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// Copyright 2015 Google Inc. All rights reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
package s2
import (
"fmt"
"io"
"math"
"github.com/golang/geo/r1"
"github.com/golang/geo/r3"
"github.com/golang/geo/s1"
)
// Loop represents a simple spherical polygon. It consists of a sequence
// of vertices where the first vertex is implicitly connected to the
// last. All loops are defined to have a CCW orientation, i.e. the interior of
// the loop is on the left side of the edges. This implies that a clockwise
// loop enclosing a small area is interpreted to be a CCW loop enclosing a
// very large area.
//
// Loops are not allowed to have any duplicate vertices (whether adjacent or
// not). Non-adjacent edges are not allowed to intersect, and furthermore edges
// of length 180 degrees are not allowed (i.e., adjacent vertices cannot be
// antipodal). Loops must have at least 3 vertices (except for the "empty" and
// "full" loops discussed below).
//
// There are two special loops: the "empty" loop contains no points and the
// "full" loop contains all points. These loops do not have any edges, but to
// preserve the invariant that every loop can be represented as a vertex
// chain, they are defined as having exactly one vertex each (see EmptyLoop
// and FullLoop).
type Loop struct {
vertices []Point
// originInside keeps a precomputed value whether this loop contains the origin
// versus computing from the set of vertices every time.
originInside bool
// depth is the nesting depth of this Loop if it is contained by a Polygon
// or other shape and is used to determine if this loop represents a hole
// or a filled in portion.
depth int
// bound is a conservative bound on all points contained by this loop.
// If l.ContainsPoint(P), then l.bound.ContainsPoint(P).
bound Rect
// Since bound is not exact, it is possible that a loop A contains
// another loop B whose bounds are slightly larger. subregionBound
// has been expanded sufficiently to account for this error, i.e.
// if A.Contains(B), then A.subregionBound.Contains(B.bound).
subregionBound Rect
// index is the spatial index for this Loop.
index *ShapeIndex
}
// LoopFromPoints constructs a loop from the given points.
func LoopFromPoints(pts []Point) *Loop {
l := &Loop{
vertices: pts,
}
l.initOriginAndBound()
return l
}
// LoopFromCell constructs a loop corresponding to the given cell.
//
// Note that the loop and cell *do not* contain exactly the same set of
// points, because Loop and Cell have slightly different definitions of
// point containment. For example, a Cell vertex is contained by all
// four neighboring Cells, but it is contained by exactly one of four
// Loops constructed from those cells. As another example, the cell
// coverings of cell and LoopFromCell(cell) will be different, because the
// loop contains points on its boundary that actually belong to other cells
// (i.e., the covering will include a layer of neighboring cells).
func LoopFromCell(c Cell) *Loop {
l := &Loop{
vertices: []Point{
c.Vertex(0),
c.Vertex(1),
c.Vertex(2),
c.Vertex(3),
},
}
l.initOriginAndBound()
return l
}
// These two points are used for the special Empty and Full loops.
var (
emptyLoopPoint = Point{r3.Vector{X: 0, Y: 0, Z: 1}}
fullLoopPoint = Point{r3.Vector{X: 0, Y: 0, Z: -1}}
)
// EmptyLoop returns a special "empty" loop.
func EmptyLoop() *Loop {
return LoopFromPoints([]Point{emptyLoopPoint})
}
// FullLoop returns a special "full" loop.
func FullLoop() *Loop {
return LoopFromPoints([]Point{fullLoopPoint})
}
// initOriginAndBound sets the origin containment for the given point and then calls
// the initialization for the bounds objects and the internal index.
func (l *Loop) initOriginAndBound() {
if len(l.vertices) < 3 {
// Check for the special "empty" and "full" loops (which have one vertex).
if !l.isEmptyOrFull() {
l.originInside = false
return
}
// This is the special empty or full loop, so the origin depends on if
// the vertex is in the southern hemisphere or not.
l.originInside = l.vertices[0].Z < 0
} else {
// Point containment testing is done by counting edge crossings starting
// at a fixed point on the sphere (OriginPoint). We need to know whether
// the reference point (OriginPoint) is inside or outside the loop before
// we can construct the ShapeIndex. We do this by first guessing that
// it is outside, and then seeing whether we get the correct containment
// result for vertex 1. If the result is incorrect, the origin must be
// inside the loop.
//
// A loop with consecutive vertices A,B,C contains vertex B if and only if
// the fixed vector R = B.Ortho is contained by the wedge ABC. The
// wedge is closed at A and open at C, i.e. the point B is inside the loop
// if A = R but not if C = R. This convention is required for compatibility
// with VertexCrossing. (Note that we can't use OriginPoint
// as the fixed vector because of the possibility that B == OriginPoint.)
l.originInside = false
v1Inside := OrderedCCW(Point{l.vertices[1].Ortho()}, l.vertices[0], l.vertices[2], l.vertices[1])
if v1Inside != l.ContainsPoint(l.vertices[1]) {
l.originInside = true
}
}
// We *must* call initBound before initializing the index, because
// initBound calls ContainsPoint which does a bounds check before using
// the index.
l.initBound()
// Create a new index and add us to it.
l.index = NewShapeIndex()
l.index.Add(l)
}
// initBound sets up the approximate bounding Rects for this loop.
func (l *Loop) initBound() {
// Check for the special "empty" and "full" loops.
if l.isEmptyOrFull() {
if l.IsEmpty() {
l.bound = EmptyRect()
} else {
l.bound = FullRect()
}
l.subregionBound = l.bound
return
}
// The bounding rectangle of a loop is not necessarily the same as the
// bounding rectangle of its vertices. First, the maximal latitude may be
// attained along the interior of an edge. Second, the loop may wrap
// entirely around the sphere (e.g. a loop that defines two revolutions of a
// candy-cane stripe). Third, the loop may include one or both poles.
// Note that a small clockwise loop near the equator contains both poles.
bounder := NewRectBounder()
for i := 0; i <= len(l.vertices); i++ { // add vertex 0 twice
bounder.AddPoint(l.Vertex(i))
}
b := bounder.RectBound()
if l.ContainsPoint(Point{r3.Vector{0, 0, 1}}) {
b = Rect{r1.Interval{b.Lat.Lo, math.Pi / 2}, s1.FullInterval()}
}
// If a loop contains the south pole, then either it wraps entirely
// around the sphere (full longitude range), or it also contains the
// north pole in which case b.Lng.IsFull() due to the test above.
// Either way, we only need to do the south pole containment test if
// b.Lng.IsFull().
if b.Lng.IsFull() && l.ContainsPoint(Point{r3.Vector{0, 0, -1}}) {
b.Lat.Lo = -math.Pi / 2
}
l.bound = b
l.subregionBound = ExpandForSubregions(l.bound)
}
// Validate checks whether this is a valid loop.
func (l *Loop) Validate() error {
if err := l.findValidationErrorNoIndex(); err != nil {
return err
}
// Check for intersections between non-adjacent edges (including at vertices)
// TODO(roberts): Once shapeutil gets findAnyCrossing uncomment this.
// return findAnyCrossing(l.index)
return nil
}
// findValidationErrorNoIndex reports whether this is not a valid loop, but
// skips checks that would require a ShapeIndex to be built for the loop. This
// is primarily used by Polygon to do validation so it doesn't trigger the
// creation of unneeded ShapeIndices.
func (l *Loop) findValidationErrorNoIndex() error {
// All vertices must be unit length.
for i, v := range l.vertices {
if !v.IsUnit() {
return fmt.Errorf("vertex %d is not unit length", i)
}
}
// Loops must have at least 3 vertices (except for empty and full).
if len(l.vertices) < 3 {
if l.isEmptyOrFull() {
return nil // Skip remaining tests.
}
return fmt.Errorf("non-empty, non-full loops must have at least 3 vertices")
}
// Loops are not allowed to have any duplicate vertices or edge crossings.
// We split this check into two parts. First we check that no edge is
// degenerate (identical endpoints). Then we check that there are no
// intersections between non-adjacent edges (including at vertices). The
// second check needs the ShapeIndex, so it does not fall within the scope
// of this method.
for i, v := range l.vertices {
if v == l.Vertex(i+1) {
return fmt.Errorf("edge %d is degenerate (duplicate vertex)", i)
}
// Antipodal vertices are not allowed.
if other := (Point{l.Vertex(i + 1).Mul(-1)}); v == other {
return fmt.Errorf("vertices %d and %d are antipodal", i,
(i+1)%len(l.vertices))
}
}
return nil
}
// Contains reports whether the region contained by this loop is a superset of the
// region contained by the given other loop.
func (l *Loop) Contains(o *Loop) bool {
// For a loop A to contain the loop B, all of the following must
// be true:
//
// (1) There are no edge crossings between A and B except at vertices.
//
// (2) At every vertex that is shared between A and B, the local edge
// ordering implies that A contains B.
//
// (3) If there are no shared vertices, then A must contain a vertex of B
// and B must not contain a vertex of A. (An arbitrary vertex may be
// chosen in each case.)
//
// The second part of (3) is necessary to detect the case of two loops whose
// union is the entire sphere, i.e. two loops that contains each other's
// boundaries but not each other's interiors.
if !l.subregionBound.Contains(o.bound) {
return false
}
// Special cases to handle either loop being empty or full.
if l.isEmptyOrFull() || o.isEmptyOrFull() {
return l.IsFull() || o.IsEmpty()
}
// Check whether there are any edge crossings, and also check the loop
// relationship at any shared vertices.
relation := &containsRelation{}
if hasCrossingRelation(l, o, relation) {
return false
}
// There are no crossings, and if there are any shared vertices then A
// contains B locally at each shared vertex.
if relation.foundSharedVertex {
return true
}
// Since there are no edge intersections or shared vertices, we just need to
// test condition (3) above. We can skip this test if we discovered that A
// contains at least one point of B while checking for edge crossings.
if !l.ContainsPoint(o.Vertex(0)) {
return false
}
// We still need to check whether (A union B) is the entire sphere.
// Normally this check is very cheap due to the bounding box precondition.
if (o.subregionBound.Contains(l.bound) || o.bound.Union(l.bound).IsFull()) &&
o.ContainsPoint(l.Vertex(0)) {
return false
}
return true
}
// Intersects reports whether the region contained by this loop intersects the region
// contained by the other loop.
func (l *Loop) Intersects(o *Loop) bool {
// Given two loops, A and B, A.Intersects(B) if and only if !A.Complement().Contains(B).
//
// This code is similar to Contains, but is optimized for the case
// where both loops enclose less than half of the sphere.
if !l.bound.Intersects(o.bound) {
return false
}
// Check whether there are any edge crossings, and also check the loop
// relationship at any shared vertices.
relation := &intersectsRelation{}
if hasCrossingRelation(l, o, relation) {
return true
}
if relation.foundSharedVertex {
return false
}
// Since there are no edge intersections or shared vertices, the loops
// intersect only if A contains B, B contains A, or the two loops contain
// each other's boundaries. These checks are usually cheap because of the
// bounding box preconditions. Note that neither loop is empty (because of
// the bounding box check above), so it is safe to access vertex(0).
// Check whether A contains B, or A and B contain each other's boundaries.
// (Note that A contains all the vertices of B in either case.)
if l.subregionBound.Contains(o.bound) || l.bound.Union(o.bound).IsFull() {
if l.ContainsPoint(o.Vertex(0)) {
return true
}
}
// Check whether B contains A.
if o.subregionBound.Contains(l.bound) {
if o.ContainsPoint(l.Vertex(0)) {
return true
}
}
return false
}
// Equal reports whether two loops have the same vertices in the same linear order
// (i.e., cyclic rotations are not allowed).
func (l *Loop) Equal(other *Loop) bool {
if len(l.vertices) != len(other.vertices) {
return false
}
for i, v := range l.vertices {
if v != other.Vertex(i) {
return false
}
}
return true
}
// BoundaryEqual reports whether the two loops have the same boundary. This is
// true if and only if the loops have the same vertices in the same cyclic order
// (i.e., the vertices may be cyclically rotated). The empty and full loops are
// considered to have different boundaries.
func (l *Loop) BoundaryEqual(o *Loop) bool {
if len(l.vertices) != len(o.vertices) {
return false
}
// Special case to handle empty or full loops. Since they have the same
// number of vertices, if one loop is empty/full then so is the other.
if l.isEmptyOrFull() {
return l.IsEmpty() == o.IsEmpty()
}
// Loop through the vertices to find the first of ours that matches the
// starting vertex of the other loop. Use that offset to then 'align' the
// vertices for comparison.
for offset, vertex := range l.vertices {
if vertex == o.Vertex(0) {
// There is at most one starting offset since loop vertices are unique.
for i := 0; i < len(l.vertices); i++ {
if l.Vertex(i+offset) != o.Vertex(i) {
return false
}
}
return true
}
}
return false
}
// compareBoundary returns +1 if this loop contains the boundary of the other loop,
// -1 if it excludes the boundary of the other, and 0 if the boundaries of the two
// loops cross. Shared edges are handled as follows:
//
// If XY is a shared edge, define Reversed(XY) to be true if XY
// appears in opposite directions in both loops.
// Then this loop contains XY if and only if Reversed(XY) == the other loop is a hole.
// (Intuitively, this checks whether this loop contains a vanishingly small region
// extending from the boundary of the other toward the interior of the polygon to
// which the other belongs.)
//
// This function is used for testing containment and intersection of
// multi-loop polygons. Note that this method is not symmetric, since the
// result depends on the direction of this loop but not on the direction of
// the other loop (in the absence of shared edges).
//
// This requires that neither loop is empty, and if other loop IsFull, then it must not
// be a hole.
func (l *Loop) compareBoundary(o *Loop) int {
// The bounds must intersect for containment or crossing.
if !l.bound.Intersects(o.bound) {
return -1
}
// Full loops are handled as though the loop surrounded the entire sphere.
if l.IsFull() {
return 1
}
if o.IsFull() {
return -1
}
// Check whether there are any edge crossings, and also check the loop
// relationship at any shared vertices.
relation := newCompareBoundaryRelation(o.IsHole())
if hasCrossingRelation(l, o, relation) {
return 0
}
if relation.foundSharedVertex {
if relation.containsEdge {
return 1
}
return -1
}
// There are no edge intersections or shared vertices, so we can check
// whether A contains an arbitrary vertex of B.
if l.ContainsPoint(o.Vertex(0)) {
return 1
}
return -1
}
// ContainsOrigin reports true if this loop contains s2.OriginPoint().
func (l *Loop) ContainsOrigin() bool {
return l.originInside
}
// ReferencePoint returns the reference point for this loop.
func (l *Loop) ReferencePoint() ReferencePoint {
return OriginReferencePoint(l.originInside)
}
// NumEdges returns the number of edges in this shape.
func (l *Loop) NumEdges() int {
if l.isEmptyOrFull() {
return 0
}
return len(l.vertices)
}
// Edge returns the endpoints for the given edge index.
func (l *Loop) Edge(i int) Edge {
return Edge{l.Vertex(i), l.Vertex(i + 1)}
}
// NumChains reports the number of contiguous edge chains in the Loop.
func (l *Loop) NumChains() int {
if l.IsEmpty() {
return 0
}
return 1
}
// Chain returns the i-th edge chain in the Shape.
func (l *Loop) Chain(chainID int) Chain {
return Chain{0, l.NumEdges()}
}
// ChainEdge returns the j-th edge of the i-th edge chain.
func (l *Loop) ChainEdge(chainID, offset int) Edge {
return Edge{l.Vertex(offset), l.Vertex(offset + 1)}
}
// ChainPosition returns a ChainPosition pair (i, j) such that edgeID is the
// j-th edge of the Loop.
func (l *Loop) ChainPosition(edgeID int) ChainPosition {
return ChainPosition{0, edgeID}
}
// Dimension returns the dimension of the geometry represented by this Loop.
func (l *Loop) Dimension() int { return 2 }
func (l *Loop) typeTag() typeTag { return typeTagNone }
func (l *Loop) privateInterface() {}
// IsEmpty reports true if this is the special empty loop that contains no points.
func (l *Loop) IsEmpty() bool {
return l.isEmptyOrFull() && !l.ContainsOrigin()
}
// IsFull reports true if this is the special full loop that contains all points.
func (l *Loop) IsFull() bool {
return l.isEmptyOrFull() && l.ContainsOrigin()
}
// isEmptyOrFull reports true if this loop is either the "empty" or "full" special loops.
func (l *Loop) isEmptyOrFull() bool {
return len(l.vertices) == 1
}
// Vertices returns the vertices in the loop.
func (l *Loop) Vertices() []Point {
return l.vertices
}
// RectBound returns a tight bounding rectangle. If the loop contains the point,
// the bound also contains it.
func (l *Loop) RectBound() Rect {
return l.bound
}
// CapBound returns a bounding cap that may have more padding than the corresponding
// RectBound. The bound is conservative such that if the loop contains a point P,
// the bound also contains it.
func (l *Loop) CapBound() Cap {
return l.bound.CapBound()
}
// Vertex returns the vertex for the given index. For convenience, the vertex indices
// wrap automatically for methods that do index math such as Edge.
// i.e., Vertex(NumEdges() + n) is the same as Vertex(n).
func (l *Loop) Vertex(i int) Point {
return l.vertices[i%len(l.vertices)]
}
// OrientedVertex returns the vertex in reverse order if the loop represents a polygon
// hole. For example, arguments 0, 1, 2 are mapped to vertices n-1, n-2, n-3, where
// n == len(vertices). This ensures that the interior of the polygon is always to
// the left of the vertex chain.
//
// This requires: 0 <= i < 2 * len(vertices)
func (l *Loop) OrientedVertex(i int) Point {
j := i - len(l.vertices)
if j < 0 {
j = i
}
if l.IsHole() {
j = len(l.vertices) - 1 - j
}
return l.Vertex(j)
}
// NumVertices returns the number of vertices in this loop.
func (l *Loop) NumVertices() int {
return len(l.vertices)
}
// bruteForceContainsPoint reports if the given point is contained by this loop.
// This method does not use the ShapeIndex, so it is only preferable below a certain
// size of loop.
func (l *Loop) bruteForceContainsPoint(p Point) bool {
origin := OriginPoint()
inside := l.originInside
crosser := NewChainEdgeCrosser(origin, p, l.Vertex(0))
for i := 1; i <= len(l.vertices); i++ { // add vertex 0 twice
inside = inside != crosser.EdgeOrVertexChainCrossing(l.Vertex(i))
}
return inside
}
// ContainsPoint returns true if the loop contains the point.
func (l *Loop) ContainsPoint(p Point) bool {
// Empty and full loops don't need a special case, but invalid loops with
// zero vertices do, so we might as well handle them all at once.
if len(l.vertices) < 3 {
return l.originInside
}
// For small loops, and during initial construction, it is faster to just
// check all the crossing.
const maxBruteForceVertices = 32
if len(l.vertices) < maxBruteForceVertices || l.index == nil {
return l.bruteForceContainsPoint(p)
}
// Otherwise, look up the point in the index.
it := l.index.Iterator()
if !it.LocatePoint(p) {
return false
}
return l.iteratorContainsPoint(it, p)
}
// ContainsCell reports whether the given Cell is contained by this Loop.
func (l *Loop) ContainsCell(target Cell) bool {
it := l.index.Iterator()
relation := it.LocateCellID(target.ID())
// If "target" is disjoint from all index cells, it is not contained.
// Similarly, if "target" is subdivided into one or more index cells then it
// is not contained, since index cells are subdivided only if they (nearly)
// intersect a sufficient number of edges. (But note that if "target" itself
// is an index cell then it may be contained, since it could be a cell with
// no edges in the loop interior.)
if relation != Indexed {
return false
}
// Otherwise check if any edges intersect "target".
if l.boundaryApproxIntersects(it, target) {
return false
}
// Otherwise check if the loop contains the center of "target".
return l.iteratorContainsPoint(it, target.Center())
}
// IntersectsCell reports whether this Loop intersects the given cell.
func (l *Loop) IntersectsCell(target Cell) bool {
it := l.index.Iterator()
relation := it.LocateCellID(target.ID())
// If target does not overlap any index cell, there is no intersection.
if relation == Disjoint {
return false
}
// If target is subdivided into one or more index cells, there is an
// intersection to within the ShapeIndex error bound (see Contains).
if relation == Subdivided {
return true
}
// If target is an index cell, there is an intersection because index cells
// are created only if they have at least one edge or they are entirely
// contained by the loop.
if it.CellID() == target.id {
return true
}
// Otherwise check if any edges intersect target.
if l.boundaryApproxIntersects(it, target) {
return true
}
// Otherwise check if the loop contains the center of target.
return l.iteratorContainsPoint(it, target.Center())
}
// CellUnionBound computes a covering of the Loop.
func (l *Loop) CellUnionBound() []CellID {
return l.CapBound().CellUnionBound()
}
// boundaryApproxIntersects reports if the loop's boundary intersects target.
// It may also return true when the loop boundary does not intersect target but
// some edge comes within the worst-case error tolerance.
//
// This requires that it.Locate(target) returned Indexed.
func (l *Loop) boundaryApproxIntersects(it *ShapeIndexIterator, target Cell) bool {
aClipped := it.IndexCell().findByShapeID(0)
// If there are no edges, there is no intersection.
if len(aClipped.edges) == 0 {
return false
}
// We can save some work if target is the index cell itself.
if it.CellID() == target.ID() {
return true
}
// Otherwise check whether any of the edges intersect target.
maxError := (faceClipErrorUVCoord + intersectsRectErrorUVDist)
bound := target.BoundUV().ExpandedByMargin(maxError)
for _, ai := range aClipped.edges {
v0, v1, ok := ClipToPaddedFace(l.Vertex(ai), l.Vertex(ai+1), target.Face(), maxError)
if ok && edgeIntersectsRect(v0, v1, bound) {
return true
}
}
return false
}
// iteratorContainsPoint reports if the iterator that is positioned at the ShapeIndexCell
// that may contain p, contains the point p.
func (l *Loop) iteratorContainsPoint(it *ShapeIndexIterator, p Point) bool {
// Test containment by drawing a line segment from the cell center to the
// given point and counting edge crossings.
aClipped := it.IndexCell().findByShapeID(0)
inside := aClipped.containsCenter
if len(aClipped.edges) > 0 {
center := it.Center()
crosser := NewEdgeCrosser(center, p)
aiPrev := -2
for _, ai := range aClipped.edges {
if ai != aiPrev+1 {
crosser.RestartAt(l.Vertex(ai))
}
aiPrev = ai
inside = inside != crosser.EdgeOrVertexChainCrossing(l.Vertex(ai+1))
}
}
return inside
}
// RegularLoop creates a loop with the given number of vertices, all
// located on a circle of the specified radius around the given center.
func RegularLoop(center Point, radius s1.Angle, numVertices int) *Loop {
return RegularLoopForFrame(getFrame(center), radius, numVertices)
}
// RegularLoopForFrame creates a loop centered around the z-axis of the given
// coordinate frame, with the first vertex in the direction of the positive x-axis.
func RegularLoopForFrame(frame matrix3x3, radius s1.Angle, numVertices int) *Loop {
return LoopFromPoints(regularPointsForFrame(frame, radius, numVertices))
}
// CanonicalFirstVertex returns a first index and a direction (either +1 or -1)
// such that the vertex sequence (first, first+dir, ..., first+(n-1)*dir) does
// not change when the loop vertex order is rotated or inverted. This allows the
// loop vertices to be traversed in a canonical order. The return values are
// chosen such that (first, ..., first+n*dir) are in the range [0, 2*n-1] as
// expected by the Vertex method.
func (l *Loop) CanonicalFirstVertex() (firstIdx, direction int) {
firstIdx = 0
n := len(l.vertices)
for i := 1; i < n; i++ {
if l.Vertex(i).Cmp(l.Vertex(firstIdx).Vector) == -1 {
firstIdx = i
}
}
// 0 <= firstIdx <= n-1, so (firstIdx+n*dir) <= 2*n-1.
if l.Vertex(firstIdx+1).Cmp(l.Vertex(firstIdx+n-1).Vector) == -1 {
return firstIdx, 1
}
// n <= firstIdx <= 2*n-1, so (firstIdx+n*dir) >= 0.
firstIdx += n
return firstIdx, -1
}
// TurningAngle returns the sum of the turning angles at each vertex. The return
// value is positive if the loop is counter-clockwise, negative if the loop is
// clockwise, and zero if the loop is a great circle. Degenerate and
// nearly-degenerate loops are handled consistently with Sign. So for example,
// if a loop has zero area (i.e., it is a very small CCW loop) then the turning
// angle will always be negative.
//
// This quantity is also called the "geodesic curvature" of the loop.
func (l *Loop) TurningAngle() float64 {
// For empty and full loops, we return the limit value as the loop area
// approaches 0 or 4*Pi respectively.
if l.isEmptyOrFull() {
if l.ContainsOrigin() {
return -2 * math.Pi
}
return 2 * math.Pi
}
// Don't crash even if the loop is not well-defined.
if len(l.vertices) < 3 {
return 0
}
// To ensure that we get the same result when the vertex order is rotated,
// and that the result is negated when the vertex order is reversed, we need
// to add up the individual turn angles in a consistent order. (In general,
// adding up a set of numbers in a different order can change the sum due to
// rounding errors.)
//
// Furthermore, if we just accumulate an ordinary sum then the worst-case
// error is quadratic in the number of vertices. (This can happen with
// spiral shapes, where the partial sum of the turning angles can be linear
// in the number of vertices.) To avoid this we use the Kahan summation
// algorithm (http://en.wikipedia.org/wiki/Kahan_summation_algorithm).
n := len(l.vertices)
i, dir := l.CanonicalFirstVertex()
sum := TurnAngle(l.Vertex((i+n-dir)%n), l.Vertex(i), l.Vertex((i+dir)%n))
compensation := s1.Angle(0)
for n-1 > 0 {
i += dir
angle := TurnAngle(l.Vertex(i-dir), l.Vertex(i), l.Vertex(i+dir))
oldSum := sum
angle += compensation
sum += angle
compensation = (oldSum - sum) + angle
n--
}
return float64(dir) * float64(sum+compensation)
}
// turningAngleMaxError return the maximum error in TurningAngle. The value is not
// constant; it depends on the loop.
func (l *Loop) turningAngleMaxError() float64 {
// The maximum error can be bounded as follows:
// 2.24 * dblEpsilon for RobustCrossProd(b, a)
// 2.24 * dblEpsilon for RobustCrossProd(c, b)
// 3.25 * dblEpsilon for Angle()
// 2.00 * dblEpsilon for each addition in the Kahan summation
// ------------------
// 9.73 * dblEpsilon
maxErrorPerVertex := 9.73 * dblEpsilon
return maxErrorPerVertex * float64(len(l.vertices))
}
// IsHole reports whether this loop represents a hole in its containing polygon.
func (l *Loop) IsHole() bool { return l.depth&1 != 0 }
// Sign returns -1 if this Loop represents a hole in its containing polygon, and +1 otherwise.
func (l *Loop) Sign() int {
if l.IsHole() {
return -1
}
return 1
}
// IsNormalized reports whether the loop area is at most 2*pi. Degenerate loops are
// handled consistently with Sign, i.e., if a loop can be
// expressed as the union of degenerate or nearly-degenerate CCW triangles,
// then it will always be considered normalized.
func (l *Loop) IsNormalized() bool {
// Optimization: if the longitude span is less than 180 degrees, then the
// loop covers less than half the sphere and is therefore normalized.
if l.bound.Lng.Length() < math.Pi {
return true
}
// We allow some error so that hemispheres are always considered normalized.
// TODO(roberts): This is no longer required by the Polygon implementation,
// so alternatively we could create the invariant that a loop is normalized
// if and only if its complement is not normalized.
return l.TurningAngle() >= -l.turningAngleMaxError()
}
// Normalize inverts the loop if necessary so that the area enclosed by the loop
// is at most 2*pi.
func (l *Loop) Normalize() {
if !l.IsNormalized() {
l.Invert()
}
}
// Invert reverses the order of the loop vertices, effectively complementing the
// region represented by the loop. For example, the loop ABCD (with edges
// AB, BC, CD, DA) becomes the loop DCBA (with edges DC, CB, BA, AD).
// Notice that the last edge is the same in both cases except that its
// direction has been reversed.
func (l *Loop) Invert() {
l.index.Reset()
if l.isEmptyOrFull() {
if l.IsFull() {
l.vertices[0] = emptyLoopPoint
} else {
l.vertices[0] = fullLoopPoint
}
} else {
// For non-special loops, reverse the slice of vertices.
for i := len(l.vertices)/2 - 1; i >= 0; i-- {
opp := len(l.vertices) - 1 - i
l.vertices[i], l.vertices[opp] = l.vertices[opp], l.vertices[i]
}
}
// originInside must be set correctly before building the ShapeIndex.
l.originInside = !l.originInside
if l.bound.Lat.Lo > -math.Pi/2 && l.bound.Lat.Hi < math.Pi/2 {
// The complement of this loop contains both poles.
l.bound = FullRect()
l.subregionBound = l.bound
} else {
l.initBound()
}
l.index.Add(l)
}
// findVertex returns the index of the vertex at the given Point in the range
// 1..numVertices, and a boolean indicating if a vertex was found.
func (l *Loop) findVertex(p Point) (index int, ok bool) {
const notFound = 0
if len(l.vertices) < 10 {
// Exhaustive search for loops below a small threshold.
for i := 1; i <= len(l.vertices); i++ {
if l.Vertex(i) == p {
return i, true
}
}
return notFound, false
}
it := l.index.Iterator()
if !it.LocatePoint(p) {
return notFound, false
}
aClipped := it.IndexCell().findByShapeID(0)
for i := aClipped.numEdges() - 1; i >= 0; i-- {
ai := aClipped.edges[i]
if l.Vertex(ai) == p {
if ai == 0 {
return len(l.vertices), true
}
return ai, true
}
if l.Vertex(ai+1) == p {
return ai + 1, true
}
}
return notFound, false
}
// ContainsNested reports whether the given loops is contained within this loop.
// This function does not test for edge intersections. The two loops must meet
// all of the Polygon requirements; for example this implies that their
// boundaries may not cross or have any shared edges (although they may have
// shared vertices).
func (l *Loop) ContainsNested(other *Loop) bool {
if !l.subregionBound.Contains(other.bound) {
return false
}
// Special cases to handle either loop being empty or full. Also bail out
// when B has no vertices to avoid heap overflow on the vertex(1) call
// below. (This method is called during polygon initialization before the
// client has an opportunity to call IsValid().)
if l.isEmptyOrFull() || other.NumVertices() < 2 {
return l.IsFull() || other.IsEmpty()
}
// We are given that A and B do not share any edges, and that either one
// loop contains the other or they do not intersect.
m, ok := l.findVertex(other.Vertex(1))
if !ok {
// Since other.vertex(1) is not shared, we can check whether A contains it.
return l.ContainsPoint(other.Vertex(1))
}
// Check whether the edge order around other.Vertex(1) is compatible with
// A containing B.
return WedgeContains(l.Vertex(m-1), l.Vertex(m), l.Vertex(m+1), other.Vertex(0), other.Vertex(2))
}
// surfaceIntegralFloat64 computes the oriented surface integral of some quantity f(x)
// over the loop interior, given a function f(A,B,C) that returns the
// corresponding integral over the spherical triangle ABC. Here "oriented
// surface integral" means:
//
// (1) f(A,B,C) must be the integral of f if ABC is counterclockwise,
// and the integral of -f if ABC is clockwise.
//
// (2) The result of this function is *either* the integral of f over the
// loop interior, or the integral of (-f) over the loop exterior.
//
// Note that there are at least two common situations where it easy to work
// around property (2) above:
//
// - If the integral of f over the entire sphere is zero, then it doesn't
// matter which case is returned because they are always equal.
//
// - If f is non-negative, then it is easy to detect when the integral over
// the loop exterior has been returned, and the integral over the loop
// interior can be obtained by adding the integral of f over the entire
// unit sphere (a constant) to the result.
//
// Any changes to this method may need corresponding changes to surfaceIntegralPoint as well.
func (l *Loop) surfaceIntegralFloat64(f func(a, b, c Point) float64) float64 {
// We sum f over a collection T of oriented triangles, possibly
// overlapping. Let the sign of a triangle be +1 if it is CCW and -1
// otherwise, and let the sign of a point x be the sum of the signs of the
// triangles containing x. Then the collection of triangles T is chosen
// such that either:
//
// (1) Each point in the loop interior has sign +1, and sign 0 otherwise; or
// (2) Each point in the loop exterior has sign -1, and sign 0 otherwise.
//
// The triangles basically consist of a fan from vertex 0 to every loop
// edge that does not include vertex 0. These triangles will always satisfy
// either (1) or (2). However, what makes this a bit tricky is that
// spherical edges become numerically unstable as their length approaches
// 180 degrees. Of course there is not much we can do if the loop itself
// contains such edges, but we would like to make sure that all the triangle
// edges under our control (i.e., the non-loop edges) are stable. For
// example, consider a loop around the equator consisting of four equally
// spaced points. This is a well-defined loop, but we cannot just split it
// into two triangles by connecting vertex 0 to vertex 2.
//
// We handle this type of situation by moving the origin of the triangle fan
// whenever we are about to create an unstable edge. We choose a new
// location for the origin such that all relevant edges are stable. We also
// create extra triangles with the appropriate orientation so that the sum
// of the triangle signs is still correct at every point.
// The maximum length of an edge for it to be considered numerically stable.
// The exact value is fairly arbitrary since it depends on the stability of
// the function f. The value below is quite conservative but could be
// reduced further if desired.
const maxLength = math.Pi - 1e-5
var sum float64
origin := l.Vertex(0)
for i := 1; i+1 < len(l.vertices); i++ {
// Let V_i be vertex(i), let O be the current origin, and let length(A,B)
// be the length of edge (A,B). At the start of each loop iteration, the
// "leading edge" of the triangle fan is (O,V_i), and we want to extend
// the triangle fan so that the leading edge is (O,V_i+1).
//
// Invariants:
// 1. length(O,V_i) < maxLength for all (i > 1).
// 2. Either O == V_0, or O is approximately perpendicular to V_0.
// 3. "sum" is the oriented integral of f over the area defined by
// (O, V_0, V_1, ..., V_i).
if l.Vertex(i+1).Angle(origin.Vector) > maxLength {
// We are about to create an unstable edge, so choose a new origin O'
// for the triangle fan.
oldOrigin := origin
if origin == l.Vertex(0) {
// The following point is well-separated from V_i and V_0 (and
// therefore V_i+1 as well).
origin = Point{l.Vertex(0).PointCross(l.Vertex(i)).Normalize()}
} else if l.Vertex(i).Angle(l.Vertex(0).Vector) < maxLength {
// All edges of the triangle (O, V_0, V_i) are stable, so we can
// revert to using V_0 as the origin.
origin = l.Vertex(0)
} else {
// (O, V_i+1) and (V_0, V_i) are antipodal pairs, and O and V_0 are
// perpendicular. Therefore V_0.CrossProd(O) is approximately
// perpendicular to all of {O, V_0, V_i, V_i+1}, and we can choose
// this point O' as the new origin.
origin = Point{l.Vertex(0).Cross(oldOrigin.Vector)}
// Advance the edge (V_0,O) to (V_0,O').
sum += f(l.Vertex(0), oldOrigin, origin)
}
// Advance the edge (O,V_i) to (O',V_i).
sum += f(oldOrigin, l.Vertex(i), origin)
}
// Advance the edge (O,V_i) to (O,V_i+1).
sum += f(origin, l.Vertex(i), l.Vertex(i+1))
}
// If the origin is not V_0, we need to sum one more triangle.
if origin != l.Vertex(0) {
// Advance the edge (O,V_n-1) to (O,V_0).
sum += f(origin, l.Vertex(len(l.vertices)-1), l.Vertex(0))
}
return sum
}
// surfaceIntegralPoint mirrors the surfaceIntegralFloat64 method but over Points;
// see that method for commentary. The C++ version uses a templated method.
// Any changes to this method may need corresponding changes to surfaceIntegralFloat64 as well.
func (l *Loop) surfaceIntegralPoint(f func(a, b, c Point) Point) Point {
const maxLength = math.Pi - 1e-5
var sum r3.Vector
origin := l.Vertex(0)
for i := 1; i+1 < len(l.vertices); i++ {
if l.Vertex(i+1).Angle(origin.Vector) > maxLength {
oldOrigin := origin
if origin == l.Vertex(0) {
origin = Point{l.Vertex(0).PointCross(l.Vertex(i)).Normalize()}
} else if l.Vertex(i).Angle(l.Vertex(0).Vector) < maxLength {
origin = l.Vertex(0)
} else {
origin = Point{l.Vertex(0).Cross(oldOrigin.Vector)}
sum = sum.Add(f(l.Vertex(0), oldOrigin, origin).Vector)
}
sum = sum.Add(f(oldOrigin, l.Vertex(i), origin).Vector)
}
sum = sum.Add(f(origin, l.Vertex(i), l.Vertex(i+1)).Vector)
}
if origin != l.Vertex(0) {
sum = sum.Add(f(origin, l.Vertex(len(l.vertices)-1), l.Vertex(0)).Vector)
}
return Point{sum}
}
// Area returns the area of the loop interior, i.e. the region on the left side of
// the loop. The return value is between 0 and 4*pi. (Note that the return
// value is not affected by whether this loop is a "hole" or a "shell".)
func (l *Loop) Area() float64 {
// It is surprisingly difficult to compute the area of a loop robustly. The
// main issues are (1) whether degenerate loops are considered to be CCW or
// not (i.e., whether their area is close to 0 or 4*pi), and (2) computing
// the areas of small loops with good relative accuracy.
//
// With respect to degeneracies, we would like Area to be consistent
// with ContainsPoint in that loops that contain many points
// should have large areas, and loops that contain few points should have
// small areas. For example, if a degenerate triangle is considered CCW
// according to s2predicates Sign, then it will contain very few points and
// its area should be approximately zero. On the other hand if it is
// considered clockwise, then it will contain virtually all points and so
// its area should be approximately 4*pi.
//
// More precisely, let U be the set of Points for which IsUnitLength
// is true, let P(U) be the projection of those points onto the mathematical
// unit sphere, and let V(P(U)) be the Voronoi diagram of the projected
// points. Then for every loop x, we would like Area to approximately
// equal the sum of the areas of the Voronoi regions of the points p for
// which x.ContainsPoint(p) is true.
//
// The second issue is that we want to compute the area of small loops
// accurately. This requires having good relative precision rather than
// good absolute precision. For example, if the area of a loop is 1e-12 and
// the error is 1e-15, then the area only has 3 digits of accuracy. (For
// reference, 1e-12 is about 40 square meters on the surface of the earth.)
// We would like to have good relative accuracy even for small loops.
//
// To achieve these goals, we combine two different methods of computing the
// area. This first method is based on the Gauss-Bonnet theorem, which says
// that the area enclosed by the loop equals 2*pi minus the total geodesic
// curvature of the loop (i.e., the sum of the "turning angles" at all the
// loop vertices). The big advantage of this method is that as long as we
// use Sign to compute the turning angle at each vertex, then
// degeneracies are always handled correctly. In other words, if a
// degenerate loop is CCW according to the symbolic perturbations used by
// Sign, then its turning angle will be approximately 2*pi.
//
// The disadvantage of the Gauss-Bonnet method is that its absolute error is
// about 2e-15 times the number of vertices (see turningAngleMaxError).
// So, it cannot compute the area of small loops accurately.
//
// The second method is based on splitting the loop into triangles and
// summing the area of each triangle. To avoid the difficulty and expense
// of decomposing the loop into a union of non-overlapping triangles,
// instead we compute a signed sum over triangles that may overlap (see the
// comments for surfaceIntegral). The advantage of this method
// is that the area of each triangle can be computed with much better
// relative accuracy (using l'Huilier's theorem). The disadvantage is that
// the result is a signed area: CCW loops may yield a small positive value,
// while CW loops may yield a small negative value (which is converted to a
// positive area by adding 4*pi). This means that small errors in computing
// the signed area may translate into a very large error in the result (if
// the sign of the sum is incorrect).
//
// So, our strategy is to combine these two methods as follows. First we
// compute the area using the "signed sum over triangles" approach (since it
// is generally more accurate). We also estimate the maximum error in this
// result. If the signed area is too close to zero (i.e., zero is within
// the error bounds), then we double-check the sign of the result using the
// Gauss-Bonnet method. (In fact we just call IsNormalized, which is
// based on this method.) If the two methods disagree, we return either 0
// or 4*pi based on the result of IsNormalized. Otherwise we return the
// area that we computed originally.
if l.isEmptyOrFull() {
if l.ContainsOrigin() {
return 4 * math.Pi
}
return 0
}
area := l.surfaceIntegralFloat64(SignedArea)
// TODO(roberts): This error estimate is very approximate. There are two
// issues: (1) SignedArea needs some improvements to ensure that its error
// is actually never higher than GirardArea, and (2) although the number of
// triangles in the sum is typically N-2, in theory it could be as high as
// 2*N for pathological inputs. But in other respects this error bound is
// very conservative since it assumes that the maximum error is achieved on
// every triangle.
maxError := l.turningAngleMaxError()
// The signed area should be between approximately -4*pi and 4*pi.
if area < 0 {
// We have computed the negative of the area of the loop exterior.
area += 4 * math.Pi
}
if area > 4*math.Pi {
area = 4 * math.Pi
}
if area < 0 {
area = 0
}
// If the area is close enough to zero or 4*pi so that the loop orientation
// is ambiguous, then we compute the loop orientation explicitly.
if area < maxError && !l.IsNormalized() {
return 4 * math.Pi
} else if area > (4*math.Pi-maxError) && l.IsNormalized() {
return 0
}
return area
}
// Centroid returns the true centroid of the loop multiplied by the area of the
// loop. The result is not unit length, so you may want to normalize it. Also
// note that in general, the centroid may not be contained by the loop.
//
// We prescale by the loop area for two reasons: (1) it is cheaper to
// compute this way, and (2) it makes it easier to compute the centroid of
// more complicated shapes (by splitting them into disjoint regions and
// adding their centroids).
//
// Note that the return value is not affected by whether this loop is a
// "hole" or a "shell".
func (l *Loop) Centroid() Point {
// surfaceIntegralPoint() returns either the integral of position over loop
// interior, or the negative of the integral of position over the loop
// exterior. But these two values are the same (!), because the integral of
// position over the entire sphere is (0, 0, 0).
return l.surfaceIntegralPoint(TrueCentroid)
}
// Encode encodes the Loop.
func (l Loop) Encode(w io.Writer) error {
e := &encoder{w: w}
l.encode(e)
return e.err
}
func (l Loop) encode(e *encoder) {
e.writeInt8(encodingVersion)
e.writeUint32(uint32(len(l.vertices)))
for _, v := range l.vertices {
e.writeFloat64(v.X)
e.writeFloat64(v.Y)
e.writeFloat64(v.Z)
}
e.writeBool(l.originInside)
e.writeInt32(int32(l.depth))
// Encode the bound.
l.bound.encode(e)
}
// Decode decodes a loop.
func (l *Loop) Decode(r io.Reader) error {
*l = Loop{}
d := &decoder{r: asByteReader(r)}
l.decode(d)
return d.err
}
func (l *Loop) decode(d *decoder) {
version := int8(d.readUint8())
if d.err != nil {
return
}
if version != encodingVersion {
d.err = fmt.Errorf("cannot decode version %d", version)
return
}
// Empty loops are explicitly allowed here: a newly created loop has zero vertices
// and such loops encode and decode properly.
nvertices := d.readUint32()
if nvertices > maxEncodedVertices {
if d.err == nil {
d.err = fmt.Errorf("too many vertices (%d; max is %d)", nvertices, maxEncodedVertices)
}
return
}
l.vertices = make([]Point, nvertices)
for i := range l.vertices {
l.vertices[i].X = d.readFloat64()
l.vertices[i].Y = d.readFloat64()
l.vertices[i].Z = d.readFloat64()
}
l.originInside = d.readBool()
l.depth = int(d.readUint32())
l.bound.decode(d)
l.subregionBound = ExpandForSubregions(l.bound)
l.index = NewShapeIndex()
l.index.Add(l)
}
// Bitmasks to read from properties.
const (
originInside = 1 << iota
boundEncoded
)
func (l *Loop) xyzFaceSiTiVertices() []xyzFaceSiTi {
ret := make([]xyzFaceSiTi, len(l.vertices))
for i, v := range l.vertices {
ret[i].xyz = v
ret[i].face, ret[i].si, ret[i].ti, ret[i].level = xyzToFaceSiTi(v)
}
return ret
}
func (l *Loop) encodeCompressed(e *encoder, snapLevel int, vertices []xyzFaceSiTi) {
if len(l.vertices) != len(vertices) {
panic("encodeCompressed: vertices must be the same length as l.vertices")
}
if len(vertices) > maxEncodedVertices {
if e.err == nil {
e.err = fmt.Errorf("too many vertices (%d; max is %d)", len(vertices), maxEncodedVertices)
}
return
}
e.writeUvarint(uint64(len(vertices)))
encodePointsCompressed(e, vertices, snapLevel)
props := l.compressedEncodingProperties()
e.writeUvarint(props)
e.writeUvarint(uint64(l.depth))
if props&boundEncoded != 0 {
l.bound.encode(e)
}
}
func (l *Loop) compressedEncodingProperties() uint64 {
var properties uint64
if l.originInside {
properties |= originInside
}
// Write whether there is a bound so we can change the threshold later.
// Recomputing the bound multiplies the decode time taken per vertex
// by a factor of about 3.5. Without recomputing the bound, decode
// takes approximately 125 ns / vertex. A loop with 63 vertices
// encoded without the bound will take ~30us to decode, which is
// acceptable. At ~3.5 bytes / vertex without the bound, adding
// the bound will increase the size by <15%, which is also acceptable.
const minVerticesForBound = 64
if len(l.vertices) >= minVerticesForBound {
properties |= boundEncoded
}
return properties
}
func (l *Loop) decodeCompressed(d *decoder, snapLevel int) {
nvertices := d.readUvarint()
if d.err != nil {
return
}
if nvertices > maxEncodedVertices {
d.err = fmt.Errorf("too many vertices (%d; max is %d)", nvertices, maxEncodedVertices)
return
}
l.vertices = make([]Point, nvertices)
decodePointsCompressed(d, snapLevel, l.vertices)
properties := d.readUvarint()
// Make sure values are valid before using.
if d.err != nil {
return
}
l.originInside = (properties & originInside) != 0
l.depth = int(d.readUvarint())
if (properties & boundEncoded) != 0 {
l.bound.decode(d)
if d.err != nil {
return
}
l.subregionBound = ExpandForSubregions(l.bound)
} else {
l.initBound()
}
l.index = NewShapeIndex()
l.index.Add(l)
}
// crossingTarget is an enum representing the possible crossing target cases for relations.
type crossingTarget int
const (
crossingTargetDontCare crossingTarget = iota
crossingTargetDontCross
crossingTargetCross
)
// loopRelation defines the interface for checking a type of relationship between two loops.
// Some examples of relations are Contains, Intersects, or CompareBoundary.
type loopRelation interface {
// Optionally, aCrossingTarget and bCrossingTarget can specify an early-exit
// condition for the loop relation. If any point P is found such that
//
// A.ContainsPoint(P) == aCrossingTarget() &&
// B.ContainsPoint(P) == bCrossingTarget()
//
// then the loop relation is assumed to be the same as if a pair of crossing
// edges were found. For example, the ContainsPoint relation has
//
// aCrossingTarget() == crossingTargetDontCross
// bCrossingTarget() == crossingTargetCross
//
// because if A.ContainsPoint(P) == false and B.ContainsPoint(P) == true
// for any point P, then it is equivalent to finding an edge crossing (i.e.,
// since Contains returns false in both cases).
//
// Loop relations that do not have an early-exit condition of this form
// should return crossingTargetDontCare for both crossing targets.
// aCrossingTarget reports whether loop A crosses the target point with
// the given relation type.
aCrossingTarget() crossingTarget
// bCrossingTarget reports whether loop B crosses the target point with
// the given relation type.
bCrossingTarget() crossingTarget
// wedgesCross reports if a shared vertex ab1 and the two associated wedges
// (a0, ab1, b2) and (b0, ab1, b2) are equivalent to an edge crossing.
// The loop relation is also allowed to maintain its own internal state, and
// can return true if it observes any sequence of wedges that are equivalent
// to an edge crossing.
wedgesCross(a0, ab1, a2, b0, b2 Point) bool
}
// loopCrosser is a helper type for determining whether two loops cross.
// It is instantiated twice for each pair of loops to be tested, once for the
// pair (A,B) and once for the pair (B,A), in order to be able to process
// edges in either loop nesting order.
type loopCrosser struct {
a, b *Loop
relation loopRelation
swapped bool
aCrossingTarget crossingTarget
bCrossingTarget crossingTarget
// state maintained by startEdge and edgeCrossesCell.
crosser *EdgeCrosser
aj, bjPrev int
// temporary data declared here to avoid repeated memory allocations.
bQuery *CrossingEdgeQuery
bCells []*ShapeIndexCell
}
// newLoopCrosser creates a loopCrosser from the given values. If swapped is true,
// the loops A and B have been swapped. This affects how arguments are passed to
// the given loop relation, since for example A.Contains(B) is not the same as
// B.Contains(A).
func newLoopCrosser(a, b *Loop, relation loopRelation, swapped bool) *loopCrosser {
l := &loopCrosser{
a: a,
b: b,
relation: relation,
swapped: swapped,
aCrossingTarget: relation.aCrossingTarget(),
bCrossingTarget: relation.bCrossingTarget(),
bQuery: NewCrossingEdgeQuery(b.index),
}
if swapped {
l.aCrossingTarget, l.bCrossingTarget = l.bCrossingTarget, l.aCrossingTarget
}
return l
}
// startEdge sets the crossers state for checking the given edge of loop A.
func (l *loopCrosser) startEdge(aj int) {
l.crosser = NewEdgeCrosser(l.a.Vertex(aj), l.a.Vertex(aj+1))
l.aj = aj
l.bjPrev = -2
}
// edgeCrossesCell reports whether the current edge of loop A has any crossings with
// edges of the index cell of loop B.
func (l *loopCrosser) edgeCrossesCell(bClipped *clippedShape) bool {
// Test the current edge of A against all edges of bClipped
bNumEdges := bClipped.numEdges()
for j := 0; j < bNumEdges; j++ {
bj := bClipped.edges[j]
if bj != l.bjPrev+1 {
l.crosser.RestartAt(l.b.Vertex(bj))
}
l.bjPrev = bj
if crossing := l.crosser.ChainCrossingSign(l.b.Vertex(bj + 1)); crossing == DoNotCross {
continue
} else if crossing == Cross {
return true
}
// We only need to check each shared vertex once, so we only
// consider the case where l.aVertex(l.aj+1) == l.b.Vertex(bj+1).
if l.a.Vertex(l.aj+1) == l.b.Vertex(bj+1) {
if l.swapped {
if l.relation.wedgesCross(l.b.Vertex(bj), l.b.Vertex(bj+1), l.b.Vertex(bj+2), l.a.Vertex(l.aj), l.a.Vertex(l.aj+2)) {
return true
}
} else {
if l.relation.wedgesCross(l.a.Vertex(l.aj), l.a.Vertex(l.aj+1), l.a.Vertex(l.aj+2), l.b.Vertex(bj), l.b.Vertex(bj+2)) {
return true
}
}
}
}
return false
}
// cellCrossesCell reports whether there are any edge crossings or wedge crossings
// within the two given cells.
func (l *loopCrosser) cellCrossesCell(aClipped, bClipped *clippedShape) bool {
// Test all edges of aClipped against all edges of bClipped.
for _, edge := range aClipped.edges {
l.startEdge(edge)
if l.edgeCrossesCell(bClipped) {
return true
}
}
return false
}
// cellCrossesAnySubcell reports whether given an index cell of A, if there are any
// edge or wedge crossings with any index cell of B contained within bID.
func (l *loopCrosser) cellCrossesAnySubcell(aClipped *clippedShape, bID CellID) bool {
// Test all edges of aClipped against all edges of B. The relevant B
// edges are guaranteed to be children of bID, which lets us find the
// correct index cells more efficiently.
bRoot := PaddedCellFromCellID(bID, 0)
for _, aj := range aClipped.edges {
// Use an CrossingEdgeQuery starting at bRoot to find the index cells
// of B that might contain crossing edges.
l.bCells = l.bQuery.getCells(l.a.Vertex(aj), l.a.Vertex(aj+1), bRoot)
if len(l.bCells) == 0 {
continue
}
l.startEdge(aj)
for c := 0; c < len(l.bCells); c++ {
if l.edgeCrossesCell(l.bCells[c].shapes[0]) {
return true
}
}
}
return false
}
// hasCrossing reports whether given two iterators positioned such that
// ai.cellID().ContainsCellID(bi.cellID()), there is an edge or wedge crossing
// anywhere within ai.cellID(). This function advances bi only past ai.cellID().
func (l *loopCrosser) hasCrossing(ai, bi *rangeIterator) bool {
// If ai.CellID() intersects many edges of B, then it is faster to use
// CrossingEdgeQuery to narrow down the candidates. But if it intersects
// only a few edges, it is faster to check all the crossings directly.
// We handle this by advancing bi and keeping track of how many edges we
// would need to test.
const edgeQueryMinEdges = 20 // Tuned from benchmarks.
var totalEdges int
l.bCells = nil
for {
if n := bi.it.IndexCell().shapes[0].numEdges(); n > 0 {
totalEdges += n
if totalEdges >= edgeQueryMinEdges {
// There are too many edges to test them directly, so use CrossingEdgeQuery.
if l.cellCrossesAnySubcell(ai.it.IndexCell().shapes[0], ai.cellID()) {
return true
}
bi.seekBeyond(ai)
return false
}
l.bCells = append(l.bCells, bi.indexCell())
}
bi.next()
if bi.cellID() > ai.rangeMax {
break
}
}
// Test all the edge crossings directly.
for _, c := range l.bCells {
if l.cellCrossesCell(ai.it.IndexCell().shapes[0], c.shapes[0]) {
return true
}
}
return false
}
// containsCenterMatches reports if the clippedShapes containsCenter boolean corresponds
// to the crossing target type given. (This is to work around C++ allowing false == 0,
// true == 1 type implicit conversions and comparisons)
func containsCenterMatches(a *clippedShape, target crossingTarget) bool {
return (!a.containsCenter && target == crossingTargetDontCross) ||
(a.containsCenter && target == crossingTargetCross)
}
// hasCrossingRelation reports whether given two iterators positioned such that
// ai.cellID().ContainsCellID(bi.cellID()), there is a crossing relationship
// anywhere within ai.cellID(). Specifically, this method returns true if there
// is an edge crossing, a wedge crossing, or a point P that matches both relations
// crossing targets. This function advances both iterators past ai.cellID.
func (l *loopCrosser) hasCrossingRelation(ai, bi *rangeIterator) bool {
aClipped := ai.it.IndexCell().shapes[0]
if aClipped.numEdges() != 0 {
// The current cell of A has at least one edge, so check for crossings.
if l.hasCrossing(ai, bi) {
return true
}
ai.next()
return false
}
if containsCenterMatches(aClipped, l.aCrossingTarget) {
// The crossing target for A is not satisfied, so we skip over these cells of B.
bi.seekBeyond(ai)
ai.next()
return false
}
// All points within ai.cellID() satisfy the crossing target for A, so it's
// worth iterating through the cells of B to see whether any cell
// centers also satisfy the crossing target for B.
for bi.cellID() <= ai.rangeMax {
bClipped := bi.it.IndexCell().shapes[0]
if containsCenterMatches(bClipped, l.bCrossingTarget) {
return true
}
bi.next()
}
ai.next()
return false
}
// hasCrossingRelation checks all edges of loop A for intersection against all edges
// of loop B and reports if there are any that satisfy the given relation. If there
// is any shared vertex, the wedges centered at this vertex are sent to the given
// relation to be tested.
//
// If the two loop boundaries cross, this method is guaranteed to return
// true. It also returns true in certain cases if the loop relationship is
// equivalent to crossing. For example, if the relation is Contains and a
// point P is found such that B contains P but A does not contain P, this
// method will return true to indicate that the result is the same as though
// a pair of crossing edges were found (since Contains returns false in
// both cases).
//
// See Contains, Intersects and CompareBoundary for the three uses of this function.
func hasCrossingRelation(a, b *Loop, relation loopRelation) bool {
// We look for CellID ranges where the indexes of A and B overlap, and
// then test those edges for crossings.
ai := newRangeIterator(a.index)
bi := newRangeIterator(b.index)
ab := newLoopCrosser(a, b, relation, false) // Tests edges of A against B
ba := newLoopCrosser(b, a, relation, true) // Tests edges of B against A
for !ai.done() || !bi.done() {
if ai.rangeMax < bi.rangeMin {
// The A and B cells don't overlap, and A precedes B.
ai.seekTo(bi)
} else if bi.rangeMax < ai.rangeMin {
// The A and B cells don't overlap, and B precedes A.
bi.seekTo(ai)
} else {
// One cell contains the other. Determine which cell is larger.
abRelation := int64(ai.it.CellID().lsb() - bi.it.CellID().lsb())
if abRelation > 0 {
// A's index cell is larger.
if ab.hasCrossingRelation(ai, bi) {
return true
}
} else if abRelation < 0 {
// B's index cell is larger.
if ba.hasCrossingRelation(bi, ai) {
return true
}
} else {
// The A and B cells are the same. Since the two cells
// have the same center point P, check whether P satisfies
// the crossing targets.
aClipped := ai.it.IndexCell().shapes[0]
bClipped := bi.it.IndexCell().shapes[0]
if containsCenterMatches(aClipped, ab.aCrossingTarget) &&
containsCenterMatches(bClipped, ab.bCrossingTarget) {
return true
}
// Otherwise test all the edge crossings directly.
if aClipped.numEdges() > 0 && bClipped.numEdges() > 0 && ab.cellCrossesCell(aClipped, bClipped) {
return true
}
ai.next()
bi.next()
}
}
}
return false
}
// containsRelation implements loopRelation for a contains operation. If
// A.ContainsPoint(P) == false && B.ContainsPoint(P) == true, it is equivalent
// to having an edge crossing (i.e., Contains returns false).
type containsRelation struct {
foundSharedVertex bool
}
func (c *containsRelation) aCrossingTarget() crossingTarget { return crossingTargetDontCross }
func (c *containsRelation) bCrossingTarget() crossingTarget { return crossingTargetCross }
func (c *containsRelation) wedgesCross(a0, ab1, a2, b0, b2 Point) bool {
c.foundSharedVertex = true
return !WedgeContains(a0, ab1, a2, b0, b2)
}
// intersectsRelation implements loopRelation for an intersects operation. Given
// two loops, A and B, if A.ContainsPoint(P) == true && B.ContainsPoint(P) == true,
// it is equivalent to having an edge crossing (i.e., Intersects returns true).
type intersectsRelation struct {
foundSharedVertex bool
}
func (i *intersectsRelation) aCrossingTarget() crossingTarget { return crossingTargetCross }
func (i *intersectsRelation) bCrossingTarget() crossingTarget { return crossingTargetCross }
func (i *intersectsRelation) wedgesCross(a0, ab1, a2, b0, b2 Point) bool {
i.foundSharedVertex = true
return WedgeIntersects(a0, ab1, a2, b0, b2)
}
// compareBoundaryRelation implements loopRelation for comparing boundaries.
//
// The compare boundary relation does not have a useful early-exit condition,
// so we return crossingTargetDontCare for both crossing targets.
//
// Aside: A possible early exit condition could be based on the following.
// If A contains a point of both B and ~B, then A intersects Boundary(B).
// If ~A contains a point of both B and ~B, then ~A intersects Boundary(B).
// So if the intersections of {A, ~A} with {B, ~B} are all non-empty,
// the return value is 0, i.e., Boundary(A) intersects Boundary(B).
// Unfortunately it isn't worth detecting this situation because by the
// time we have seen a point in all four intersection regions, we are also
// guaranteed to have seen at least one pair of crossing edges.
type compareBoundaryRelation struct {
reverse bool // True if the other loop should be reversed.
foundSharedVertex bool // True if any wedge was processed.
containsEdge bool // True if any edge of the other loop is contained by this loop.
excludesEdge bool // True if any edge of the other loop is excluded by this loop.
}
func newCompareBoundaryRelation(reverse bool) *compareBoundaryRelation {
return &compareBoundaryRelation{reverse: reverse}
}
func (c *compareBoundaryRelation) aCrossingTarget() crossingTarget { return crossingTargetDontCare }
func (c *compareBoundaryRelation) bCrossingTarget() crossingTarget { return crossingTargetDontCare }
func (c *compareBoundaryRelation) wedgesCross(a0, ab1, a2, b0, b2 Point) bool {
// Because we don't care about the interior of the other, only its boundary,
// it is sufficient to check whether this one contains the semiwedge (ab1, b2).
c.foundSharedVertex = true
if wedgeContainsSemiwedge(a0, ab1, a2, b2, c.reverse) {
c.containsEdge = true
} else {
c.excludesEdge = true
}
return c.containsEdge && c.excludesEdge
}
// wedgeContainsSemiwedge reports whether the wedge (a0, ab1, a2) contains the
// "semiwedge" defined as any non-empty open set of rays immediately CCW from
// the edge (ab1, b2). If reverse is true, then substitute clockwise for CCW;
// this simulates what would happen if the direction of the other loop was reversed.
func wedgeContainsSemiwedge(a0, ab1, a2, b2 Point, reverse bool) bool {
if b2 == a0 || b2 == a2 {
// We have a shared or reversed edge.
return (b2 == a0) == reverse
}
return OrderedCCW(a0, a2, b2, ab1)
}
// containsNonCrossingBoundary reports whether given two loops whose boundaries
// do not cross (see compareBoundary), if this loop contains the boundary of the
// other loop. If reverse is true, the boundary of the other loop is reversed
// first (which only affects the result when there are shared edges). This method
// is cheaper than compareBoundary because it does not test for edge intersections.
//
// This function requires that neither loop is empty, and that if the other is full,
// then reverse == false.
func (l *Loop) containsNonCrossingBoundary(other *Loop, reverseOther bool) bool {
// The bounds must intersect for containment.
if !l.bound.Intersects(other.bound) {
return false
}
// Full loops are handled as though the loop surrounded the entire sphere.
if l.IsFull() {
return true
}
if other.IsFull() {
return false
}
m, ok := l.findVertex(other.Vertex(0))
if !ok {
// Since the other loops vertex 0 is not shared, we can check if this contains it.
return l.ContainsPoint(other.Vertex(0))
}
// Otherwise check whether the edge (b0, b1) is contained by this loop.
return wedgeContainsSemiwedge(l.Vertex(m-1), l.Vertex(m), l.Vertex(m+1),
other.Vertex(1), reverseOther)
}
// TODO(roberts): Differences from the C++ version:
// DistanceToPoint
// DistanceToBoundary
// Project
// ProjectToBoundary
// BoundaryApproxEqual
// BoundaryNear
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