region
 //   extending from the boundary of the other toward the interior of the polygon to
 //   which the other belongs.)
 //
 // This function is used for testing containment and intersection of
 // multi-loop polygons. Note that this method is not symmetric, since the
 // result depends on the direction of this loop but not on the direction of
 // the other loop (in the absence of shared edges).
 //
 // This requires that neither loop is empty, and if other loop IsFull, then it must not
 // be a hole.
 func (l *Loop) compareBoundary(o *Loop) int {
 	// The bounds must intersect for containment or crossing.
 	if !l.bound.Intersects(o.bound) {
 		return -1
 	}
 
 	// Full loops are handled as though the loop surrounded the entire sphere.
 	if l.IsFull() {
 		return 1
 	}
 	if o.IsFull() {
 		return -1
 	}
 
 	// Check whether there are any edge crossings, and also check the loop
 	// relationship at any shared vertices.
 	relation := newCompareBoundaryRelation(o.IsHole())
 	if hasCrossingRelation(l, o, relation) {
 		return 0
 	}
 	if relation.foundSharedVertex {
 		if relation.containsEdge {
 			return 1
 		}
 		return -1
 	}
 
 	// There are no edge intersections or shared vertices, so we can check
 	// whether A contains an arbitrary vertex of B.
 	if l.ContainsPoint(o.Vertex(0)) {
 		return 1
 	}
 	return -1
 }
 
 // ContainsOrigin reports true if this loop contains s2.OriginPoint().
 func (l *Loop) ContainsOrigin() bool {
 	return l.originInside
 }
 
 // ReferencePoint returns the reference point for this loop.
 func (l *Loop) ReferencePoint() ReferencePoint {
 	return OriginReferencePoint(l.originInside)
 }
 
 // NumEdges returns the number of edges in this shape.
 func (l *Loop) NumEdges() int {
 	if l.isEmptyOrFull() {
 		return 0
 	}
 	return len(l.vertices)
 }
 
 // Edge returns the endpoints for the given edge index.
 func (l *Loop) Edge(i int) Edge {
 	return Edge{l.Vertex(i), l.Vertex(i + 1)}
 }
 
 // NumChains reports the number of contiguous edge chains in the Loop.
 func (l *Loop) NumChains() int {
 	if l.IsEmpty() {
 		return 0
 	}
 	return 1
 }
 
 // Chain returns the i-th edge chain in the Shape.
 func (l *Loop) Chain(chainID int) Chain {
 	return Chain{0, l.NumEdges()}
 }
 
 // ChainEdge returns the j-th edge of the i-th edge chain.
 func (l *Loop) ChainEdge(chainID, offset int) Edge {
 	return Edge{l.Vertex(offset), l.Vertex(offset + 1)}
 }
 
 // ChainPosition returns a ChainPosition pair (i, j) such that edgeID is the
 // j-th edge of the Loop.
 func (l *Loop) ChainPosition(edgeID int) ChainPosition {
 	return ChainPosition{0, edgeID}
 }
 
 // Dimension returns the dimension of the geometry represented by this Loop.
 func (l *Loop) Dimension() int { return 2 }
 
 func (l *Loop) typeTag() typeTag { return typeTagNone }
 
 func (l *Loop) privateInterface() {}
 
 // IsEmpty reports true if this is the special empty loop that contains no points.
 func (l *Loop) IsEmpty() bool {
 	return l.isEmptyOrFull() && !l.ContainsOrigin()
 }
 
 // IsFull reports true if this is the special full loop that contains all points.
 func (l *Loop) IsFull() bool {
 	return l.isEmptyOrFull() && l.ContainsOrigin()
 }
 
 // isEmptyOrFull reports true if this loop is either the "empty" or "full" special loops.
 func (l *Loop) isEmptyOrFull() bool {
 	return len(l.vertices) == 1
 }
 
 // Vertices returns the vertices in the loop.
 func (l *Loop) Vertices() []Point {
 	return l.vertices
 }
 
 // RectBound returns a tight bounding rectangle. If the loop contains the point,
 // the bound also contains it.
 func (l *Loop) RectBound() Rect {
 	return l.bound
 }
 
 // CapBound returns a bounding cap that may have more padding than the corresponding
 // RectBound. The bound is conservative such that if the loop contains a point P,
 // the bound also contains it.
 func (l *Loop) CapBound() Cap {
 	return l.bound.CapBound()
 }
 
 // Vertex returns the vertex for the given index. For convenience, the vertex indices
 // wrap automatically for methods that do index math such as Edge.
 // i.e., Vertex(NumEdges() + n) is the same as Vertex(n).
 func (l *Loop) Vertex(i int) Point {
 	return l.vertices[i%len(l.vertices)]
 }
 
 // OrientedVertex returns the vertex in reverse order if the loop represents a polygon
 // hole. For example, arguments 0, 1, 2 are mapped to vertices n-1, n-2, n-3, where
 // n == len(vertices). This ensures that the interior of the polygon is always to
 // the left of the vertex chain.
 //
 // This requires: 0 <= i < 2 * len(vertices)
 func (l *Loop) OrientedVertex(i int) Point {
 	j := i - len(l.vertices)
 	if j < 0 {
 		j = i
 	}
 	if l.IsHole() {
 		j = len(l.vertices) - 1 - j
 	}
 	return l.Vertex(j)
 }
 
 // NumVertices returns the number of vertices in this loop.
 func (l *Loop) NumVertices() int {
 	return len(l.vertices)
 }
 
 // bruteForceContainsPoint reports if the given point is contained by this loop.
 // This method does not use the ShapeIndex, so it is only preferable below a certain
 // size of loop.
 func (l *Loop) bruteForceContainsPoint(p Point) bool {
 	origin := OriginPoint()
 	inside := l.originInside
 	crosser := NewChainEdgeCrosser(origin, p, l.Vertex(0))
 	for i := 1; i <= len(l.vertices); i++ { // add vertex 0 twice
 		inside = inside != crosser.EdgeOrVertexChainCrossing(l.Vertex(i))
 	}
 	return inside
 }
 
 // ContainsPoint returns true if the loop contains the point.
 func (l *Loop) ContainsPoint(p Point) bool {
 	if !l.index.IsFresh() && !l.bound.ContainsPoint(p) {
 		return false
 	}
 
 	// For small loops it is faster to just check all the crossings.  We also
 	// use this method during loop initialization because InitOriginAndBound()
 	// calls Contains() before InitIndex().  Otherwise, we keep track of the
 	// number of calls to Contains() and only build the index when enough calls
 	// have been made so that we think it is worth the effort.  Note that the
 	// code below is structured so that if many calls are made in parallel only
 	// one thread builds the index, while the rest continue using brute force
 	// until the index is actually available.
 
 	const maxBruteForceVertices = 32
 	// TODO(roberts): add unindexed contains calls tracking
 
 	if len(l.index.shapes) == 0 || // Index has not been initialized yet.
 		len(l.vertices) <= maxBruteForceVertices {
 		return l.bruteForceContainsPoint(p)
 	}
 
 	// Otherwise, look up the point in the index.
 	it := l.index.Iterator()
 	if !it.LocatePoint(p) {
 		return false
 	}
 	return l.iteratorContainsPoint(it, p)
 }
 
 // ContainsCell reports whether the given Cell is contained by this Loop.
 func (l *Loop) ContainsCell(target Cell) bool {
 	it := l.index.Iterator()
 	relation := it.LocateCellID(target.ID())
 
 	// If "target" is disjoint from all index cells, it is not contained.
 	// Similarly, if "target" is subdivided into one or more index cells then it
 	// is not contained, since index cells are subdivided only if they (nearly)
 	// intersect a sufficient number of edges.  (But note that if "target" itself
 	// is an index cell then it may be contained, since it could be a cell with
 	// no edges in the loop interior.)
 	if relation != Indexed {
 		return false
 	}
 
 	// Otherwise check if any edges intersect "target".
 	if l.boundaryApproxIntersects(it, target) {
 		return false
 	}
 
 	// Otherwise check if the loop contains the center of "target".
 	return l.iteratorContainsPoint(it, target.Center())
 }
 
 // IntersectsCell reports whether this Loop intersects the given cell.
 func (l *Loop) IntersectsCell(target Cell) bool {
 	it := l.index.Iterator()
 	relation := it.LocateCellID(target.ID())
 
 	// If target does not overlap any index cell, there is no intersection.
 	if relation == Disjoint {
 		return false
 	}
 	// If target is subdivided into one or more index cells, there is an
 	// intersection to within the ShapeIndex error bound (see Contains).
 	if relation == Subdivided {
 		return true
 	}
 	// If target is an index cell, there is an intersection because index cells
 	// are created only if they have at least one edge or they are entirely
 	// contained by the loop.
 	if it.CellID() == target.id {
 		return true
 	}
 	// Otherwise check if any edges intersect target.
 	if l.boundaryApproxIntersects(it, target) {
 		return true
 	}
 	// Otherwise check if the loop contains the center of target.
 	return l.iteratorContainsPoint(it, target.Center())
 }
 
 // CellUnionBound computes a covering of the Loop.
 func (l *Loop) CellUnionBound() []CellID {
 	return l.CapBound().CellUnionBound()
 }
 
 // boundaryApproxIntersects reports if the loop's boundary intersects target.
 // It may also return true when the loop boundary does not intersect target but
 // some edge comes within the worst-case error tolerance.
 //
 // This requires that it.Locate(target) returned Indexed.
 func (l *Loop) boundaryApproxIntersects(it *ShapeIndexIterator, target Cell) bool {
 	aClipped := it.IndexCell().findByShapeID(0)
 
 	// If there are no edges, there is no intersection.
 	if len(aClipped.edges) == 0 {
 		return false
 	}
 
 	// We can save some work if target is the index cell itself.
 	if it.CellID() == target.ID() {
 		return true
 	}
 
 	// Otherwise check whether any of the edges intersect target.
 	maxError := (faceClipErrorUVCoord + intersectsRectErrorUVDist)
 	bound := target.BoundUV().ExpandedByMargin(maxError)
 	for _, ai := range aClipped.edges {
 		v0, v1, ok := ClipToPaddedFace(l.Vertex(ai), l.Vertex(ai+1), target.Face(), maxError)
 		if ok && edgeIntersectsRect(v0, v1, bound) {
 			return true
 		}
 	}
 	return false
 }
 
 // iteratorContainsPoint reports if the iterator that is positioned at the ShapeIndexCell
 // that may contain p, contains the point p.
 func (l *Loop) iteratorContainsPoint(it *ShapeIndexIterator, p Point) bool {
 	// Test containment by drawing a line segment from the cell center to the
 	// given point and counting edge crossings.
 	aClipped := it.IndexCell().findByShapeID(0)
 	inside := aClipped.containsCenter
 	if len(aClipped.edges) > 0 {
 		center := it.Center()
 		crosser := NewEdgeCrosser(center, p)
 		aiPrev := -2
 		for _, ai := range aClipped.edges {
 			if ai != aiPrev+1 {
 				crosser.RestartAt(l.Vertex(ai))
 			}
 			aiPrev = ai
 			inside = inside != crosser.EdgeOrVertexChainCrossing(l.Vertex(ai+1))
 		}
 	}
 	return inside
 }
 
 // RegularLoop creates a loop with the given number of vertices, all
 // located on a circle of the specified radius around the given center.
 func RegularLoop(center Point, radius s1.Angle, numVertices int) *Loop {
 	return RegularLoopForFrame(getFrame(center), radius, numVertices)
 }
 
 // RegularLoopForFrame creates a loop centered around the z-axis of the given
 // coordinate frame, with the first vertex in the direction of the positive x-axis.
 func RegularLoopForFrame(frame matrix3x3, radius s1.Angle, numVertices int) *Loop {
 	return LoopFromPoints(regularPointsForFrame(frame, radius, numVertices))
 }
 
 // CanonicalFirstVertex returns a first index and a direction (either +1 or -1)
 // such that the vertex sequence (first, first+dir, ..., first+(n-1)*dir) does
 // not change when the loop vertex order is rotated or inverted. This allows the
 // loop vertices to be traversed in a canonical order. The return values are
 // chosen such that (first, ..., first+n*dir) are in the range [0, 2*n-1] as
 // expected by the Vertex method.
 func (l *Loop) CanonicalFirstVertex() (firstIdx, direction int) {
 	firstIdx = 0
 	n := len(l.vertices)
 	for i := 1; i < n; i++ {
 		if l.Vertex(i).Cmp(l.Vertex(firstIdx).Vector) == -1 {
 			firstIdx = i
 		}
 	}
 
 	// 0 <= firstIdx <= n-1, so (firstIdx+n*dir) <= 2*n-1.
 	if l.Vertex(firstIdx+1).Cmp(l.Vertex(firstIdx+n-1).Vector) == -1 {
 		return firstIdx, 1
 	}
 
 	// n <= firstIdx <= 2*n-1, so (firstIdx+n*dir) >= 0.
 	firstIdx += n
 	return firstIdx, -1
 }
 
 // TurningAngle returns the sum of the turning angles at each vertex. The return
 // value is positive if the loop is counter-clockwise, negative if the loop is
 // clockwise, and zero if the loop is a great circle. Degenerate and
 // nearly-degenerate loops are handled consistently with Sign. So for example,
 // if a loop has zero area (i.e., it is a very small CCW loop) then the turning
 // angle will always be negative.
 //
 // This quantity is also called the "geodesic curvature" of the loop.
 func (l *Loop) TurningAngle() float64 {
 	// For empty and full loops, we return the limit value as the loop area
 	// approaches 0 or 4*Pi respectively.
 	if l.isEmptyOrFull() {
 		if l.ContainsOrigin() {
 			return -2 * math.Pi
 		}
 		return 2 * math.Pi
 	}
 
 	// Don't crash even if the loop is not well-defined.
 	if len(l.vertices) < 3 {
 		return 0
 	}
 
 	// To ensure that we get the same result when the vertex order is rotated,
 	// and that the result is negated when the vertex order is reversed, we need
 	// to add up the individual turn angles in a consistent order. (In general,
 	// adding up a set of numbers in a different order can change the sum due to
 	// rounding errors.)
 	//
 	// Furthermore, if we just accumulate an ordinary sum then the worst-case
 	// error is quadratic in the number of vertices. (This can happen with
 	// spiral shapes, where the partial sum of the turning angles can be linear
 	// in the number of vertices.) To avoid this we use the Kahan summation
 	// algorithm (http://en.wikipedia.org/wiki/Kahan_summation_algorithm).
 	n := len(l.vertices)
 	i, dir := l.CanonicalFirstVertex()
 	sum := TurnAngle(l.Vertex((i+n-dir)%n), l.Vertex(i), l.Vertex((i+dir)%n))
 
 	compensation := s1.Angle(0)
 	for n-1 > 0 {
 		i += dir
 		angle := TurnAngle(l.Vertex(i-dir), l.Vertex(i), l.Vertex(i+dir))
 		oldSum := sum
 		angle += compensation
 		sum += angle
 		compensation = (oldSum - sum) + angle
 		n--
 	}
 
 	const maxCurvature = 2*math.Pi - 4*dblEpsilon
 
 	return math.Max(-maxCurvature, math.Min(maxCurvature, float64(dir)*float64(sum+compensation)))
 }
 
 // turningAngleMaxError return the maximum error in TurningAngle. The value is not
 // constant; it depends on the loop.
 func (l *Loop) turningAngleMaxError() float64 {
 	// The maximum error can be bounded as follows:
 	//   3.00 * dblEpsilon    for RobustCrossProd(b, a)
 	//   3.00 * dblEpsilon    for RobustCrossProd(c, b)
 	//   3.25 * dblEpsilon    for Angle()
 	//   2.00 * dblEpsilon    for each addition in the Kahan summation
 	//   ------------------
 	//  11.25 * dblEpsilon
 	maxErrorPerVertex := 11.25 * dblEpsilon
 	return maxErrorPerVertex * float64(len(l.vertices))
 }
 
 // IsHole reports whether this loop represents a hole in its containing polygon.
 func (l *Loop) IsHole() bool { return l.depth&1 != 0 }
 
 // Sign returns -1 if this Loop represents a hole in its containing polygon, and +1 otherwise.
 func (l *Loop) Sign() int {
 	if l.IsHole() {
 		return -1
 	}
 	return 1
 }
 
 // IsNormalized reports whether the loop area is at most 2*pi. Degenerate loops are
 // handled consistently with Sign, i.e., if a loop can be
 // expressed as the union of degenerate or nearly-degenerate CCW triangles,
 // then it will always be considered normalized.
 func (l *Loop) IsNormalized() bool {
 	// Optimization: if the longitude span is less than 180 degrees, then the
 	// loop covers less than half the sphere and is therefore normalized.
 	if l.bound.Lng.Length() < math.Pi {
 		return true
 	}
 
 	// We allow some error so that hemispheres are always considered normalized.
 	// TODO(roberts): This is no longer required by the Polygon implementation,
 	// so alternatively we could create the invariant that a loop is normalized
 	// if and only if its complement is not normalized.
 	return l.TurningAngle() >= -l.turningAngleMaxError()
 }
 
 // Normalize inverts the loop if necessary so that the area enclosed by the loop
 // is at most 2*pi.
 func (l *Loop) Normalize() {
 	if !l.IsNormalized() {
 		l.Invert()
 	}
 }
 
 // Invert reverses the order of the loop vertices, effectively complementing the
 // region represented by the loop. For example, the loop ABCD (with edges
 // AB, BC, CD, DA) becomes the loop DCBA (with edges DC, CB, BA, AD).
 // Notice that the last edge is the same in both cases except that its
 // direction has been reversed.
 func (l *Loop) Invert() {
 	l.index.Reset()
 	if l.isEmptyOrFull() {
 		if l.IsFull() {
 			l.vertices[0] = emptyLoopPoint
 		} else {
 			l.vertices[0] = fullLoopPoint
 		}
 	} else {
 		// For non-special loops, reverse the slice of vertices.
 		for i := len(l.vertices)/2 - 1; i >= 0; i-- {
 			opp := len(l.vertices) - 1 - i
 			l.vertices[i], l.vertices[opp] = l.vertices[opp], l.vertices[i]
 		}
 	}
 
 	// originInside must be set correctly before building the ShapeIndex.
 	l.originInside = !l.originInside
 	if l.bound.Lat.Lo > -math.Pi/2 && l.bound.Lat.Hi < math.Pi/2 {
 		// The complement of this loop contains both poles.
 		l.bound = FullRect()
 		l.subregionBound = l.bound
 	} else {
 		l.initBound()
 	}
 	l.index.Add(l)
 }
 
 // findVertex returns the index of the vertex at the given Point in the range
 // 1..numVertices, and a boolean indicating if a vertex was found.
 func (l *Loop) findVertex(p Point) (index int, ok bool) {
 	const notFound = 0
 	if len(l.vertices) < 10 {
 		// Exhaustive search for loops below a small threshold.
 		for i := 1; i <= len(l.vertices); i++ {
 			if l.Vertex(i) == p {
 				return i, true
 			}
 		}
 		return notFound, false
 	}
 
 	it := l.index.Iterator()
 	if !it.LocatePoint(p) {
 		return notFound, false
 	}
 
 	aClipped := it.IndexCell().findByShapeID(0)
 	for i := aClipped.numEdges() - 1; i >= 0; i-- {
 		ai := aClipped.edges[i]
 		if l.Vertex(ai) == p {
 			if ai == 0 {
 				return len(l.vertices), true
 			}
 			return ai, true
 		}
 
 		if l.Vertex(ai+1) == p {
 			return ai + 1, true
 		}
 	}
 	return notFound, false
 }
 
 // ContainsNested reports whether the given loops is contained within this loop.
 // This function does not test for edge intersections. The two loops must meet
 // all of the Polygon requirements; for example this implies that their
 // boundaries may not cross or have any shared edges (although they may have
 // shared vertices).
 func (l *Loop) ContainsNested(other *Loop) bool {
 	if !l.subregionBound.Contains(other.bound) {
 		return false
 	}
 
 	// Special cases to handle either loop being empty or full.  Also bail out
 	// when B has no vertices to avoid heap overflow on the vertex(1) call
 	// below.  (This method is called during polygon initialization before the
 	// client has an opportunity to call IsValid().)
 	if l.isEmptyOrFull() || other.NumVertices() < 2 {
 		return l.IsFull() || other.IsEmpty()
 	}
 
 	// We are given that A and B do not share any edges, and that either one
 	// loop contains the other or they do not intersect.
 	m, ok := l.findVertex(other.Vertex(1))
 	if !ok {
 		// Since other.vertex(1) is not shared, we can check whether A contains it.
 		return l.ContainsPoint(other.Vertex(1))
 	}
 
 	// Check whether the edge order around other.Vertex(1) is compatible with
 	// A containing B.
 	return WedgeContains(l.Vertex(m-1), l.Vertex(m), l.Vertex(m+1), other.Vertex(0), other.Vertex(2))
 }
 
 // surfaceIntegralFloat64 computes the oriented surface integral of some quantity f(x)
 // over the loop interior, given a function f(A,B,C) that returns the
 // corresponding integral over the spherical triangle ABC. Here "oriented
 // surface integral" means:
 //
 // (1) f(A,B,C) must be the integral of f if ABC is counterclockwise,
 //     and the integral of -f if ABC is clockwise.
 //
 // (2) The result of this function is *either* the integral of f over the
 //     loop interior, or the integral of (-f) over the loop exterior.
 //
 // Note that there are at least two common situations where it easy to work
 // around property (2) above:
 //
 //  - If the integral of f over the entire sphere is zero, then it doesn't
 //    matter which case is returned because they are always equal.
 //
 //  - If f is non-negative, then it is easy to detect when the integral over
 //    the loop exterior has been returned, and the integral over the loop
 //    interior can be obtained by adding the integral of f over the entire
 //    unit sphere (a constant) to the result.
 //
 // Any changes to this method may need corresponding changes to surfaceIntegralPoint as well.
 func (l *Loop) surfaceIntegralFloat64(f func(a, b, c Point) float64) float64 {
 	// We sum f over a collection T of oriented triangles, possibly
 	// overlapping. Let the sign of a triangle be +1 if it is CCW and -1
 	// otherwise, and let the sign of a point x be the sum of the signs of the
 	// triangles containing x. Then the collection of triangles T is chosen
 	// such that either:
 	//
 	//  (1) Each point in the loop interior has sign +1, and sign 0 otherwise; or
 	//  (2) Each point in the loop exterior has sign -1, and sign 0 otherwise.
 	//
 	// The triangles basically consist of a fan from vertex 0 to every loop
 	// edge that does not include vertex 0. These triangles will always satisfy
 	// either (1) or (2). However, what makes this a bit tricky is that
 	// spherical edges become numerically unstable as their length approaches
 	// 180 degrees. Of course there is not much we can do if the loop itself
 	// contains such edges, but we would like to make sure that all the triangle
 	// edges under our control (i.e., the non-loop edges) are stable. For
 	// example, consider a loop around the equator consisting of four equally
 	// spaced points. This is a well-defined loop, but we cannot just split it
 	// into two triangles by connecting vertex 0 to vertex 2.
 	//
 	// We handle this type of situation by moving the origin of the triangle fan
 	// whenever we are about to create an unstable edge. We choose a new
 	// location for the origin such that all relevant edges are stable. We also
 	// create extra triangles with the appropriate orientation so that the sum
 	// of the triangle signs is still correct at every point.
 
 	// The maximum length of an edge for it to be considered numerically stable.
 	// The exact value is fairly arbitrary since it depends on the stability of
 	// the function f. The value below is quite conservative but could be
 	// reduced further if desired.
 	const maxLength = math.Pi - 1e-5
 
 	var sum float64
 	origin := l.Vertex(0)
 	for i := 1; i+1 < len(l.vertices); i++ {
 		// Let V_i be vertex(i), let O be the current origin, and let length(A,B)
 		// be the length of edge (A,B). At the start of each loop iteration, the
 		// "leading edge" of the triangle fan is (O,V_i), and we want to extend
 		// the triangle fan so that the leading edge is (O,V_i+1).
 		//
 		// Invariants:
 		//  1. length(O,V_i) < maxLength for all (i > 1).
 		//  2. Either O == V_0, or O is approximately perpendicular to V_0.
 		//  3. "sum" is the oriented integral of f over the area defined by
 		//     (O, V_0, V_1, ..., V_i).
 		if l.Vertex(i+1).Angle(origin.Vector) > maxLength {
 			// We are about to create an unstable edge, so choose a new origin O'
 			// for the triangle fan.
 			oldOrigin := origin
 			if origin == l.Vertex(0) {
 				// The following point is well-separated from V_i and V_0 (and
 				// therefore V_i+1 as well).
 				origin = Point{l.Vertex(0).PointCross(l.Vertex(i)).Normalize()}
 			} else if l.Vertex(i).Angle(l.Vertex(0).Vector) < maxLength {
 				// All edges of the triangle (O, V_0, V_i) are stable, so we can
 				// revert to using V_0 as the origin.
 				origin = l.Vertex(0)
 			} else {
 				// (O, V_i+1) and (V_0, V_i) are antipodal pairs, and O and V_0 are
 				// perpendicular. Therefore V_0.CrossProd(O) is approximately
 				// perpendicular to all of {O, V_0, V_i, V_i+1}, and we can choose
 				// this point O' as the new origin.
 				origin = Point{l.Vertex(0).Cross(oldOrigin.Vector)}
 
 				// Advance the edge (V_0,O) to (V_0,O').
 				sum += f(l.Vertex(0), oldOrigin, origin)
 			}
 			// Advance the edge (O,V_i) to (O',V_i).
 			sum += f(oldOrigin, l.Vertex(i), origin)
 		}
 		// Advance the edge (O,V_i) to (O,V_i+1).
 		sum += f(origin, l.Vertex(i), l.Vertex(i+1))
 	}
 	// If the origin is not V_0, we need to sum one more triangle.
 	if origin != l.Vertex(0) {
 		// Advance the edge (O,V_n-1) to (O,V_0).
 		sum += f(origin, l.Vertex(len(l.vertices)-1), l.Vertex(0))
 	}
 	return sum
 }
 
 // surfaceIntegralPoint mirrors the surfaceIntegralFloat64 method but over Points;
 // see that method for commentary. The C++ version uses a templated method.
 // Any changes to this method may need corresponding changes to surfaceIntegralFloat64 as well.
 func (l *Loop) surfaceIntegralPoint(f func(a, b, c Point) Point) Point {
 	const maxLength = math.Pi - 1e-5
 	var sum r3.Vector
 
 	origin := l.Vertex(0)
 	for i := 1; i+1 < len(l.vertices); i++ {
 		if l.Vertex(i+1).Angle(origin.Vector) > maxLength {
 			oldOrigin := origin
 			if origin == l.Vertex(0) {
 				origin = Point{l.Vertex(0).PointCross(l.Vertex(i)).Normalize()}
 			} else if l.Vertex(i).Angle(l.Vertex(0).Vector) < maxLength {
 				origin = l.Vertex(0)
 			} else {
 				origin = Point{l.Vertex(0).Cross(oldOrigin.Vector)}
 				sum = sum.Add(f(l.Vertex(0), oldOrigin, origin).Vector)
 			}
 			sum = sum.Add(f(oldOrigin, l.Vertex(i), origin).Vector)
 		}
 		sum = sum.Add(f(origin, l.Vertex(i), l.Vertex(i+1)).Vector)
 	}
 	if origin != l.Vertex(0) {
 		sum = sum.Add(f(origin, l.Vertex(len(l.vertices)-1), l.Vertex(0)).Vector)
 	}
 	return Point{sum}
 }
 
 // Area returns the area of the loop interior, i.e. the region on the left side of
 // the loop. The return value is between 0 and 4*pi. (Note that the return
 // value is not affected by whether this loop is a "hole" or a "shell".)
 func (l *Loop) Area() float64 {
 	// It is surprisingly difficult to compute the area of a loop robustly. The
 	// main issues are (1) whether degenerate loops are considered to be CCW or
 	// not (i.e., whether their area is close to 0 or 4*pi), and (2) computing
 	// the areas of small loops with good relative accuracy.
 	//
 	// With respect to degeneracies, we would like Area to be consistent
 	// with ContainsPoint in that loops that contain many points
 	// should have large areas, and loops that contain few points should have
 	// small areas. For example, if a degenerate triangle is considered CCW
 	// according to s2predicates Sign, then it will contain very few points and
 	// its area should be approximately zero. On the other hand if it is
 	// considered clockwise, then it will contain virtually all points and so
 	// its area should be approximately 4*pi.
 	//
 	// More precisely, let U be the set of Points for which IsUnitLength
 	// is true, let P(U) be the projection of those points onto the mathematical
 	// unit sphere, and let V(P(U)) be the Voronoi diagram of the projected
 	// points. Then for every loop x, we would like Area to approximately
 	// equal the sum of the areas of the Voronoi regions of the points p for
 	// which x.ContainsPoint(p) is true.
 	//
 	// The second issue is that we want to compute the area of small loops
 	// accurately. This requires having good relative precision rather than
 	// good absolute precision. For example, if the area of a loop is 1e-12 and
 	// the error is 1e-15, then the area only has 3 digits of accuracy. (For
 	// reference, 1e-12 is about 40 square meters on the surface of the earth.)
 	// We would like to have good relative accuracy even for small loops.
 	//
 	// To achieve these goals, we combine two different methods of computing the
 	// area. This first method is based on the Gauss-Bonnet theorem, which says
 	// that the area enclosed by the loop equals 2*pi minus the total geodesic
 	// curvature of the loop (i.e., the sum of the "turning angles" at all the
 	// loop vertices). The big advantage of this method is that as long as we
 	// use Sign to compute the turning angle at each vertex, then
 	// degeneracies are always handled correctly. In other words, if a
 	// degenerate loop is CCW according to the symbolic perturbations used by
 	// Sign, then its turning angle will be approximately 2*pi.
 	//
 	// The disadvantage of the Gauss-Bonnet method is that its absolute error is
 	// about 2e-15 times the number of vertices (see turningAngleMaxError).
 	// So, it cannot compute the area of small loops accurately.
 	//
 	// The second method is based on splitting the loop into triangles and
 	// summing the area of each triangle. To avoid the difficulty and expense
 	// of decomposing the loop into a union of non-overlapping triangles,
 	// instead we compute a signed sum over triangles that may overlap (see the
 	// comments for surfaceIntegral). The advantage of this method
 	// is that the area of each triangle can be computed with much better
 	// relative accuracy (using l'Huilier's theorem). The disadvantage is that
 	// the result is a signed area: CCW loops may yield a small positive value,
 	// while CW loops may yield a small negative value (which is converted to a
 	// positive area by adding 4*pi). This means that small errors in computing
 	// the signed area may translate into a very large error in the result (if
 	// the sign of the sum is incorrect).
 	//
 	// So, our strategy is to combine these two methods as follows. First we
 	// compute the area using the "signed sum over triangles" approach (since it
 	// is generally more accurate). We also estimate the maximum error in this
 	// result. If the signed area is too close to zero (i.e., zero is within
 	// the error bounds), then we double-check the sign of the result using the
 	// Gauss-Bonnet method. (In fact we just call IsNormalized, which is
 	// based on this method.) If the two methods disagree, we return either 0
 	// or 4*pi based on the result of IsNormalized. Otherwise we return the
 	// area that we computed originally.
 	if l.isEmptyOrFull() {
 		if l.ContainsOrigin() {
 			return 4 * math.Pi
 		}
 		return 0
 	}
 	area := l.surfaceIntegralFloat64(SignedArea)
 
 	// TODO(roberts): This error estimate is very approximate. There are two
 	// issues: (1) SignedArea needs some improvements to ensure that its error
 	// is actually never higher than GirardArea, and (2) although the number of
 	// triangles in the sum is typically N-2, in theory it could be as high as
 	// 2*N for pathological inputs. But in other respects this error bound is
 	// very conservative since it assumes that the maximum error is achieved on
 	// every triangle.
 	maxError := l.turningAngleMaxError()
 
 	// The signed area should be between approximately -4*pi and 4*pi.
 	if area < 0 {
 		// We have computed the negative of the area of the loop exterior.
 		area += 4 * math.Pi
 	}
 
 	if area > 4*math.Pi {
 		area = 4 * math.Pi
 	}
 	if area < 0 {
 		area = 0
 	}
 
 	// If the area is close enough to zero or 4*pi so that the loop orientation
 	// is ambiguous, then we compute the loop orientation explicitly.
 	if area < maxError && !l.IsNormalized() {
 		return 4 * math.Pi
 	} else if area > (4*math.Pi-maxError) && l.IsNormalized() {
 		return 0
 	}
 
 	return area
 }
 
 // Centroid returns the true centroid of the loop multiplied by the area of the
 // loop. The result is not unit length, so you may want to normalize it. Also
 // note that in general, the centroid may not be contained by the loop.
 //
 // We prescale by the loop area for two reasons: (1) it is cheaper to
 // compute this way, and (2) it makes it easier to compute the centroid of
 // more complicated shapes (by splitting them into disjoint regions and
 // adding their centroids).
 //
 // Note that the return value is not affected by whether this loop is a
 // "hole" or a "shell".
 func (l *Loop) Centroid() Point {
 	// surfaceIntegralPoint() returns either the integral of position over loop
 	// interior, or the negative of the integral of position over the loop
 	// exterior. But these two values are the same (!), because the integral of
 	// position over the entire sphere is (0, 0, 0).
 	return l.surfaceIntegralPoint(TrueCentroid)
 }
 
 // Encode encodes the Loop.
 func (l Loop) Encode(w io.Writer) error {
 	e := &encoder{w: w}
 	l.encode(e)
 	return e.err
 }
 
 func (l Loop) encode(e *encoder) {
 	e.writeInt8(encodingVersion)
 	e.writeUint32(uint32(len(l.vertices)))
 	for _, v := range l.vertices {
 		e.writeFloat64(v.X)
 		e.writeFloat64(v.Y)
 		e.writeFloat64(v.Z)
 	}
 
 	e.writeBool(l.originInside)
 	e.writeInt32(int32(l.depth))
 
 	// Encode the bound.
 	l.bound.encode(e)
 }
 
 // Decode decodes a loop.
 func (l *Loop) Decode(r io.Reader) error {
 	*l = Loop{}
 	d := &decoder{r: asByteReader(r)}
 	l.decode(d)
 	return d.err
 }
 
 func (l *Loop) decode(d *decoder) {
 	version := int8(d.readUint8())
 	if d.err != nil {
 		return
 	}
 	if version != encodingVersion {
 		d.err = fmt.Errorf("cannot decode version %d", version)
 		return
 	}
 
 	// Empty loops are explicitly allowed here: a newly created loop has zero vertices
 	// and such loops encode and decode properly.
 	nvertices := d.readUint32()
 	if nvertices > maxEncodedVertices {
 		if d.err == nil {
 			d.err = fmt.Errorf("too many vertices (%d; max is %d)", nvertices, maxEncodedVertices)
 
 		}
 		return
 	}
 	l.vertices = make([]Point, nvertices)
 	for i := range l.vertices {
 		l.vertices[i].X = d.readFloat64()
 		l.vertices[i].Y = d.readFloat64()
 		l.vertices[i].Z = d.readFloat64()
 	}
 	l.index = NewShapeIndex()
 	l.originInside = d.readBool()
 	l.depth = int(d.readUint32())
 	l.bound.decode(d)
 	l.subregionBound = ExpandForSubregions(l.bound)
 
 	l.index.Add(l)
 }
 
 // Bitmasks to read from properties.
 const (
 	originInside = 1 << iota
 	boundEncoded
 )
 
 func (l *Loop) xyzFaceSiTiVertices() []xyzFaceSiTi {
 	ret := make([]xyzFaceSiTi, len(l.vertices))
 	for i, v := range l.vertices {
 		ret[i].xyz = v
 		ret[i].face, ret[i].si, ret[i].ti, ret[i].level = xyzToFaceSiTi(v)
 	}
 	return ret
 }
 
 func (l *Loop) encodeCompressed(e *encoder, snapLevel int, vertices []xyzFaceSiTi) {
 	if len(l.vertices) != len(vertices) {
 		panic("encodeCompressed: vertices must be the same length as l.vertices")
 	}
 	if len(vertices) > maxEncodedVertices {
 		if e.err == nil {
 			e.err = fmt.Errorf("too many vertices (%d; max is %d)", len(vertices), maxEncodedVertices)
 		}
 		return
 	}
 	e.writeUvarint(uint64(len(vertices)))
 	encodePointsCompressed(e, vertices, snapLevel)
 
 	props := l.compressedEncodingProperties()
 	e.writeUvarint(props)
 	e.writeUvarint(uint64(l.depth))
 	if props&boundEncoded != 0 {
 		l.bound.encode(e)
 	}
 }
 
 func (l *Loop) compressedEncodingProperties() uint64 {
 	var properties uint64
 	if l.originInside {
 		properties |= originInside
 	}
 
 	// Write whether there is a bound so we can change the threshold later.
 	// Recomputing the bound multiplies the decode time taken per vertex
 	// by a factor of about 3.5.  Without recomputing the bound, decode
 	// takes approximately 125 ns / vertex.  A loop with 63 vertices
 	// encoded without the bound will take ~30us to decode, which is
 	// acceptable.  At ~3.5 bytes / vertex without the bound, adding
 	// the bound will increase the size by <15%, which is also acceptable.
 	const minVerticesForBound = 64
 	if len(l.vertices) >= minVerticesForBound {
 		properties |= boundEncoded
 	}
 
 	return properties
 }
 
 func (l *Loop) decodeCompressed(d *decoder, snapLevel int) {
 	nvertices := d.readUvarint()
 	if d.err != nil {
 		return
 	}
 	if nvertices > maxEncodedVertices {
 		d.err = fmt.Errorf("too many vertices (%d; max is %d)", nvertices, maxEncodedVertices)
 		return
 	}
 	l.vertices = make([]Point, nvertices)
 	decodePointsCompressed(d, snapLevel, l.vertices)
 	properties := d.readUvarint()
 
 	// Make sure values are valid before using.
 	if d.err != nil {
 		return
 	}
 
 	l.index = NewShapeIndex()
 	l.originInside = (properties & originInside) != 0
 
 	l.depth = int(d.readUvarint())
 
 	if (properties & boundEncoded) != 0 {
 		l.bound.decode(d)
 		if d.err != nil {
 			return
 		}
 		l.subregionBound = ExpandForSubregions(l.bound)
 	} else {
 		l.initBound()
 	}
 
 	l.index.Add(l)
 }
 
 // crossingTarget is an enum representing the possible crossing target cases for relations.
 type crossingTarget int
 
 const (
 	crossingTargetDontCare crossingTarget = iota
 	crossingTargetDontCross
 	crossingTargetCross
 )
 
 // loopRelation defines the interface for checking a type of relationship between two loops.
 // Some examples of relations are Contains, Intersects, or CompareBoundary.
 type loopRelation interface {
 	// Optionally, aCrossingTarget and bCrossingTarget can specify an early-exit
 	// condition for the loop relation. If any point P is found such that
 	//
 	//   A.ContainsPoint(P) == aCrossingTarget() &&
 	//   B.ContainsPoint(P) == bCrossingTarget()
 	//
 	// then the loop relation is assumed to be the same as if a pair of crossing
 	// edges were found. For example, the ContainsPoint relation has
 	//
 	//   aCrossingTarget() == crossingTargetDontCross
 	//   bCrossingTarget() == crossingTargetCross
 	//
 	// because if A.ContainsPoint(P) == false and B.ContainsPoint(P) == true
 	// for any point P, then it is equivalent to finding an edge crossing (i.e.,
 	// since Contains returns false in both cases).
 	//
 	// Loop relations that do not have an early-exit condition of this form
 	// should return crossingTargetDontCare for both crossing targets.
 
 	// aCrossingTarget reports whether loop A crosses the target point with
 	// the given relation type.
 	aCrossingTarget() crossingTarget
 	// bCrossingTarget reports whether loop B crosses the target point with
 	// the given relation type.
 	bCrossingTarget() crossingTarget
 
 	// wedgesCross reports if a shared vertex ab1 and the two associated wedges
 	// (a0, ab1, b2) and (b0, ab1, b2) are equivalent to an edge crossing.
 	// The loop relation is also allowed to maintain its own internal state, and
 	// can return true if it observes any sequence of wedges that are equivalent
 	// to an edge crossing.
 	wedgesCross(a0, ab1, a2, b0, b2 Point) bool
 }
 
 // loopCrosser is a helper type for determining whether two loops cross.
 // It is instantiated twice for each pair of loops to be tested, once for the
 // pair (A,B) and once for the pair (B,A), in order to be able to process
 // edges in either loop nesting order.
 type loopCrosser struct {
 	a, b            *Loop
 	relation        loopRelation
 	swapped         bool
 	aCrossingTarget crossingTarget
 	bCrossingTarget crossingTarget
 
 	// state maintained by startEdge and edgeCrossesCell.
 	crosser    *EdgeCrosser
 	aj, bjPrev int
 
 	// temporary data declared here to avoid repeated memory allocations.
 	bQuery *CrossingEdgeQuery
 	bCells []*ShapeIndexCell
 }
 
 // newLoopCrosser creates a loopCrosser from the given values. If swapped is true,
 // the loops A and B have been swapped. This affects how arguments are passed to
 // the given loop relation, since for example A.Contains(B) is not the same as
 // B.Contains(A).
 func newLoopCrosser(a, b *Loop, relation loopRelation, swapped bool) *loopCrosser {
 	l := &loopCrosser{
 		a:               a,
 		b:               b,
 		relation:        relation,
 		swapped:         swapped,
 		aCrossingTarget: relation.aCrossingTarget(),
 		bCrossingTarget: relation.bCrossingTarget(),
 		bQuery:          NewCrossingEdgeQuery(b.index),
 	}
 	if swapped {
 		l.aCrossingTarget, l.bCrossingTarget = l.bCrossingTarget, l.aCrossingTarget
 	}
 
 	return l
 }
 
 // startEdge sets the crossers state for checking the given edge of loop A.
 func (l *loopCrosser) startEdge(aj int) {
 	l.crosser = NewEdgeCrosser(l.a.Vertex(aj), l.a.Vertex(aj+1))
 	l.aj = aj
 	l.bjPrev = -2
 }
 
 // edgeCrossesCell reports whether the current edge of loop A has any crossings with
 // edges of the index cell of loop B.
 func (l *loopCrosser) edgeCrossesCell(bClipped *clippedShape) bool {
 	// Test the current edge of A against all edges of bClipped
 	bNumEdges := bClipped.numEdges()
 	for j := 0; j < bNumEdges; j++ {
 		bj := bClipped.edges[j]
 		if bj != l.bjPrev+1 {
 			l.crosser.RestartAt(l.b.Vertex(bj))
 		}
 		l.bjPrev = bj
 		if crossing := l.crosser.ChainCrossingSign(l.b.Vertex(bj + 1)); crossing == DoNotCross {
 			continue
 		} else if crossing == Cross {
 			return true
 		}
 
 		// We only need to check each shared vertex once, so we only
 		// consider the case where l.aVertex(l.aj+1) == l.b.Vertex(bj+1).
 		if l.a.Vertex(l.aj+1) == l.b.Vertex(bj+1) {
 			if l.swapped {
 				if l.relation.wedgesCross(l.b.Vertex(bj), l.b.Vertex(bj+1), l.b.Vertex(bj+2), l.a.Vertex(l.aj), l.a.Vertex(l.aj+2)) {
 					return true
 				}
 			} else {
 				if l.relation.wedgesCross(l.a.Vertex(l.aj), l.a.Vertex(l.aj+1), l.a.Vertex(l.aj+2), l.b.Vertex(bj), l.b.Vertex(bj+2)) {
 					return true
 				}
 			}
 		}
 	}
 
 	return false
 }
 
 // cellCrossesCell reports whether there are any edge crossings or wedge crossings
 // within the two given cells.
 func (l *loopCrosser) cellCrossesCell(aClipped, bClipped *clippedShape) bool {
 	// Test all edges of aClipped against all edges of bClipped.
 	for _, edge := range aClipped.edges {
 		l.startEdge(edge)
 		if l.edgeCrossesCell(bClipped) {
 			return true
 		}
 	}
 
 	return false
 }
 
 // cellCrossesAnySubcell reports whether given an index cell of A, if there are any
 // edge or wedge crossings with any index cell of B contained within bID.
 func (l *loopCrosser) cellCrossesAnySubcell(aClipped *clippedShape, bID CellID) bool {
 	// Test all edges of aClipped against all edges of B. The relevant B
 	// edges are guaranteed to be children of bID, which lets us find the
 	// correct index cells more efficiently.
 	bRoot := PaddedCellFromCellID(bID, 0)
 	for _, aj := range aClipped.edges {
 		// Use an CrossingEdgeQuery starting at bRoot to find the index cells
 		// of B that might contain crossing edges.
 		l.bCells = l.bQuery.getCells(l.a.Vertex(aj), l.a.Vertex(aj+1), bRoot)
 		if len(l.bCells) == 0 {
 			continue
 		}
 		l.startEdge(aj)
 		for c := 0; c < len(l.bCells); c++ {
 			if l.edgeCrossesCell(l.bCells[c].shapes[0]) {
 				return true
 			}
 		}
 	}
 
 	return false
 }
 
 // hasCrossing reports whether given two iterators positioned such that
 // ai.cellID().ContainsCellID(bi.cellID()), there is an edge or wedge crossing
 // anywhere within ai.cellID(). This function advances bi only past ai.cellID().
 func (l *loopCrosser) hasCrossing(ai, bi *rangeIterator) bool {
 	// If ai.CellID() intersects many edges of B, then it is faster to use
 	// CrossingEdgeQuery to narrow down the candidates. But if it intersects
 	// only a few edges, it is faster to check all the crossings directly.
 	// We handle this by advancing bi and keeping track of how many edges we
 	// would need to test.
 	const edgeQueryMinEdges = 20 // Tuned from benchmarks.
 	var totalEdges int
 	l.bCells = nil
 
 	for {
 		if n := bi.it.IndexCell().shapes[0].numEdges(); n > 0 {
 			totalEdges += n
 			if totalEdges >= edgeQueryMinEdges {
 				// There are too many edges to test them directly, so use CrossingEdgeQuery.
 				if l.cellCrossesAnySubcell(ai.it.IndexCell().shapes[0], ai.cellID()) {
 					return true
 				}
 				bi.seekBeyond(ai)
 				return false
 			}
 			l.bCells = append(l.bCells, bi.indexCell())
 		}
 		bi.next()
 		if bi.cellID() > ai.rangeMax {
 			break
 		}
 	}
 
 	// Test all the edge crossings directly.
 	for _, c := range l.bCells {
 		if l.cellCrossesCell(ai.it.IndexCell().shapes[0], c.shapes[0]) {
 			return true
 		}
 	}
 
 	return false
 }
 
 // containsCenterMatches reports if the clippedShapes containsCenter boolean corresponds
 // to the crossing target type given. (This is to work around C++ allowing false == 0,
 // true == 1 type implicit conversions and comparisons)
 func containsCenterMatches(a *clippedShape, target crossingTarget) bool {
 	return (!a.containsCenter && target == crossingTargetDontCross) ||
 		(a.containsCenter && target == crossingTargetCross)
 }
 
 // hasCrossingRelation reports whether given two iterators positioned such that
 // ai.cellID().ContainsCellID(bi.cellID()), there is a crossing relationship
 // anywhere within ai.cellID(). Specifically, this method returns true if there
 // is an edge crossing, a wedge crossing, or a point P that matches both relations
 // crossing targets. This function advances both iterators past ai.cellID.
 func (l *loopCrosser) hasCrossingRelation(ai, bi *rangeIterator) bool {
 	aClipped := ai.it.IndexCell().shapes[0]
 	if aClipped.numEdges() != 0 {
 		// The current cell of A has at least one edge, so check for crossings.
 		if l.hasCrossing(ai, bi) {
 			return true
 		}
 		ai.next()
 		return false
 	}
 
 	if containsCenterMatches(aClipped, l.aCrossingTarget) {
 		// The crossing target for A is not satisfied, so we skip over these cells of B.
 		bi.seekBeyond(ai)
 		ai.next()
 		return false
 	}
 
 	// All points within ai.cellID() satisfy the crossing target for A, so it's
 	// worth iterating through the cells of B to see whether any cell
 	// centers also satisfy the crossing target for B.
 	for bi.cellID() <= ai.rangeMax {
 		bClipped := bi.it.IndexCell().shapes[0]
 		if containsCenterMatches(bClipped, l.bCrossingTarget) {
 			return true
 		}
 		bi.next()
 	}
 	ai.next()
 	return false
 }
 
 // hasCrossingRelation checks all edges of loop A for intersection against all edges
 // of loop B and reports if there are any that satisfy the given relation. If there
 // is any shared vertex, the wedges centered at this vertex are sent to the given
 // relation to be tested.
 //
 // If the two loop boundaries cross, this method is guaranteed to return
 // true. It also returns true in certain cases if the loop relationship is
 // equivalent to crossing. For example, if the relation is Contains and a
 // point P is found such that B contains P but A does not contain P, this
 // method will return true to indicate that the result is the same as though
 // a pair of crossing edges were found (since Contains returns false in
 // both cases).
 //
 // See Contains, Intersects and CompareBoundary for the three uses of this function.
 func hasCrossingRelation(a, b *Loop, relation loopRelation) bool {
 	// We look for CellID ranges where the indexes of A and B overlap, and
 	// then test those edges for crossings.
 	ai := newRangeIterator(a.index)
 	bi := newRangeIterator(b.index)
 
 	ab := newLoopCrosser(a, b, relation, false) // Tests edges of A against B
 	ba := newLoopCrosser(b, a, relation, true)  // Tests edges of B against A
 
 	for !ai.done() || !bi.done() {
 		if ai.rangeMax < bi.rangeMin {
 			// The A and B cells don't overlap, and A precedes B.
 			ai.seekTo(bi)
 		} else if bi.rangeMax < ai.rangeMin {
 			// The A and B cells don't overlap, and B precedes A.
 			bi.seekTo(ai)
 		} else {
 			// One cell contains the other. Determine which cell is larger.
 			abRelation := int64(ai.it.CellID().lsb() - bi.it.CellID().lsb())
 			if abRelation > 0 {
 				// A's index cell is larger.
 				if ab.hasCrossingRelation(ai, bi) {
 					return true
 				}
 			} else if abRelation < 0 {
 				// B's index cell is larger.
 				if ba.hasCrossingRelation(bi, ai) {
 					return true
 				}
 			} else {
 				// The A and B cells are the same. Since the two cells
 				// have the same center point P, check whether P satisfies
 				// the crossing targets.
 				aClipped := ai.it.IndexCell().shapes[0]
 				bClipped := bi.it.IndexCell().shapes[0]
 				if containsCenterMatches(aClipped, ab.aCrossingTarget) &&
 					containsCenterMatches(bClipped, ab.bCrossingTarget) {
 					return true
 				}
 				// Otherwise test all the edge crossings directly.
 				if aClipped.numEdges() > 0 && bClipped.numEdges() > 0 && ab.cellCrossesCell(aClipped, bClipped) {
 					return true
 				}
 				ai.next()
 				bi.next()
 			}
 		}
 	}
 	return false
 }
 
 // containsRelation implements loopRelation for a contains operation. If
 // A.ContainsPoint(P) == false && B.ContainsPoint(P) == true, it is equivalent
 // to having an edge crossing (i.e., Contains returns false).
 type containsRelation struct {
 	foundSharedVertex bool
 }
 
 func (c *containsRelation) aCrossingTarget() crossingTarget { return crossingTargetDontCross }
 func (c *containsRelation) bCrossingTarget() crossingTarget { return crossingTargetCross }
 func (c *containsRelation) wedgesCross(a0, ab1, a2, b0, b2 Point) bool {
 	c.foundSharedVertex = true
 	return !WedgeContains(a0, ab1, a2, b0, b2)
 }
 
 // intersectsRelation implements loopRelation for an intersects operation. Given
 // two loops, A and B, if A.ContainsPoint(P) == true && B.ContainsPoint(P) == true,
 // it is equivalent to having an edge crossing (i.e., Intersects returns true).
 type intersectsRelation struct {
 	foundSharedVertex bool
 }
 
 func (i *intersectsRelation) aCrossingTarget() crossingTarget { return crossingTargetCross }
 func (i *intersectsRelation) bCrossingTarget() crossingTarget { return crossingTargetCross }
 func (i *intersectsRelation) wedgesCross(a0, ab1, a2, b0, b2 Point) bool {
 	i.foundSharedVertex = true
 	return WedgeIntersects(a0, ab1, a2, b0, b2)
 }
 
 // compareBoundaryRelation implements loopRelation for comparing boundaries.
 //
 // The compare boundary relation does not have a useful early-exit condition,
 // so we return crossingTargetDontCare for both crossing targets.
 //
 // Aside: A possible early exit condition could be based on the following.
 //   If A contains a point of both B and ~B, then A intersects Boundary(B).
 //   If ~A contains a point of both B and ~B, then ~A intersects Boundary(B).
 //   So if the intersections of {A, ~A} with {B, ~B} are all non-empty,
 //   the return value is 0, i.e., Boundary(A) intersects Boundary(B).
 // Unfortunately it isn't worth detecting this situation because by the
 // time we have seen a point in all four intersection regions, we are also
 // guaranteed to have seen at least one pair of crossing edges.
 type compareBoundaryRelation struct {
 	reverse           bool // True if the other loop should be reversed.
 	foundSharedVertex bool // True if any wedge was processed.
 	containsEdge      bool // True if any edge of the other loop is contained by this loop.
 	excludesEdge      bool // True if any edge of the other loop is excluded by this loop.
 }
 
 func newCompareBoundaryRelation(reverse bool) *compareBoundaryRelation {
 	return &compareBoundaryRelation{reverse: reverse}
 }
 
 func (c *compareBoundaryRelation) aCrossingTarget() crossingTarget { return crossingTargetDontCare }
 func (c *compareBoundaryRelation) bCrossingTarget() crossingTarget { return crossingTargetDontCare }
 func (c *compareBoundaryRelation) wedgesCross(a0, ab1, a2, b0, b2 Point) bool {
 	// Because we don't care about the interior of the other, only its boundary,
 	// it is sufficient to check whether this one contains the semiwedge (ab1, b2).
 	c.foundSharedVertex = true
 	if wedgeContainsSemiwedge(a0, ab1, a2, b2, c.reverse) {
 		c.containsEdge = true
 	} else {
 		c.excludesEdge = true
 	}
 	return c.containsEdge && c.excludesEdge
 }
 
 // wedgeContainsSemiwedge reports whether the wedge (a0, ab1, a2) contains the
 // "semiwedge" defined as any non-empty open set of rays immediately CCW from
 // the edge (ab1, b2). If reverse is true, then substitute clockwise for CCW;
 // this simulates what would happen if the direction of the other loop was reversed.
 func wedgeContainsSemiwedge(a0, ab1, a2, b2 Point, reverse bool) bool {
 	if b2 == a0 || b2 == a2 {
 		// We have a shared or reversed edge.
 		return (b2 == a0) == reverse
 	}
 	return OrderedCCW(a0, a2, b2, ab1)
 }
 
 // containsNonCrossingBoundary reports whether given two loops whose boundaries
 // do not cross (see compareBoundary), if this loop contains the boundary of the
 // other loop. If reverse is true, the boundary of the other loop is reversed
 // first (which only affects the result when there are shared edges). This method
 // is cheaper than compareBoundary because it does not test for edge intersections.
 //
 // This function requires that neither loop is empty, and that if the other is full,
 // then reverse == false.
 func (l *Loop) containsNonCrossingBoundary(other *Loop, reverseOther bool) bool {
 	// The bounds must intersect for containment.
 	if !l.bound.Intersects(other.bound) {
 		return false
 	}
 
 	// Full loops are handled as though the loop surrounded the entire sphere.
 	if l.IsFull() {
 		return true
 	}
 	if other.IsFull() {
 		return false
 	}
 
 	m, ok := l.findVertex(other.Vertex(0))
 	if !ok {
 		// Since the other loops vertex 0 is not shared, we can check if this contains it.
 		return l.ContainsPoint(other.Vertex(0))
 	}
 	// Otherwise check whether the edge (b0, b1) is contained by this loop.
 	return wedgeContainsSemiwedge(l.Vertex(m-1), l.Vertex(m), l.Vertex(m+1),
 		other.Vertex(1), reverseOther)
 }
 
 // TODO(roberts): Differences from the C++ version:
 // DistanceToPoint
 // DistanceToBoundary
 // Project
 // ProjectToBoundary
 // BoundaryApproxEqual
 // BoundaryNear