gotosocial/vendor/github.com/golang/geo/s2/edge_crosser.go

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// Copyright 2017 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 (
"math"
)
// EdgeCrosser allows edges to be efficiently tested for intersection with a
// given fixed edge AB. It is especially efficient when testing for
// intersection with an edge chain connecting vertices v0, v1, v2, ...
//
// Example usage:
//
// func CountIntersections(a, b Point, edges []Edge) int {
// count := 0
// crosser := NewEdgeCrosser(a, b)
// for _, edge := range edges {
// if crosser.CrossingSign(&edge.First, &edge.Second) != DoNotCross {
// count++
// }
// }
// return count
// }
//
type EdgeCrosser struct {
a Point
b Point
aXb Point
// To reduce the number of calls to expensiveSign, we compute an
// outward-facing tangent at A and B if necessary. If the plane
// perpendicular to one of these tangents separates AB from CD (i.e., one
// edge on each side) then there is no intersection.
aTangent Point // Outward-facing tangent at A.
bTangent Point // Outward-facing tangent at B.
// The fields below are updated for each vertex in the chain.
c Point // Previous vertex in the vertex chain.
acb Direction // The orientation of triangle ACB.
}
// NewEdgeCrosser returns an EdgeCrosser with the fixed edge AB.
func NewEdgeCrosser(a, b Point) *EdgeCrosser {
norm := a.PointCross(b)
return &EdgeCrosser{
a: a,
b: b,
aXb: Point{a.Cross(b.Vector)},
aTangent: Point{a.Cross(norm.Vector)},
bTangent: Point{norm.Cross(b.Vector)},
}
}
// CrossingSign reports whether the edge AB intersects the edge CD. If any two
// vertices from different edges are the same, returns MaybeCross. If either edge
// is degenerate (A == B or C == D), returns either DoNotCross or MaybeCross.
//
// Properties of CrossingSign:
//
// (1) CrossingSign(b,a,c,d) == CrossingSign(a,b,c,d)
// (2) CrossingSign(c,d,a,b) == CrossingSign(a,b,c,d)
// (3) CrossingSign(a,b,c,d) == MaybeCross if a==c, a==d, b==c, b==d
// (3) CrossingSign(a,b,c,d) == DoNotCross or MaybeCross if a==b or c==d
//
// Note that if you want to check an edge against a chain of other edges,
// it is slightly more efficient to use the single-argument version
// ChainCrossingSign below.
func (e *EdgeCrosser) CrossingSign(c, d Point) Crossing {
if c != e.c {
e.RestartAt(c)
}
return e.ChainCrossingSign(d)
}
// EdgeOrVertexCrossing reports whether if CrossingSign(c, d) > 0, or AB and
// CD share a vertex and VertexCrossing(a, b, c, d) is true.
//
// This method extends the concept of a "crossing" to the case where AB
// and CD have a vertex in common. The two edges may or may not cross,
// according to the rules defined in VertexCrossing above. The rules
// are designed so that point containment tests can be implemented simply
// by counting edge crossings. Similarly, determining whether one edge
// chain crosses another edge chain can be implemented by counting.
func (e *EdgeCrosser) EdgeOrVertexCrossing(c, d Point) bool {
if c != e.c {
e.RestartAt(c)
}
return e.EdgeOrVertexChainCrossing(d)
}
// NewChainEdgeCrosser is a convenience constructor that uses AB as the fixed edge,
// and C as the first vertex of the vertex chain (equivalent to calling RestartAt(c)).
//
// You don't need to use this or any of the chain functions unless you're trying to
// squeeze out every last drop of performance. Essentially all you are saving is a test
// whether the first vertex of the current edge is the same as the second vertex of the
// previous edge.
func NewChainEdgeCrosser(a, b, c Point) *EdgeCrosser {
e := NewEdgeCrosser(a, b)
e.RestartAt(c)
return e
}
// RestartAt sets the current point of the edge crosser to be c.
// Call this method when your chain 'jumps' to a new place.
// The argument must point to a value that persists until the next call.
func (e *EdgeCrosser) RestartAt(c Point) {
e.c = c
e.acb = -triageSign(e.a, e.b, e.c)
}
// ChainCrossingSign is like CrossingSign, but uses the last vertex passed to one of
// the crossing methods (or RestartAt) as the first vertex of the current edge.
func (e *EdgeCrosser) ChainCrossingSign(d Point) Crossing {
// For there to be an edge crossing, the triangles ACB, CBD, BDA, DAC must
// all be oriented the same way (CW or CCW). We keep the orientation of ACB
// as part of our state. When each new point D arrives, we compute the
// orientation of BDA and check whether it matches ACB. This checks whether
// the points C and D are on opposite sides of the great circle through AB.
// Recall that triageSign is invariant with respect to rotating its
// arguments, i.e. ABD has the same orientation as BDA.
bda := triageSign(e.a, e.b, d)
if e.acb == -bda && bda != Indeterminate {
// The most common case -- triangles have opposite orientations. Save the
// current vertex D as the next vertex C, and also save the orientation of
// the new triangle ACB (which is opposite to the current triangle BDA).
e.c = d
e.acb = -bda
return DoNotCross
}
return e.crossingSign(d, bda)
}
// EdgeOrVertexChainCrossing is like EdgeOrVertexCrossing, but uses the last vertex
// passed to one of the crossing methods (or RestartAt) as the first vertex of the current edge.
func (e *EdgeCrosser) EdgeOrVertexChainCrossing(d Point) bool {
// We need to copy e.c since it is clobbered by ChainCrossingSign.
c := e.c
switch e.ChainCrossingSign(d) {
case DoNotCross:
return false
case Cross:
return true
}
return VertexCrossing(e.a, e.b, c, d)
}
// crossingSign handle the slow path of CrossingSign.
func (e *EdgeCrosser) crossingSign(d Point, bda Direction) Crossing {
// Compute the actual result, and then save the current vertex D as the next
// vertex C, and save the orientation of the next triangle ACB (which is
// opposite to the current triangle BDA).
defer func() {
e.c = d
e.acb = -bda
}()
// At this point, a very common situation is that A,B,C,D are four points on
// a line such that AB does not overlap CD. (For example, this happens when
// a line or curve is sampled finely, or when geometry is constructed by
// computing the union of S2CellIds.) Most of the time, we can determine
// that AB and CD do not intersect using the two outward-facing
// tangents at A and B (parallel to AB) and testing whether AB and CD are on
// opposite sides of the plane perpendicular to one of these tangents. This
// is moderately expensive but still much cheaper than expensiveSign.
// The error in RobustCrossProd is insignificant. The maximum error in
// the call to CrossProd (i.e., the maximum norm of the error vector) is
// (0.5 + 1/sqrt(3)) * dblEpsilon. The maximum error in each call to
// DotProd below is dblEpsilon. (There is also a small relative error
// term that is insignificant because we are comparing the result against a
// constant that is very close to zero.)
maxError := (1.5 + 1/math.Sqrt(3)) * dblEpsilon
if (e.c.Dot(e.aTangent.Vector) > maxError && d.Dot(e.aTangent.Vector) > maxError) || (e.c.Dot(e.bTangent.Vector) > maxError && d.Dot(e.bTangent.Vector) > maxError) {
return DoNotCross
}
// Otherwise, eliminate the cases where two vertices from different edges are
// equal. (These cases could be handled in the code below, but we would rather
// avoid calling ExpensiveSign if possible.)
if e.a == e.c || e.a == d || e.b == e.c || e.b == d {
return MaybeCross
}
// Eliminate the cases where an input edge is degenerate. (Note that in
// most cases, if CD is degenerate then this method is not even called
// because acb and bda have different signs.)
if e.a == e.b || e.c == d {
return DoNotCross
}
// Otherwise it's time to break out the big guns.
if e.acb == Indeterminate {
e.acb = -expensiveSign(e.a, e.b, e.c)
}
if bda == Indeterminate {
bda = expensiveSign(e.a, e.b, d)
}
if bda != e.acb {
return DoNotCross
}
cbd := -RobustSign(e.c, d, e.b)
if cbd != e.acb {
return DoNotCross
}
dac := RobustSign(e.c, d, e.a)
if dac != e.acb {
return DoNotCross
}
return Cross
}