staticmap/vendor/github.com/golang/geo/s2/polygon.go

// 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"
)

// 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

}

// TODO(roberts): Implement initOriented
/*
// PolygonFromOrientedLoops, like PolygonFromLoops, returns a Polygon from the
// given set of loops. 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 {
	panic("PolygonFromOrientedLoops not yet implemented")
	p := &Polygon{
		loops: loops,
	}
	p.initOriented()
	return p
}
*/

// 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()
}

// initOriented takes the loops in this polygon and performs the nesting
// computations. It expects the 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 (p *Polygon) initOriented() {
	// 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.
	panic("initOriented not yet implemented")
}

// 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() {
	// the loops depths are set by initNested/initOriented prior to this.

	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() {
	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, look up the ShapeIndex cell containing this point.
	it := p.index.Iterator()
	if !it.LocatePoint(point) {
		return false
	}

	return p.iteratorContainsPoint(it, 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)}
}

// HasInterior reports whether this Polygon has an interior.
func (p *Polygon) HasInterior() bool {
	return p.dimension() == polygonGeometry
}

// 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() dimension { return polygonGeometry }

// 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)
		// TODO(roberts): Update this bound.Union call when initLoopProperties is implemented.
		p.bound = p.bound.Union(p.loops[i].bound)
		p.numVertices += len(p.loops[i].vertices)
	}
	if d.err != nil {
		return
	}
	if p.numVertices == 0 {
		p.bound = EmptyRect()
	}
	p.subregionBound = ExpandForSubregions(p.bound)
	p.initEdgesAndIndex()
}

// 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