gtsocial-umbx

polygon.go (38825B)

---

 // 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"
 )
 
 // Polygon represents a sequence of zero or more loops; recall that the
 // interior of a loop is defined to be its left-hand side (see Loop).
 //
 // When the polygon is initialized, the given loops are automatically converted
 // into a canonical form consisting of "shells" and "holes". Shells and holes
 // are both oriented CCW, and are nested hierarchically. The loops are
 // reordered to correspond to a pre-order traversal of the nesting hierarchy.
 //
 // Polygons may represent any region of the sphere with a polygonal boundary,
 // including the entire sphere (known as the "full" polygon). The full polygon
 // consists of a single full loop (see Loop), whereas the empty polygon has no
 // loops at all.
 //
 // Use FullPolygon() to construct a full polygon. The zero value of Polygon is
 // treated as the empty polygon.
 //
 // Polygons have the following restrictions:
 //
 //  - Loops may not cross, i.e. the boundary of a loop may not intersect
 //    both the interior and exterior of any other loop.
 //
 //  - Loops may not share edges, i.e. if a loop contains an edge AB, then
 //    no other loop may contain AB or BA.
 //
 //  - Loops may share vertices, however no vertex may appear twice in a
 //    single loop (see Loop).
 //
 //  - No loop may be empty. The full loop may appear only in the full polygon.
 type Polygon struct {
 	loops []*Loop
 
 	// index is a spatial index of all the polygon loops.
 	index *ShapeIndex
 
 	// hasHoles tracks if this polygon has at least one hole.
 	hasHoles bool
 
 	// numVertices keeps the running total of all of the vertices of the contained loops.
 	numVertices int
 
 	// numEdges tracks the total number of edges in all the loops in this polygon.
 	numEdges 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
 
 	// A slice where element i is the cumulative number of edges in the
 	// preceding loops in the polygon. This field is used for polygons that
 	// have a large number of loops, and may be empty for polygons with few loops.
 	cumulativeEdges []int
 }
 
 // PolygonFromLoops constructs a polygon from the given set of loops. The polygon
 // interior consists of the points contained by an odd number of loops. (Recall
 // that a loop contains the set of points on its left-hand side.)
 //
 // This method determines the loop nesting hierarchy and assigns every loop a
 // depth. Shells have even depths, and holes have odd depths.
 //
 // Note: The given set of loops are reordered by this method so that the hierarchy
 // can be traversed using Parent, LastDescendant and the loops depths.
 func PolygonFromLoops(loops []*Loop) *Polygon {
 	p := &Polygon{}
 	// Empty polygons do not contain any loops, even the Empty loop.
 	if len(loops) == 1 && loops[0].IsEmpty() {
 		p.initLoopProperties()
 		return p
 	}
 	p.loops = loops
 	p.initNested()
 	return p
 }
 
 // PolygonFromOrientedLoops returns a Polygon from the given set of loops,
 // like PolygonFromLoops. It expects loops to be oriented such that the polygon
 // interior is on the left-hand side of all loops. This implies that shells
 // and holes should have opposite orientations in the input to this method.
 // (During initialization, loops representing holes will automatically be
 // inverted.)
 func PolygonFromOrientedLoops(loops []*Loop) *Polygon {
 	// Here is the algorithm:
 	//
 	// 1. Remember which of the given loops contain OriginPoint.
 	//
 	// 2. Invert loops as necessary to ensure that they are nestable (i.e., no
 	//    loop contains the complement of any other loop). This may result in a
 	//    set of loops corresponding to the complement of the given polygon, but
 	//    we will fix that problem later.
 	//
 	//    We make the loops nestable by first normalizing all the loops (i.e.,
 	//    inverting any loops whose turning angle is negative). This handles
 	//    all loops except those whose turning angle is very close to zero
 	//    (within the maximum error tolerance). Any such loops are inverted if
 	//    and only if they contain OriginPoint(). (In theory this step is only
 	//    necessary if there are at least two such loops.) The resulting set of
 	//    loops is guaranteed to be nestable.
 	//
 	// 3. Build the polygon. This yields either the desired polygon or its
 	//    complement.
 	//
 	// 4. If there is at least one loop, we find a loop L that is adjacent to
 	//    OriginPoint() (where "adjacent" means that there exists a path
 	//    connecting OriginPoint() to some vertex of L such that the path does
 	//    not cross any loop). There may be a single such adjacent loop, or
 	//    there may be several (in which case they should all have the same
 	//    contains_origin() value). We choose L to be the loop containing the
 	//    origin whose depth is greatest, or loop(0) (a top-level shell) if no
 	//    such loop exists.
 	//
 	// 5. If (L originally contained origin) != (polygon contains origin), we
 	//    invert the polygon. This is done by inverting a top-level shell whose
 	//    turning angle is minimal and then fixing the nesting hierarchy. Note
 	//    that because we normalized all the loops initially, this step is only
 	//    necessary if the polygon requires at least one non-normalized loop to
 	//    represent it.
 
 	containedOrigin := make(map[*Loop]bool)
 	for _, l := range loops {
 		containedOrigin[l] = l.ContainsOrigin()
 	}
 
 	for _, l := range loops {
 		angle := l.TurningAngle()
 		if math.Abs(angle) > l.turningAngleMaxError() {
 			// Normalize the loop.
 			if angle < 0 {
 				l.Invert()
 			}
 		} else {
 			// Ensure that the loop does not contain the origin.
 			if l.ContainsOrigin() {
 				l.Invert()
 			}
 		}
 	}
 
 	p := PolygonFromLoops(loops)
 
 	if p.NumLoops() > 0 {
 		originLoop := p.Loop(0)
 		polygonContainsOrigin := false
 		for _, l := range p.Loops() {
 			if l.ContainsOrigin() {
 				polygonContainsOrigin = !polygonContainsOrigin
 
 				originLoop = l
 			}
 		}
 		if containedOrigin[originLoop] != polygonContainsOrigin {
 			p.Invert()
 		}
 	}
 
 	return p
 }
 
 // Invert inverts the polygon (replaces it by its complement).
 func (p *Polygon) Invert() {
 	// Inverting any one loop will invert the polygon.  The best loop to invert
 	// is the one whose area is largest, since this yields the smallest area
 	// after inversion. The loop with the largest area is always at depth 0.
 	// The descendents of this loop all have their depth reduced by 1, while the
 	// former siblings of this loop all have their depth increased by 1.
 
 	// The empty and full polygons are handled specially.
 	if p.IsEmpty() {
 		*p = *FullPolygon()
 		p.initLoopProperties()
 		return
 	}
 	if p.IsFull() {
 		*p = Polygon{}
 		p.initLoopProperties()
 		return
 	}
 
 	// Find the loop whose area is largest (i.e., whose turning angle is
 	// smallest), minimizing calls to TurningAngle(). In particular, for
 	// polygons with a single shell at level 0 there is no need to call
 	// TurningAngle() at all. (This method is relatively expensive.)
 	best := 0
 	const none = 10.0 // Flag that means "not computed yet"
 	bestAngle := none
 	for i := 1; i < p.NumLoops(); i++ {
 		if p.Loop(i).depth != 0 {
 			continue
 		}
 		// We defer computing the turning angle of loop 0 until we discover
 		// that the polygon has another top-level shell.
 		if bestAngle == none {
 			bestAngle = p.Loop(best).TurningAngle()
 		}
 		angle := p.Loop(i).TurningAngle()
 		// We break ties deterministically in order to avoid having the output
 		// depend on the input order of the loops.
 		if angle < bestAngle || (angle == bestAngle && compareLoops(p.Loop(i), p.Loop(best)) < 0) {
 			best = i
 			bestAngle = angle
 		}
 	}
 	// Build the new loops vector, starting with the inverted loop.
 	p.Loop(best).Invert()
 	newLoops := make([]*Loop, 0, p.NumLoops())
 	// Add the former siblings of this loop as descendants.
 	lastBest := p.LastDescendant(best)
 	newLoops = append(newLoops, p.Loop(best))
 	for i, l := range p.Loops() {
 		if i < best || i > lastBest {
 			l.depth++
 			newLoops = append(newLoops, l)
 		}
 	}
 	// Add the former children of this loop as siblings.
 	for i, l := range p.Loops() {
 		if i > best && i <= lastBest {
 			l.depth--
 			newLoops = append(newLoops, l)
 		}
 	}
 
 	p.loops = newLoops
 	p.initLoopProperties()
 }
 
 // Defines a total ordering on Loops that does not depend on the cyclic
 // order of loop vertices. This function is used to choose which loop to
 // invert in the case where several loops have exactly the same area.
 func compareLoops(a, b *Loop) int {
 	if na, nb := a.NumVertices(), b.NumVertices(); na != nb {
 		return na - nb
 	}
 	ai, aDir := a.CanonicalFirstVertex()
 	bi, bDir := b.CanonicalFirstVertex()
 	if aDir != bDir {
 		return aDir - bDir
 	}
 	for n := a.NumVertices() - 1; n >= 0; n, ai, bi = n-1, ai+aDir, bi+bDir {
 		if cmp := a.Vertex(ai).Cmp(b.Vertex(bi).Vector); cmp != 0 {
 			return cmp
 		}
 	}
 	return 0
 }
 
 // PolygonFromCell returns a Polygon from a single loop created from the given Cell.
 func PolygonFromCell(cell Cell) *Polygon {
 	return PolygonFromLoops([]*Loop{LoopFromCell(cell)})
 }
 
 // initNested takes the set of loops in this polygon and performs the nesting
 // computations to set the proper nesting and parent/child relationships.
 func (p *Polygon) initNested() {
 	if len(p.loops) == 1 {
 		p.initOneLoop()
 		return
 	}
 
 	lm := make(loopMap)
 
 	for _, l := range p.loops {
 		lm.insertLoop(l, nil)
 	}
 	// The loops have all been added to the loopMap for ordering. Clear the
 	// loops slice because we add all the loops in-order in initLoops.
 	p.loops = nil
 
 	// Reorder the loops in depth-first traversal order.
 	p.initLoops(lm)
 	p.initLoopProperties()
 }
 
 // loopMap is a map of a loop to its immediate children with respect to nesting.
 // It is used to determine which loops are shells and which are holes.
 type loopMap map[*Loop][]*Loop
 
 // insertLoop adds the given loop to the loop map under the specified parent.
 // All children of the new entry are checked to see if the need to move up to
 // a different level.
 func (lm loopMap) insertLoop(newLoop, parent *Loop) {
 	var children []*Loop
 	for done := false; !done; {
 		children = lm[parent]
 		done = true
 		for _, child := range children {
 			if child.ContainsNested(newLoop) {
 				parent = child
 				done = false
 				break
 			}
 		}
 	}
 
 	// Now, we have found a parent for this loop, it may be that some of the
 	// children of the parent of this loop may now be children of the new loop.
 	newChildren := lm[newLoop]
 	for i := 0; i < len(children); {
 		child := children[i]
 		if newLoop.ContainsNested(child) {
 			newChildren = append(newChildren, child)
 			children = append(children[0:i], children[i+1:]...)
 		} else {
 			i++
 		}
 	}
 
 	lm[newLoop] = newChildren
 	lm[parent] = append(children, newLoop)
 }
 
 // loopStack simplifies access to the loops while being initialized.
 type loopStack []*Loop
 
 func (s *loopStack) push(v *Loop) {
 	*s = append(*s, v)
 }
 func (s *loopStack) pop() *Loop {
 	l := len(*s)
 	r := (*s)[l-1]
 	*s = (*s)[:l-1]
 	return r
 }
 
 // initLoops walks the mapping of loops to all of their children, and adds them in
 // order into to the polygons set of loops.
 func (p *Polygon) initLoops(lm loopMap) {
 	var stack loopStack
 	stack.push(nil)
 	depth := -1
 
 	for len(stack) > 0 {
 		loop := stack.pop()
 		if loop != nil {
 			depth = loop.depth
 			p.loops = append(p.loops, loop)
 		}
 		children := lm[loop]
 		for i := len(children) - 1; i >= 0; i-- {
 			child := children[i]
 			child.depth = depth + 1
 			stack.push(child)
 		}
 	}
 }
 
 // initOneLoop set the properties for a polygon made of a single loop.
 // TODO(roberts): Can this be merged with initLoopProperties
 func (p *Polygon) initOneLoop() {
 	p.hasHoles = false
 	p.numVertices = len(p.loops[0].vertices)
 	p.bound = p.loops[0].RectBound()
 	p.subregionBound = ExpandForSubregions(p.bound)
 	// Ensure the loops depth is set correctly.
 	p.loops[0].depth = 0
 
 	p.initEdgesAndIndex()
 }
 
 // initLoopProperties sets the properties for polygons with multiple loops.
 func (p *Polygon) initLoopProperties() {
 	p.numVertices = 0
 	// the loops depths are set by initNested/initOriented prior to this.
 	p.bound = EmptyRect()
 	p.hasHoles = false
 	for _, l := range p.loops {
 		if l.IsHole() {
 			p.hasHoles = true
 		} else {
 			p.bound = p.bound.Union(l.RectBound())
 		}
 		p.numVertices += l.NumVertices()
 	}
 	p.subregionBound = ExpandForSubregions(p.bound)
 
 	p.initEdgesAndIndex()
 }
 
 // initEdgesAndIndex performs the shape related initializations and adds the final
 // polygon to the index.
 func (p *Polygon) initEdgesAndIndex() {
 	p.numEdges = 0
 	p.cumulativeEdges = nil
 	if p.IsFull() {
 		return
 	}
 	const maxLinearSearchLoops = 12 // Based on benchmarks.
 	if len(p.loops) > maxLinearSearchLoops {
 		p.cumulativeEdges = make([]int, 0, len(p.loops))
 	}
 
 	for _, l := range p.loops {
 		if p.cumulativeEdges != nil {
 			p.cumulativeEdges = append(p.cumulativeEdges, p.numEdges)
 		}
 		p.numEdges += len(l.vertices)
 	}
 
 	p.index = NewShapeIndex()
 	p.index.Add(p)
 }
 
 // FullPolygon returns a special "full" polygon.
 func FullPolygon() *Polygon {
 	ret := &Polygon{
 		loops: []*Loop{
 			FullLoop(),
 		},
 		numVertices:    len(FullLoop().Vertices()),
 		bound:          FullRect(),
 		subregionBound: FullRect(),
 	}
 	ret.initEdgesAndIndex()
 	return ret
 }
 
 // Validate checks whether this is a valid polygon,
 // including checking whether all the loops are themselves valid.
 func (p *Polygon) Validate() error {
 	for i, l := range p.loops {
 		// Check for loop errors that don't require building a ShapeIndex.
 		if err := l.findValidationErrorNoIndex(); err != nil {
 			return fmt.Errorf("loop %d: %v", i, err)
 		}
 		// Check that no loop is empty, and that the full loop only appears in the
 		// full polygon.
 		if l.IsEmpty() {
 			return fmt.Errorf("loop %d: empty loops are not allowed", i)
 		}
 		if l.IsFull() && len(p.loops) > 1 {
 			return fmt.Errorf("loop %d: full loop appears in non-full polygon", i)
 		}
 	}
 
 	// TODO(roberts): Uncomment the remaining checks when they are completed.
 
 	// Check for loop self-intersections and loop pairs that cross
 	// (including duplicate edges and vertices).
 	// if findSelfIntersection(p.index) {
 	//	return fmt.Errorf("polygon has loop pairs that cross")
 	// }
 
 	// Check whether initOriented detected inconsistent loop orientations.
 	// if p.hasInconsistentLoopOrientations {
 	// 	return fmt.Errorf("inconsistent loop orientations detected")
 	// }
 
 	// Finally, verify the loop nesting hierarchy.
 	return p.findLoopNestingError()
 }
 
 // findLoopNestingError reports if there is an error in the loop nesting hierarchy.
 func (p *Polygon) findLoopNestingError() error {
 	// First check that the loop depths make sense.
 	lastDepth := -1
 	for i, l := range p.loops {
 		depth := l.depth
 		if depth < 0 || depth > lastDepth+1 {
 			return fmt.Errorf("loop %d: invalid loop depth (%d)", i, depth)
 		}
 		lastDepth = depth
 	}
 	// Then check that they correspond to the actual loop nesting.  This test
 	// is quadratic in the number of loops but the cost per iteration is small.
 	for i, l := range p.loops {
 		last := p.LastDescendant(i)
 		for j, l2 := range p.loops {
 			if i == j {
 				continue
 			}
 			nested := (j >= i+1) && (j <= last)
 			const reverseB = false
 
 			if l.containsNonCrossingBoundary(l2, reverseB) != nested {
 				nestedStr := ""
 				if !nested {
 					nestedStr = "not "
 				}
 				return fmt.Errorf("invalid nesting: loop %d should %scontain loop %d", i, nestedStr, j)
 			}
 		}
 	}
 	return nil
 }
 
 // IsEmpty reports whether this is the special "empty" polygon (consisting of no loops).
 func (p *Polygon) IsEmpty() bool {
 	return len(p.loops) == 0
 }
 
 // IsFull reports whether this is the special "full" polygon (consisting of a
 // single loop that encompasses the entire sphere).
 func (p *Polygon) IsFull() bool {
 	return len(p.loops) == 1 && p.loops[0].IsFull()
 }
 
 // NumLoops returns the number of loops in this polygon.
 func (p *Polygon) NumLoops() int {
 	return len(p.loops)
 }
 
 // Loops returns the loops in this polygon.
 func (p *Polygon) Loops() []*Loop {
 	return p.loops
 }
 
 // Loop returns the loop at the given index. Note that during initialization,
 // the given loops are reordered according to a pre-order traversal of the loop
 // nesting hierarchy. This implies that every loop is immediately followed by
 // its descendants. This hierarchy can be traversed using the methods Parent,
 // LastDescendant, and Loop.depth.
 func (p *Polygon) Loop(k int) *Loop {
 	return p.loops[k]
 }
 
 // Parent returns the index of the parent of loop k.
 // If the loop does not have a parent, ok=false is returned.
 func (p *Polygon) Parent(k int) (index int, ok bool) {
 	// See where we are on the depth hierarchy.
 	depth := p.loops[k].depth
 	if depth == 0 {
 		return -1, false
 	}
 
 	// There may be several loops at the same nesting level as us that share a
 	// parent loop with us. (Imagine a slice of swiss cheese, of which we are one loop.
 	// we don't know how many may be next to us before we get back to our parent loop.)
 	// Move up one position from us, and then begin traversing back through the set of loops
 	// until we find the one that is our parent or we get to the top of the polygon.
 	for k--; k >= 0 && p.loops[k].depth <= depth; k-- {
 	}
 	return k, true
 }
 
 // LastDescendant returns the index of the last loop that is contained within loop k.
 // If k is negative, it returns the last loop in the polygon.
 // Note that loops are indexed according to a pre-order traversal of the nesting
 // hierarchy, so the immediate children of loop k can be found by iterating over
 // the loops (k+1)..LastDescendant(k) and selecting those whose depth is equal
 // to Loop(k).depth+1.
 func (p *Polygon) LastDescendant(k int) int {
 	if k < 0 {
 		return len(p.loops) - 1
 	}
 
 	depth := p.loops[k].depth
 
 	// Find the next loop immediately past us in the set of loops, and then start
 	// moving down the list until we either get to the end or find the next loop
 	// that is higher up the hierarchy than we are.
 	for k++; k < len(p.loops) && p.loops[k].depth > depth; k++ {
 	}
 	return k - 1
 }
 
 // CapBound returns a bounding spherical cap.
 func (p *Polygon) CapBound() Cap { return p.bound.CapBound() }
 
 // RectBound returns a bounding latitude-longitude rectangle.
 func (p *Polygon) RectBound() Rect { return p.bound }
 
 // ContainsPoint reports whether the polygon contains the point.
 func (p *Polygon) ContainsPoint(point Point) bool {
 	// NOTE: A bounds check slows down this function by about 50%. It is
 	// worthwhile only when it might allow us to delay building the index.
 	if !p.index.IsFresh() && !p.bound.ContainsPoint(point) {
 		return false
 	}
 
 	// For small polygons, and during initial construction, it is faster to just
 	// check all the crossing.
 	const maxBruteForceVertices = 32
 	if p.numVertices < maxBruteForceVertices || p.index == nil {
 		inside := false
 		for _, l := range p.loops {
 			// use loops bruteforce to avoid building the index on each loop.
 			inside = inside != l.bruteForceContainsPoint(point)
 		}
 		return inside
 	}
 
 	// Otherwise we look up the ShapeIndex cell containing this point.
 	return NewContainsPointQuery(p.index, VertexModelSemiOpen).Contains(point)
 }
 
 // ContainsCell reports whether the polygon contains the given cell.
 func (p *Polygon) ContainsCell(cell Cell) bool {
 	it := p.index.Iterator()
 	relation := it.LocateCellID(cell.ID())
 
 	// If "cell" is disjoint from all index cells, it is not contained.
 	// Similarly, if "cell" 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 "cell" 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 "cell".
 	if p.boundaryApproxIntersects(it, cell) {
 		return false
 	}
 
 	// Otherwise check if the loop contains the center of "cell".
 	return p.iteratorContainsPoint(it, cell.Center())
 }
 
 // IntersectsCell reports whether the polygon intersects the given cell.
 func (p *Polygon) IntersectsCell(cell Cell) bool {
 	it := p.index.Iterator()
 	relation := it.LocateCellID(cell.ID())
 
 	// If cell does not overlap any index cell, there is no intersection.
 	if relation == Disjoint {
 		return false
 	}
 	// If cell is subdivided into one or more index cells, there is an
 	// intersection to within the S2ShapeIndex error bound (see Contains).
 	if relation == Subdivided {
 		return true
 	}
 	// If cell 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() == cell.id {
 		return true
 	}
 	// Otherwise check if any edges intersect cell.
 	if p.boundaryApproxIntersects(it, cell) {
 		return true
 	}
 	// Otherwise check if the loop contains the center of cell.
 	return p.iteratorContainsPoint(it, cell.Center())
 }
 
 // CellUnionBound computes a covering of the Polygon.
 func (p *Polygon) CellUnionBound() []CellID {
 	// TODO(roberts): Use ShapeIndexRegion when it's available.
 	return p.CapBound().CellUnionBound()
 }
 
 // boundaryApproxIntersects reports whether the loop's boundary intersects cell.
 // It may also return true when the loop boundary does not intersect cell but
 // some edge comes within the worst-case error tolerance.
 //
 // This requires that it.Locate(cell) returned Indexed.
 func (p *Polygon) boundaryApproxIntersects(it *ShapeIndexIterator, cell 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 cell is the index cell itself.
 	if it.CellID() == cell.ID() {
 		return true
 	}
 
 	// Otherwise check whether any of the edges intersect cell.
 	maxError := (faceClipErrorUVCoord + intersectsRectErrorUVDist)
 	bound := cell.BoundUV().ExpandedByMargin(maxError)
 	for _, e := range aClipped.edges {
 		edge := p.index.Shape(0).Edge(e)
 		v0, v1, ok := ClipToPaddedFace(edge.V0, edge.V1, cell.Face(), maxError)
 		if ok && edgeIntersectsRect(v0, v1, bound) {
 			return true
 		}
 	}
 
 	return false
 }
 
 // iteratorContainsPoint reports whether the iterator that is positioned at the
 // ShapeIndexCell that may contain p, contains the point p.
 func (p *Polygon) iteratorContainsPoint(it *ShapeIndexIterator, point 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 {
 		return inside
 	}
 
 	// This block requires ShapeIndex.
 	crosser := NewEdgeCrosser(it.Center(), point)
 	shape := p.index.Shape(0)
 	for _, e := range aClipped.edges {
 		edge := shape.Edge(e)
 		inside = inside != crosser.EdgeOrVertexCrossing(edge.V0, edge.V1)
 	}
 
 	return inside
 }
 
 // Shape Interface
 
 // NumEdges returns the number of edges in this shape.
 func (p *Polygon) NumEdges() int {
 	return p.numEdges
 }
 
 // Edge returns endpoints for the given edge index.
 func (p *Polygon) Edge(e int) Edge {
 	var i int
 
 	if len(p.cumulativeEdges) > 0 {
 		for i = range p.cumulativeEdges {
 			if i+1 >= len(p.cumulativeEdges) || e < p.cumulativeEdges[i+1] {
 				e -= p.cumulativeEdges[i]
 				break
 			}
 		}
 	} else {
 		// When the number of loops is small, use linear search. Most often
 		// there is exactly one loop and the code below executes zero times.
 		for i = 0; e >= len(p.Loop(i).vertices); i++ {
 			e -= len(p.Loop(i).vertices)
 		}
 	}
 
 	return Edge{p.Loop(i).OrientedVertex(e), p.Loop(i).OrientedVertex(e + 1)}
 }
 
 // ReferencePoint returns the reference point for this polygon.
 func (p *Polygon) ReferencePoint() ReferencePoint {
 	containsOrigin := false
 	for _, l := range p.loops {
 		containsOrigin = containsOrigin != l.ContainsOrigin()
 	}
 	return OriginReferencePoint(containsOrigin)
 }
 
 // NumChains reports the number of contiguous edge chains in the Polygon.
 func (p *Polygon) NumChains() int {
 	return p.NumLoops()
 }
 
 // Chain returns the i-th edge Chain (loop) in the Shape.
 func (p *Polygon) Chain(chainID int) Chain {
 	if p.cumulativeEdges != nil {
 		return Chain{p.cumulativeEdges[chainID], len(p.Loop(chainID).vertices)}
 	}
 	e := 0
 	for j := 0; j < chainID; j++ {
 		e += len(p.Loop(j).vertices)
 	}
 
 	// Polygon represents a full loop as a loop with one vertex, while
 	// Shape represents a full loop as a chain with no vertices.
 	if numVertices := p.Loop(chainID).NumVertices(); numVertices != 1 {
 		return Chain{e, numVertices}
 	}
 	return Chain{e, 0}
 }
 
 // ChainEdge returns the j-th edge of the i-th edge Chain (loop).
 func (p *Polygon) ChainEdge(i, j int) Edge {
 	return Edge{p.Loop(i).OrientedVertex(j), p.Loop(i).OrientedVertex(j + 1)}
 }
 
 // ChainPosition returns a pair (i, j) such that edgeID is the j-th edge
 // of the i-th edge Chain.
 func (p *Polygon) ChainPosition(edgeID int) ChainPosition {
 	var i int
 
 	if len(p.cumulativeEdges) > 0 {
 		for i = range p.cumulativeEdges {
 			if i+1 >= len(p.cumulativeEdges) || edgeID < p.cumulativeEdges[i+1] {
 				edgeID -= p.cumulativeEdges[i]
 				break
 			}
 		}
 	} else {
 		// When the number of loops is small, use linear search. Most often
 		// there is exactly one loop and the code below executes zero times.
 		for i = 0; edgeID >= len(p.Loop(i).vertices); i++ {
 			edgeID -= len(p.Loop(i).vertices)
 		}
 	}
 	// TODO(roberts): unify this and Edge since they are mostly identical.
 	return ChainPosition{i, edgeID}
 }
 
 // Dimension returns the dimension of the geometry represented by this Polygon.
 func (p *Polygon) Dimension() int { return 2 }
 
 func (p *Polygon) typeTag() typeTag { return typeTagPolygon }
 
 func (p *Polygon) privateInterface() {}
 
 // Contains reports whether this polygon contains the other polygon.
 // Specifically, it reports whether all the points in the other polygon
 // are also in this polygon.
 func (p *Polygon) Contains(o *Polygon) bool {
 	// If both polygons have one loop, use the more efficient Loop method.
 	// Note that Loop's Contains does its own bounding rectangle check.
 	if len(p.loops) == 1 && len(o.loops) == 1 {
 		return p.loops[0].Contains(o.loops[0])
 	}
 
 	// Otherwise if neither polygon has holes, we can still use the more
 	// efficient Loop's Contains method (rather than compareBoundary),
 	// but it's worthwhile to do our own bounds check first.
 	if !p.subregionBound.Contains(o.bound) {
 		// Even though Bound(A) does not contain Bound(B), it is still possible
 		// that A contains B. This can only happen when union of the two bounds
 		// spans all longitudes. For example, suppose that B consists of two
 		// shells with a longitude gap between them, while A consists of one shell
 		// that surrounds both shells of B but goes the other way around the
 		// sphere (so that it does not intersect the longitude gap).
 		if !p.bound.Lng.Union(o.bound.Lng).IsFull() {
 			return false
 		}
 	}
 
 	if !p.hasHoles && !o.hasHoles {
 		for _, l := range o.loops {
 			if !p.anyLoopContains(l) {
 				return false
 			}
 		}
 		return true
 	}
 
 	// Polygon A contains B iff B does not intersect the complement of A. From
 	// the intersection algorithm below, this means that the complement of A
 	// must exclude the entire boundary of B, and B must exclude all shell
 	// boundaries of the complement of A. (It can be shown that B must then
 	// exclude the entire boundary of the complement of A.) The first call
 	// below returns false if the boundaries cross, therefore the second call
 	// does not need to check for any crossing edges (which makes it cheaper).
 	return p.containsBoundary(o) && o.excludesNonCrossingComplementShells(p)
 }
 
 // Intersects reports whether this polygon intersects the other polygon, i.e.
 // if there is a point that is contained by both polygons.
 func (p *Polygon) Intersects(o *Polygon) bool {
 	// If both polygons have one loop, use the more efficient Loop method.
 	// Note that Loop Intersects does its own bounding rectangle check.
 	if len(p.loops) == 1 && len(o.loops) == 1 {
 		return p.loops[0].Intersects(o.loops[0])
 	}
 
 	// Otherwise if neither polygon has holes, we can still use the more
 	// efficient Loop.Intersects method. The polygons intersect if and
 	// only if some pair of loop regions intersect.
 	if !p.bound.Intersects(o.bound) {
 		return false
 	}
 
 	if !p.hasHoles && !o.hasHoles {
 		for _, l := range o.loops {
 			if p.anyLoopIntersects(l) {
 				return true
 			}
 		}
 		return false
 	}
 
 	// Polygon A is disjoint from B if A excludes the entire boundary of B and B
 	// excludes all shell boundaries of A. (It can be shown that B must then
 	// exclude the entire boundary of A.) The first call below returns false if
 	// the boundaries cross, therefore the second call does not need to check
 	// for crossing edges.
 	return !p.excludesBoundary(o) || !o.excludesNonCrossingShells(p)
 }
 
 // compareBoundary returns +1 if this polygon contains the boundary of B, -1 if A
 // excludes the boundary of B, and 0 if the boundaries of A and B cross.
 func (p *Polygon) compareBoundary(o *Loop) int {
 	result := -1
 	for i := 0; i < len(p.loops) && result != 0; i++ {
 		// If B crosses any loop of A, the result is 0. Otherwise the result
 		// changes sign each time B is contained by a loop of A.
 		result *= -p.loops[i].compareBoundary(o)
 	}
 	return result
 }
 
 // containsBoundary reports whether this polygon contains the entire boundary of B.
 func (p *Polygon) containsBoundary(o *Polygon) bool {
 	for _, l := range o.loops {
 		if p.compareBoundary(l) <= 0 {
 			return false
 		}
 	}
 	return true
 }
 
 // excludesBoundary reports whether this polygon excludes the entire boundary of B.
 func (p *Polygon) excludesBoundary(o *Polygon) bool {
 	for _, l := range o.loops {
 		if p.compareBoundary(l) >= 0 {
 			return false
 		}
 	}
 	return true
 }
 
 // containsNonCrossingBoundary reports whether polygon A contains the boundary of
 // loop B. Shared edges are handled according to the rule described in loops
 // containsNonCrossingBoundary.
 func (p *Polygon) containsNonCrossingBoundary(o *Loop, reverse bool) bool {
 	var inside bool
 	for _, l := range p.loops {
 		x := l.containsNonCrossingBoundary(o, reverse)
 		inside = (inside != x)
 	}
 	return inside
 }
 
 // excludesNonCrossingShells reports wheterh given two polygons A and B such that the
 // boundary of A does not cross any loop of B, if A excludes all shell boundaries of B.
 func (p *Polygon) excludesNonCrossingShells(o *Polygon) bool {
 	for _, l := range o.loops {
 		if l.IsHole() {
 			continue
 		}
 		if p.containsNonCrossingBoundary(l, false) {
 			return false
 		}
 	}
 	return true
 }
 
 // excludesNonCrossingComplementShells reports whether given two polygons A and B
 // such that the boundary of A does not cross any loop of B, if A excludes all
 // shell boundaries of the complement of B.
 func (p *Polygon) excludesNonCrossingComplementShells(o *Polygon) bool {
 	// Special case to handle the complement of the empty or full polygons.
 	if o.IsEmpty() {
 		return !p.IsFull()
 	}
 	if o.IsFull() {
 		return true
 	}
 
 	// Otherwise the complement of B may be obtained by inverting loop(0) and
 	// then swapping the shell/hole status of all other loops. This implies
 	// that the shells of the complement consist of loop 0 plus all the holes of
 	// the original polygon.
 	for j, l := range o.loops {
 		if j > 0 && !l.IsHole() {
 			continue
 		}
 
 		// The interior of the complement is to the right of loop 0, and to the
 		// left of the loops that were originally holes.
 		if p.containsNonCrossingBoundary(l, j == 0) {
 			return false
 		}
 	}
 	return true
 }
 
 // anyLoopContains reports whether any loop in this polygon contains the given loop.
 func (p *Polygon) anyLoopContains(o *Loop) bool {
 	for _, l := range p.loops {
 		if l.Contains(o) {
 			return true
 		}
 	}
 	return false
 }
 
 // anyLoopIntersects reports whether any loop in this polygon intersects the given loop.
 func (p *Polygon) anyLoopIntersects(o *Loop) bool {
 	for _, l := range p.loops {
 		if l.Intersects(o) {
 			return true
 		}
 	}
 	return false
 }
 
 // Area returns the area of the polygon interior, i.e. the region on the left side
 // of an odd number of loops. The return value is between 0 and 4*Pi.
 func (p *Polygon) Area() float64 {
 	var area float64
 	for _, loop := range p.loops {
 		area += float64(loop.Sign()) * loop.Area()
 	}
 	return area
 }
 
 // Encode encodes the Polygon
 func (p *Polygon) Encode(w io.Writer) error {
 	e := &encoder{w: w}
 	p.encode(e)
 	return e.err
 }
 
 // encode only supports lossless encoding and not compressed format.
 func (p *Polygon) encode(e *encoder) {
 	if p.numVertices == 0 {
 		p.encodeCompressed(e, maxLevel, nil)
 		return
 	}
 
 	// Convert all the polygon vertices to XYZFaceSiTi format.
 	vs := make([]xyzFaceSiTi, 0, p.numVertices)
 	for _, l := range p.loops {
 		vs = append(vs, l.xyzFaceSiTiVertices()...)
 	}
 
 	// Computes a histogram of the cell levels at which the vertices are snapped.
 	// (histogram[0] is the number of unsnapped vertices, histogram[i] the number
 	// of vertices snapped at level i-1).
 	histogram := make([]int, maxLevel+2)
 	for _, v := range vs {
 		histogram[v.level+1]++
 	}
 
 	// Compute the level at which most of the vertices are snapped.
 	// If multiple levels have the same maximum number of vertices
 	// snapped to it, the first one (lowest level number / largest
 	// area / smallest encoding length) will be chosen, so this
 	// is desired.
 	var snapLevel, numSnapped int
 	for level, h := range histogram[1:] {
 		if h > numSnapped {
 			snapLevel, numSnapped = level, h
 		}
 	}
 
 	// Choose an encoding format based on the number of unsnapped vertices and a
 	// rough estimate of the encoded sizes.
 	numUnsnapped := p.numVertices - numSnapped // Number of vertices that won't be snapped at snapLevel.
 	const pointSize = 3 * 8                    // s2.Point is an r3.Vector, which is 3 float64s. That's 3*8 = 24 bytes.
 	compressedSize := 4*p.numVertices + (pointSize+2)*numUnsnapped
 	losslessSize := pointSize * p.numVertices
 	if compressedSize < losslessSize {
 		p.encodeCompressed(e, snapLevel, vs)
 	} else {
 		p.encodeLossless(e)
 	}
 }
 
 // encodeLossless encodes the polygon's Points as float64s.
 func (p *Polygon) encodeLossless(e *encoder) {
 	e.writeInt8(encodingVersion)
 	e.writeBool(true) // a legacy c++ value. must be true.
 	e.writeBool(p.hasHoles)
 	e.writeUint32(uint32(len(p.loops)))
 
 	if e.err != nil {
 		return
 	}
 	if len(p.loops) > maxEncodedLoops {
 		e.err = fmt.Errorf("too many loops (%d; max is %d)", len(p.loops), maxEncodedLoops)
 		return
 	}
 	for _, l := range p.loops {
 		l.encode(e)
 	}
 
 	// Encode the bound.
 	p.bound.encode(e)
 }
 
 func (p *Polygon) encodeCompressed(e *encoder, snapLevel int, vertices []xyzFaceSiTi) {
 	e.writeUint8(uint8(encodingCompressedVersion))
 	e.writeUint8(uint8(snapLevel))
 	e.writeUvarint(uint64(len(p.loops)))
 
 	if e.err != nil {
 		return
 	}
 	if l := len(p.loops); l > maxEncodedLoops {
 		e.err = fmt.Errorf("too many loops to encode: %d; max is %d", l, maxEncodedLoops)
 		return
 	}
 
 	for _, l := range p.loops {
 		l.encodeCompressed(e, snapLevel, vertices[:len(l.vertices)])
 		vertices = vertices[len(l.vertices):]
 	}
 	// Do not write the bound, num_vertices, or has_holes_ as they can be
 	// cheaply recomputed by decodeCompressed.  Microbenchmarks show the
 	// speed difference is inconsequential.
 }
 
 // Decode decodes the Polygon.
 func (p *Polygon) Decode(r io.Reader) error {
 	d := &decoder{r: asByteReader(r)}
 	version := int8(d.readUint8())
 	var dec func(*decoder)
 	switch version {
 	case encodingVersion:
 		dec = p.decode
 	case encodingCompressedVersion:
 		dec = p.decodeCompressed
 	default:
 		return fmt.Errorf("unsupported version %d", version)
 	}
 	dec(d)
 	return d.err
 }
 
 // maxEncodedLoops is the biggest supported number of loops in a polygon during encoding.
 // Setting a maximum guards an allocation: it prevents an attacker from easily pushing us OOM.
 const maxEncodedLoops = 10000000
 
 func (p *Polygon) decode(d *decoder) {
 	*p = Polygon{}
 	d.readUint8() // Ignore irrelevant serialized owns_loops_ value.
 
 	p.hasHoles = d.readBool()
 
 	// Polygons with no loops are explicitly allowed here: a newly created
 	// polygon has zero loops and such polygons encode and decode properly.
 	nloops := d.readUint32()
 	if d.err != nil {
 		return
 	}
 	if nloops > maxEncodedLoops {
 		d.err = fmt.Errorf("too many loops (%d; max is %d)", nloops, maxEncodedLoops)
 		return
 	}
 	p.loops = make([]*Loop, nloops)
 	for i := range p.loops {
 		p.loops[i] = new(Loop)
 		p.loops[i].decode(d)
 		p.numVertices += len(p.loops[i].vertices)
 	}
 
 	p.bound.decode(d)
 	if d.err != nil {
 		return
 	}
 	p.subregionBound = ExpandForSubregions(p.bound)
 	p.initEdgesAndIndex()
 }
 
 func (p *Polygon) decodeCompressed(d *decoder) {
 	snapLevel := int(d.readUint8())
 
 	if snapLevel > maxLevel {
 		d.err = fmt.Errorf("snaplevel too big: %d", snapLevel)
 		return
 	}
 	// Polygons with no loops are explicitly allowed here: a newly created
 	// polygon has zero loops and such polygons encode and decode properly.
 	nloops := int(d.readUvarint())
 	if nloops > maxEncodedLoops {
 		d.err = fmt.Errorf("too many loops (%d; max is %d)", nloops, maxEncodedLoops)
 	}
 	p.loops = make([]*Loop, nloops)
 	for i := range p.loops {
 		p.loops[i] = new(Loop)
 		p.loops[i].decodeCompressed(d, snapLevel)
 	}
 	p.initLoopProperties()
 }
 
 // TODO(roberts): Differences from C++
 // Centroid
 // SnapLevel
 // DistanceToPoint
 // DistanceToBoundary
 // Project
 // ProjectToBoundary
 // ApproxContains/ApproxDisjoint for Polygons
 // InitTo{Intersection/ApproxIntersection/Union/ApproxUnion/Diff/ApproxDiff}
 // InitToSimplified
 // InitToSnapped
 // IntersectWithPolyline
 // ApproxIntersectWithPolyline
 // SubtractFromPolyline
 // ApproxSubtractFromPolyline
 // DestructiveUnion
 // DestructiveApproxUnion
 // InitToCellUnionBorder
 // IsNormalized
 // Equal/BoundaryEqual/BoundaryApproxEqual/BoundaryNear Polygons
 // BreakEdgesAndAddToBuilder
 //
 // clearLoops
 // findLoopNestingError
 // initToSimplifiedInternal
 // internalClipPolyline
 // clipBoundary