gtsocial-umbx

edge_query.go (30347B)

---

 // Copyright 2019 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 (
 	"sort"
 
 	"github.com/golang/geo/s1"
 )
 
 // EdgeQueryOptions holds the options for controlling how EdgeQuery operates.
 //
 // Options can be chained together builder-style:
 //
 //	opts = NewClosestEdgeQueryOptions().
 //		MaxResults(1).
 //		DistanceLimit(s1.ChordAngleFromAngle(3 * s1.Degree)).
 //		MaxError(s1.ChordAngleFromAngle(0.001 * s1.Degree))
 //	query = NewClosestEdgeQuery(index, opts)
 //
 //  or set individually:
 //
 //	opts = NewClosestEdgeQueryOptions()
 //	opts.IncludeInteriors(true)
 //
 // or just inline:
 //
 //	query = NewClosestEdgeQuery(index, NewClosestEdgeQueryOptions().MaxResults(3))
 //
 // If you pass a nil as the options you get the default values for the options.
 type EdgeQueryOptions struct {
 	common *queryOptions
 }
 
 // DistanceLimit specifies that only edges whose distance to the target is
 // within, this distance should be returned.  Edges whose distance is equal
 // are not returned. To include values that are equal, specify the limit with
 // the next largest representable distance. i.e. limit.Successor().
 func (e *EdgeQueryOptions) DistanceLimit(limit s1.ChordAngle) *EdgeQueryOptions {
 	e.common = e.common.DistanceLimit(limit)
 	return e
 }
 
 // IncludeInteriors specifies whether polygon interiors should be
 // included when measuring distances.
 func (e *EdgeQueryOptions) IncludeInteriors(x bool) *EdgeQueryOptions {
 	e.common = e.common.IncludeInteriors(x)
 	return e
 }
 
 // UseBruteForce sets or disables the use of brute force in a query.
 func (e *EdgeQueryOptions) UseBruteForce(x bool) *EdgeQueryOptions {
 	e.common = e.common.UseBruteForce(x)
 	return e
 }
 
 // MaxError specifies that edges up to dist away than the true
 // matching edges may be substituted in the result set, as long as such
 // edges satisfy all the remaining search criteria (such as DistanceLimit).
 // This option only has an effect if MaxResults is also specified;
 // otherwise all edges closer than MaxDistance will always be returned.
 func (e *EdgeQueryOptions) MaxError(dist s1.ChordAngle) *EdgeQueryOptions {
 	e.common = e.common.MaxError(dist)
 	return e
 }
 
 // MaxResults specifies that at most MaxResults edges should be returned.
 // This must be at least 1.
 func (e *EdgeQueryOptions) MaxResults(n int) *EdgeQueryOptions {
 	e.common = e.common.MaxResults(n)
 	return e
 }
 
 // NewClosestEdgeQueryOptions returns a set of edge query options suitable
 // for performing closest edge queries.
 func NewClosestEdgeQueryOptions() *EdgeQueryOptions {
 	return &EdgeQueryOptions{
 		common: newQueryOptions(minDistance(0)),
 	}
 }
 
 // NewFurthestEdgeQueryOptions returns a set of edge query options suitable
 // for performing furthest edge queries.
 func NewFurthestEdgeQueryOptions() *EdgeQueryOptions {
 	return &EdgeQueryOptions{
 		common: newQueryOptions(maxDistance(0)),
 	}
 }
 
 // EdgeQueryResult represents an edge that meets the target criteria for the
 // query. Note the following special cases:
 //
 //  - ShapeID >= 0 && EdgeID < 0 represents the interior of a shape.
 //    Such results may be returned when the option IncludeInteriors is true.
 //
 //  - ShapeID < 0 && EdgeID < 0 is returned to indicate that no edge
 //    satisfies the requested query options.
 type EdgeQueryResult struct {
 	distance distance
 	shapeID  int32
 	edgeID   int32
 }
 
 // Distance reports the distance between the edge in this shape that satisfied
 // the query's parameters.
 func (e EdgeQueryResult) Distance() s1.ChordAngle { return e.distance.chordAngle() }
 
 // ShapeID reports the ID of the Shape this result is for.
 func (e EdgeQueryResult) ShapeID() int32 { return e.shapeID }
 
 // EdgeID reports the ID of the edge in the results Shape.
 func (e EdgeQueryResult) EdgeID() int32 { return e.edgeID }
 
 // newEdgeQueryResult returns a result instance with default values.
 func newEdgeQueryResult(target distanceTarget) EdgeQueryResult {
 	return EdgeQueryResult{
 		distance: target.distance().infinity(),
 		shapeID:  -1,
 		edgeID:   -1,
 	}
 }
 
 // IsInterior reports if this result represents the interior of a Shape.
 func (e EdgeQueryResult) IsInterior() bool {
 	return e.shapeID >= 0 && e.edgeID < 0
 }
 
 // IsEmpty reports if this has no edge that satisfies the given edge query options.
 // This result is only returned in one special case, namely when FindEdge() does
 // not find any suitable edges.
 func (e EdgeQueryResult) IsEmpty() bool {
 	return e.shapeID < 0
 }
 
 // Less reports if this results is less that the other first by distance,
 // then by (shapeID, edgeID). This is used for sorting.
 func (e EdgeQueryResult) Less(other EdgeQueryResult) bool {
 	if e.distance.chordAngle() != other.distance.chordAngle() {
 		return e.distance.less(other.distance)
 	}
 	if e.shapeID != other.shapeID {
 		return e.shapeID < other.shapeID
 	}
 	return e.edgeID < other.edgeID
 }
 
 // EdgeQuery is used to find the edge(s) between two geometries that match a
 // given set of options. It is flexible enough so that it can be adapted to
 // compute maximum distances and even potentially Hausdorff distances.
 //
 // By using the appropriate options, this type can answer questions such as:
 //
 //  - Find the minimum distance between two geometries A and B.
 //  - Find all edges of geometry A that are within a distance D of geometry B.
 //  - Find the k edges of geometry A that are closest to a given point P.
 //
 // You can also specify whether polygons should include their interiors (i.e.,
 // if a point is contained by a polygon, should the distance be zero or should
 // it be measured to the polygon boundary?)
 //
 // The input geometries may consist of any number of points, polylines, and
 // polygons (collectively referred to as "shapes"). Shapes do not need to be
 // disjoint; they may overlap or intersect arbitrarily. The implementation is
 // designed to be fast for both simple and complex geometries.
 type EdgeQuery struct {
 	index  *ShapeIndex
 	opts   *queryOptions
 	target distanceTarget
 
 	// True if opts.maxError must be subtracted from ShapeIndex cell distances
 	// in order to ensure that such distances are measured conservatively. This
 	// is true only if the target takes advantage of maxError in order to
 	// return faster results, and 0 < maxError < distanceLimit.
 	useConservativeCellDistance bool
 
 	// The decision about whether to use the brute force algorithm is based on
 	// counting the total number of edges in the index. However if the index
 	// contains a large number of shapes, this in itself might take too long.
 	// So instead we only count edges up to (maxBruteForceIndexSize() + 1)
 	// for the current target type (stored as indexNumEdgesLimit).
 	indexNumEdges      int
 	indexNumEdgesLimit int
 
 	// The distance beyond which we can safely ignore further candidate edges.
 	// (Candidates that are exactly at the limit are ignored; this is more
 	// efficient for UpdateMinDistance and should not affect clients since
 	// distance measurements have a small amount of error anyway.)
 	//
 	// Initially this is the same as the maximum distance specified by the user,
 	// but it can also be updated by the algorithm (see maybeAddResult).
 	distanceLimit distance
 
 	// The current set of results of the query.
 	results []EdgeQueryResult
 
 	// This field is true when duplicates must be avoided explicitly. This
 	// is achieved by maintaining a separate set keyed by (shapeID, edgeID)
 	// only, and checking whether each edge is in that set before computing the
 	// distance to it.
 	avoidDuplicates bool
 
 	// testedEdges tracks the set of shape and edges that have already been tested.
 	testedEdges map[ShapeEdgeID]uint32
 
 	// For the optimized algorihm we precompute the top-level CellIDs that
 	// will be added to the priority queue. There can be at most 6 of these
 	// cells. Essentially this is just a covering of the indexed edges, except
 	// that we also store pointers to the corresponding ShapeIndexCells to
 	// reduce the number of index seeks required.
 	indexCovering []CellID
 	indexCells    []*ShapeIndexCell
 
 	// The algorithm maintains a priority queue of unprocessed CellIDs, sorted
 	// in increasing order of distance from the target.
 	queue *queryQueue
 
 	iter                *ShapeIndexIterator
 	maxDistanceCovering []CellID
 	initialCells        []CellID
 }
 
 // NewClosestEdgeQuery returns an EdgeQuery that is used for finding the
 // closest edge(s) to a given Point, Edge, Cell, or geometry collection.
 //
 // You can find either the k closest edges, or all edges within a given
 // radius, or both (i.e., the k closest edges up to a given maximum radius).
 // E.g. to find all the edges within 5 kilometers, set the DistanceLimit in
 // the options.
 //
 // By default *all* edges are returned, so you should always specify either
 // MaxResults or DistanceLimit options or both.
 //
 // Note that by default, distances are measured to the boundary and interior
 // of polygons. For example, if a point is inside a polygon then its distance
 // is zero. To change this behavior, set the IncludeInteriors option to false.
 //
 // If you only need to test whether the distance is above or below a given
 // threshold (e.g., 10 km), you can use the IsDistanceLess() method.  This is
 // much faster than actually calculating the distance with FindEdge,
 // since the implementation can stop as soon as it can prove that the minimum
 // distance is either above or below the threshold.
 func NewClosestEdgeQuery(index *ShapeIndex, opts *EdgeQueryOptions) *EdgeQuery {
 	if opts == nil {
 		opts = NewClosestEdgeQueryOptions()
 	}
 	e := &EdgeQuery{
 		testedEdges: make(map[ShapeEdgeID]uint32),
 		index:       index,
 		opts:        opts.common,
 		queue:       newQueryQueue(),
 	}
 
 	return e
 }
 
 // NewFurthestEdgeQuery returns an EdgeQuery that is used for finding the
 // furthest edge(s) to a given Point, Edge, Cell, or geometry collection.
 //
 // The furthest edge is defined as the one which maximizes the
 // distance from any point on that edge to any point on the target geometry.
 //
 // Similar to the example in NewClosestEdgeQuery, to find the 5 furthest edges
 // from a given Point:
 func NewFurthestEdgeQuery(index *ShapeIndex, opts *EdgeQueryOptions) *EdgeQuery {
 	if opts == nil {
 		opts = NewFurthestEdgeQueryOptions()
 	}
 	e := &EdgeQuery{
 		testedEdges: make(map[ShapeEdgeID]uint32),
 		index:       index,
 		opts:        opts.common,
 		queue:       newQueryQueue(),
 	}
 
 	return e
 }
 
 // Reset resets the state of this EdgeQuery.
 func (e *EdgeQuery) Reset() {
 	e.indexNumEdges = 0
 	e.indexNumEdgesLimit = 0
 	e.indexCovering = nil
 	e.indexCells = nil
 }
 
 // FindEdges returns the edges for the given target that satisfy the current options.
 //
 // Note that if opts.IncludeInteriors is true, the results may include some
 // entries with edge_id == -1. This indicates that the target intersects
 // the indexed polygon with the given ShapeID.
 func (e *EdgeQuery) FindEdges(target distanceTarget) []EdgeQueryResult {
 	return e.findEdges(target, e.opts)
 }
 
 // Distance reports the distance to the target. If the index or target is empty,
 // returns the EdgeQuery's maximal sentinel.
 //
 // Use IsDistanceLess()/IsDistanceGreater() if you only want to compare the
 // distance against a threshold value, since it is often much faster.
 func (e *EdgeQuery) Distance(target distanceTarget) s1.ChordAngle {
 	return e.findEdge(target, e.opts).Distance()
 }
 
 // IsDistanceLess reports if the distance to target is less than the given limit.
 //
 // This method is usually much faster than Distance(), since it is much
 // less work to determine whether the minimum distance is above or below a
 // threshold than it is to calculate the actual minimum distance.
 //
 // If you wish to check if the distance is less than or equal to the limit, use:
 //
 //	query.IsDistanceLess(target, limit.Successor())
 //
 func (e *EdgeQuery) IsDistanceLess(target distanceTarget, limit s1.ChordAngle) bool {
 	opts := e.opts
 	opts = opts.MaxResults(1).
 		DistanceLimit(limit).
 		MaxError(s1.StraightChordAngle)
 	return !e.findEdge(target, opts).IsEmpty()
 }
 
 // IsDistanceGreater reports if the distance to target is greater than limit.
 //
 // This method is usually much faster than Distance, since it is much
 // less work to determine whether the maximum distance is above or below a
 // threshold than it is to calculate the actual maximum distance.
 // If you wish to check if the distance is less than or equal to the limit, use:
 //
 //	query.IsDistanceGreater(target, limit.Predecessor())
 //
 func (e *EdgeQuery) IsDistanceGreater(target distanceTarget, limit s1.ChordAngle) bool {
 	return e.IsDistanceLess(target, limit)
 }
 
 // IsConservativeDistanceLessOrEqual reports if the distance to target is less
 // or equal to the limit, where the limit has been expanded by the maximum error
 // for the distance calculation.
 //
 // For example, suppose that we want to test whether two geometries might
 // intersect each other after they are snapped together using Builder
 // (using the IdentitySnapFunction with a given "snap radius").  Since
 // Builder uses exact distance predicates (s2predicates), we need to
 // measure the distance between the two geometries conservatively.  If the
 // distance is definitely greater than "snap radius", then the geometries
 // are guaranteed to not intersect after snapping.
 func (e *EdgeQuery) IsConservativeDistanceLessOrEqual(target distanceTarget, limit s1.ChordAngle) bool {
 	return e.IsDistanceLess(target, limit.Expanded(minUpdateDistanceMaxError(limit)))
 }
 
 // IsConservativeDistanceGreaterOrEqual reports if the distance to the target is greater
 // than or equal to the given limit with some small tolerance.
 func (e *EdgeQuery) IsConservativeDistanceGreaterOrEqual(target distanceTarget, limit s1.ChordAngle) bool {
 	return e.IsDistanceGreater(target, limit.Expanded(-minUpdateDistanceMaxError(limit)))
 }
 
 // findEdges returns the closest edges to the given target that satisfy the given options.
 //
 // Note that if opts.includeInteriors is true, the results may include some
 // entries with edgeID == -1. This indicates that the target intersects the
 // indexed polygon with the given shapeID.
 func (e *EdgeQuery) findEdges(target distanceTarget, opts *queryOptions) []EdgeQueryResult {
 	e.findEdgesInternal(target, opts)
 	// TODO(roberts): Revisit this if there is a heap or other sorted and
 	// uniquing datastructure we can use instead of just a slice.
 	e.results = sortAndUniqueResults(e.results)
 	if len(e.results) > e.opts.maxResults {
 		e.results = e.results[:e.opts.maxResults]
 	}
 	return e.results
 }
 
 func sortAndUniqueResults(results []EdgeQueryResult) []EdgeQueryResult {
 	if len(results) <= 1 {
 		return results
 	}
 	sort.Slice(results, func(i, j int) bool { return results[i].Less(results[j]) })
 	j := 0
 	for i := 1; i < len(results); i++ {
 		if results[j] == results[i] {
 			continue
 		}
 		j++
 		results[j] = results[i]
 	}
 	return results[:j+1]
 }
 
 // findEdge is a convenience method that returns exactly one edge, and if no
 // edges satisfy the given search criteria, then a default Result is returned.
 //
 // This is primarily to ease the usage of a number of the methods in the DistanceTargets
 // and in EdgeQuery.
 func (e *EdgeQuery) findEdge(target distanceTarget, opts *queryOptions) EdgeQueryResult {
 	opts.MaxResults(1)
 	e.findEdges(target, opts)
 	if len(e.results) > 0 {
 		return e.results[0]
 	}
 
 	return newEdgeQueryResult(target)
 }
 
 // findEdgesInternal does the actual work for find edges that match the given options.
 func (e *EdgeQuery) findEdgesInternal(target distanceTarget, opts *queryOptions) {
 	e.target = target
 	e.opts = opts
 
 	e.testedEdges = make(map[ShapeEdgeID]uint32)
 	e.distanceLimit = target.distance().fromChordAngle(opts.distanceLimit)
 	e.results = make([]EdgeQueryResult, 0)
 
 	if e.distanceLimit == target.distance().zero() {
 		return
 	}
 
 	if opts.includeInteriors {
 		shapeIDs := map[int32]struct{}{}
 		e.target.visitContainingShapes(e.index, func(containingShape Shape, targetPoint Point) bool {
 			shapeIDs[e.index.idForShape(containingShape)] = struct{}{}
 			return len(shapeIDs) < opts.maxResults
 		})
 		for shapeID := range shapeIDs {
 			e.addResult(EdgeQueryResult{target.distance().zero(), shapeID, -1})
 		}
 
 		if e.distanceLimit == target.distance().zero() {
 			return
 		}
 	}
 
 	// If maxError > 0 and the target takes advantage of this, then we may
 	// need to adjust the distance estimates to ShapeIndex cells to ensure
 	// that they are always a lower bound on the true distance. For example,
 	// suppose max_distance == 100, maxError == 30, and we compute the distance
 	// to the target from some cell C0 as d(C0) == 80. Then because the target
 	// takes advantage of maxError, the true distance could be as low as 50.
 	// In order not to miss edges contained by such cells, we need to subtract
 	// maxError from the distance estimates. This behavior is controlled by
 	// the useConservativeCellDistance flag.
 	//
 	// However there is one important case where this adjustment is not
 	// necessary, namely when distanceLimit < maxError, This is because
 	// maxError only affects the algorithm once at least maxEdges edges
 	// have been found that satisfy the given distance limit. At that point,
 	// maxError is subtracted from distanceLimit in order to ensure that
 	// any further matches are closer by at least that amount. But when
 	// distanceLimit < maxError, this reduces the distance limit to 0,
 	// i.e. all remaining candidate cells and edges can safely be discarded.
 	// (This is how IsDistanceLess() and friends are implemented.)
 	targetUsesMaxError := opts.maxError != target.distance().zero().chordAngle() &&
 		e.target.setMaxError(opts.maxError)
 
 	// Note that we can't compare maxError and distanceLimit directly
 	// because one is a Delta and one is a Distance. Instead we subtract them.
 	e.useConservativeCellDistance = targetUsesMaxError &&
 		(e.distanceLimit == target.distance().infinity() ||
 			target.distance().zero().less(e.distanceLimit.sub(target.distance().fromChordAngle(opts.maxError))))
 
 	// Use the brute force algorithm if the index is small enough. To avoid
 	// spending too much time counting edges when there are many shapes, we stop
 	// counting once there are too many edges. We may need to recount the edges
 	// if we later see a target with a larger brute force edge threshold.
 	minOptimizedEdges := e.target.maxBruteForceIndexSize() + 1
 	if minOptimizedEdges > e.indexNumEdgesLimit && e.indexNumEdges >= e.indexNumEdgesLimit {
 		e.indexNumEdges = e.index.NumEdgesUpTo(minOptimizedEdges)
 		e.indexNumEdgesLimit = minOptimizedEdges
 	}
 
 	if opts.useBruteForce || e.indexNumEdges < minOptimizedEdges {
 		// The brute force algorithm already considers each edge exactly once.
 		e.avoidDuplicates = false
 		e.findEdgesBruteForce()
 	} else {
 		// If the target takes advantage of maxError then we need to avoid
 		// duplicate edges explicitly. (Otherwise it happens automatically.)
 		e.avoidDuplicates = targetUsesMaxError && opts.maxResults > 1
 		e.findEdgesOptimized()
 	}
 }
 
 func (e *EdgeQuery) addResult(r EdgeQueryResult) {
 	e.results = append(e.results, r)
 	if e.opts.maxResults == 1 {
 		// Optimization for the common case where only the closest edge is wanted.
 		e.distanceLimit = r.distance.sub(e.target.distance().fromChordAngle(e.opts.maxError))
 	}
 	// TODO(roberts): Add the other if/else cases when a different data structure
 	// is used for the results.
 }
 
 func (e *EdgeQuery) maybeAddResult(shape Shape, edgeID int32) {
 	if _, ok := e.testedEdges[ShapeEdgeID{e.index.idForShape(shape), edgeID}]; e.avoidDuplicates && !ok {
 		return
 	}
 	edge := shape.Edge(int(edgeID))
 	dist := e.distanceLimit
 
 	if dist, ok := e.target.updateDistanceToEdge(edge, dist); ok {
 		e.addResult(EdgeQueryResult{dist, e.index.idForShape(shape), edgeID})
 	}
 }
 
 func (e *EdgeQuery) findEdgesBruteForce() {
 	// Range over all shapes in the index. Does order matter here? if so
 	// switch to for i = 0 .. n?
 	for _, shape := range e.index.shapes {
 		// TODO(roberts): can this happen if we are only ranging over current entries?
 		if shape == nil {
 			continue
 		}
 		for edgeID := int32(0); edgeID < int32(shape.NumEdges()); edgeID++ {
 			e.maybeAddResult(shape, edgeID)
 		}
 	}
 }
 
 func (e *EdgeQuery) findEdgesOptimized() {
 	e.initQueue()
 	// Repeatedly find the closest Cell to "target" and either split it into
 	// its four children or process all of its edges.
 	for e.queue.size() > 0 {
 		// We need to copy the top entry before removing it, and we need to
 		// remove it before adding any new entries to the queue.
 		entry := e.queue.pop()
 
 		if !entry.distance.less(e.distanceLimit) {
 			e.queue.reset() // Clear any remaining entries.
 			break
 		}
 		// If this is already known to be an index cell, just process it.
 		if entry.indexCell != nil {
 			e.processEdges(entry)
 			continue
 		}
 		// Otherwise split the cell into its four children.  Before adding a
 		// child back to the queue, we first check whether it is empty.  We do
 		// this in two seek operations rather than four by seeking to the key
 		// between children 0 and 1 and to the key between children 2 and 3.
 		id := entry.id
 		ch := id.Children()
 		e.iter.seek(ch[1].RangeMin())
 
 		if !e.iter.Done() && e.iter.CellID() <= ch[1].RangeMax() {
 			e.processOrEnqueueCell(ch[1])
 		}
 		if e.iter.Prev() && e.iter.CellID() >= id.RangeMin() {
 			e.processOrEnqueueCell(ch[0])
 		}
 
 		e.iter.seek(ch[3].RangeMin())
 		if !e.iter.Done() && e.iter.CellID() <= id.RangeMax() {
 			e.processOrEnqueueCell(ch[3])
 		}
 		if e.iter.Prev() && e.iter.CellID() >= ch[2].RangeMin() {
 			e.processOrEnqueueCell(ch[2])
 		}
 	}
 }
 
 func (e *EdgeQuery) processOrEnqueueCell(id CellID) {
 	if e.iter.CellID() == id {
 		e.processOrEnqueue(id, e.iter.IndexCell())
 	} else {
 		e.processOrEnqueue(id, nil)
 	}
 }
 
 func (e *EdgeQuery) initQueue() {
 	if len(e.indexCovering) == 0 {
 		// We delay iterator initialization until now to make queries on very
 		// small indexes a bit faster (i.e., where brute force is used).
 		e.iter = NewShapeIndexIterator(e.index)
 	}
 
 	// Optimization: if the user is searching for just the closest edge, and the
 	// center of the target's bounding cap happens to intersect an index cell,
 	// then we try to limit the search region to a small disc by first
 	// processing the edges in that cell.  This sets distance_limit_ based on
 	// the closest edge in that cell, which we can then use to limit the search
 	// area.  This means that the cell containing "target" will be processed
 	// twice, but in general this is still faster.
 	//
 	// TODO(roberts): Even if the cap center is not contained, we could still
 	// process one or both of the adjacent index cells in CellID order,
 	// provided that those cells are closer than distanceLimit.
 	cb := e.target.capBound()
 	if cb.IsEmpty() {
 		return // Empty target.
 	}
 
 	if e.opts.maxResults == 1 && e.iter.LocatePoint(cb.Center()) {
 		e.processEdges(&queryQueueEntry{
 			distance:  e.target.distance().zero(),
 			id:        e.iter.CellID(),
 			indexCell: e.iter.IndexCell(),
 		})
 		// Skip the rest of the algorithm if we found an intersecting edge.
 		if e.distanceLimit == e.target.distance().zero() {
 			return
 		}
 	}
 	if len(e.indexCovering) == 0 {
 		e.initCovering()
 	}
 	if e.distanceLimit == e.target.distance().infinity() {
 		// Start with the precomputed index covering.
 		for i := range e.indexCovering {
 			e.processOrEnqueue(e.indexCovering[i], e.indexCells[i])
 		}
 	} else {
 		// Compute a covering of the search disc and intersect it with the
 		// precomputed index covering.
 		coverer := &RegionCoverer{MaxCells: 4, LevelMod: 1, MaxLevel: maxLevel}
 
 		radius := cb.Radius() + e.distanceLimit.chordAngleBound().Angle()
 		searchCB := CapFromCenterAngle(cb.Center(), radius)
 		maxDistCover := coverer.FastCovering(searchCB)
 		e.initialCells = CellUnionFromIntersection(e.indexCovering, maxDistCover)
 
 		// Now we need to clean up the initial cells to ensure that they all
 		// contain at least one cell of the ShapeIndex. (Some may not intersect
 		// the index at all, while other may be descendants of an index cell.)
 		i, j := 0, 0
 		for i < len(e.initialCells) {
 			idI := e.initialCells[i]
 			// Find the top-level cell that contains this initial cell.
 			for e.indexCovering[j].RangeMax() < idI {
 				j++
 			}
 
 			idJ := e.indexCovering[j]
 			if idI == idJ {
 				// This initial cell is one of the top-level cells.  Use the
 				// precomputed ShapeIndexCell pointer to avoid an index seek.
 				e.processOrEnqueue(idJ, e.indexCells[j])
 				i++
 				j++
 			} else {
 				// This initial cell is a proper descendant of a top-level cell.
 				// Check how it is related to the cells of the ShapeIndex.
 				r := e.iter.LocateCellID(idI)
 				if r == Indexed {
 					// This cell is a descendant of an index cell.
 					// Enqueue it and skip any other initial cells
 					// that are also descendants of this cell.
 					e.processOrEnqueue(e.iter.CellID(), e.iter.IndexCell())
 					lastID := e.iter.CellID().RangeMax()
 					for i < len(e.initialCells) && e.initialCells[i] <= lastID {
 						i++
 					}
 				} else {
 					// Enqueue the cell only if it contains at least one index cell.
 					if r == Subdivided {
 						e.processOrEnqueue(idI, nil)
 					}
 					i++
 				}
 			}
 		}
 	}
 }
 
 func (e *EdgeQuery) initCovering() {
 	// Find the range of Cells spanned by the index and choose a level such
 	// that the entire index can be covered with just a few cells. These are
 	// the "top-level" cells. There are two cases:
 	//
 	//  - If the index spans more than one face, then there is one top-level cell
 	// per spanned face, just big enough to cover the index cells on that face.
 	//
 	//  - If the index spans only one face, then we find the smallest cell "C"
 	// that covers the index cells on that face (just like the case above).
 	// Then for each of the 4 children of "C", if the child contains any index
 	// cells then we create a top-level cell that is big enough to just fit
 	// those index cells (i.e., shrinking the child as much as possible to fit
 	// its contents). This essentially replicates what would happen if we
 	// started with "C" as the top-level cell, since "C" would immediately be
 	// split, except that we take the time to prune the children further since
 	// this will save work on every subsequent query.
 	e.indexCovering = make([]CellID, 0, 6)
 
 	// TODO(roberts): Use a single iterator below and save position
 	// information using pair {CellID, ShapeIndexCell}.
 	next := NewShapeIndexIterator(e.index, IteratorBegin)
 	last := NewShapeIndexIterator(e.index, IteratorEnd)
 	last.Prev()
 	if next.CellID() != last.CellID() {
 		// The index has at least two cells. Choose a level such that the entire
 		// index can be spanned with at most 6 cells (if the index spans multiple
 		// faces) or 4 cells (it the index spans a single face).
 		level, ok := next.CellID().CommonAncestorLevel(last.CellID())
 		if !ok {
 			level = 0
 		} else {
 			level++
 		}
 
 		// Visit each potential top-level cell except the last (handled below).
 		lastID := last.CellID().Parent(level)
 		for id := next.CellID().Parent(level); id != lastID; id = id.Next() {
 			// Skip any top-level cells that don't contain any index cells.
 			if id.RangeMax() < next.CellID() {
 				continue
 			}
 
 			// Find the range of index cells contained by this top-level cell and
 			// then shrink the cell if necessary so that it just covers them.
 			cellFirst := next.clone()
 			next.seek(id.RangeMax().Next())
 			cellLast := next.clone()
 			cellLast.Prev()
 			e.addInitialRange(cellFirst, cellLast)
 			break
 		}
 
 	}
 	e.addInitialRange(next, last)
 }
 
 // addInitialRange adds an entry to the indexCovering and indexCells that covers the given
 // inclusive range of cells.
 //
 // This requires that first and last cells have a common ancestor.
 func (e *EdgeQuery) addInitialRange(first, last *ShapeIndexIterator) {
 	if first.CellID() == last.CellID() {
 		// The range consists of a single index cell.
 		e.indexCovering = append(e.indexCovering, first.CellID())
 		e.indexCells = append(e.indexCells, first.IndexCell())
 	} else {
 		// Add the lowest common ancestor of the given range.
 		level, _ := first.CellID().CommonAncestorLevel(last.CellID())
 		e.indexCovering = append(e.indexCovering, first.CellID().Parent(level))
 		e.indexCells = append(e.indexCells, nil)
 	}
 }
 
 // processEdges processes all the edges of the given index cell.
 func (e *EdgeQuery) processEdges(entry *queryQueueEntry) {
 	for _, clipped := range entry.indexCell.shapes {
 		shape := e.index.Shape(clipped.shapeID)
 		for j := 0; j < clipped.numEdges(); j++ {
 			e.maybeAddResult(shape, int32(clipped.edges[j]))
 		}
 	}
 }
 
 // processOrEnqueue the given cell id and indexCell.
 func (e *EdgeQuery) processOrEnqueue(id CellID, indexCell *ShapeIndexCell) {
 	if indexCell != nil {
 		// If this index cell has only a few edges, then it is faster to check
 		// them directly rather than computing the minimum distance to the Cell
 		// and inserting it into the queue.
 		const minEdgesToEnqueue = 10
 		numEdges := indexCell.numEdges()
 		if numEdges == 0 {
 			return
 		}
 		if numEdges < minEdgesToEnqueue {
 			// Set "distance" to zero to avoid the expense of computing it.
 			e.processEdges(&queryQueueEntry{
 				distance:  e.target.distance().zero(),
 				id:        id,
 				indexCell: indexCell,
 			})
 			return
 		}
 	}
 
 	// Otherwise compute the minimum distance to any point in the cell and add
 	// it to the priority queue.
 	cell := CellFromCellID(id)
 	dist := e.distanceLimit
 	var ok bool
 	if dist, ok = e.target.updateDistanceToCell(cell, dist); !ok {
 		return
 	}
 	if e.useConservativeCellDistance {
 		// Ensure that "distance" is a lower bound on the true distance to the cell.
 		dist = dist.sub(e.target.distance().fromChordAngle(e.opts.maxError))
 	}
 
 	e.queue.push(&queryQueueEntry{
 		distance:  dist,
 		id:        id,
 		indexCell: indexCell,
 	})
 }
 
 // TODO(roberts): Remaining pieces
 // GetEdge
 // Project