mirror of
https://github.com/superseriousbusiness/gotosocial.git
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94e87610c4
* add back exif-terminator and use only for jpeg,png,webp * fix arguments passed to terminateExif() * pull in latest exif-terminator * fix test * update processed img --------- Co-authored-by: tobi <tobi.smethurst@protonmail.com>
397 lines
15 KiB
Go
397 lines
15 KiB
Go
// Copyright 2017 Google Inc. All rights reserved.
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//
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// Licensed under the Apache License, Version 2.0 (the "License");
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// you may not use this file except in compliance with the License.
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// You may obtain a copy of the License at
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//
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// http://www.apache.org/licenses/LICENSE-2.0
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//
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// Unless required by applicable law or agreed to in writing, software
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// distributed under the License is distributed on an "AS IS" BASIS,
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// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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// See the License for the specific language governing permissions and
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// limitations under the License.
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package s2
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import (
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"fmt"
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"math"
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"github.com/golang/geo/r3"
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"github.com/golang/geo/s1"
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)
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const (
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// intersectionError can be set somewhat arbitrarily, because the algorithm
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// uses more precision if necessary in order to achieve the specified error.
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// The only strict requirement is that intersectionError >= dblEpsilon
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// radians. However, using a larger error tolerance makes the algorithm more
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// efficient because it reduces the number of cases where exact arithmetic is
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// needed.
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intersectionError = s1.Angle(8 * dblError)
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// intersectionMergeRadius is used to ensure that intersection points that
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// are supposed to be coincident are merged back together into a single
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// vertex. This is required in order for various polygon operations (union,
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// intersection, etc) to work correctly. It is twice the intersection error
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// because two coincident intersection points might have errors in
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// opposite directions.
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intersectionMergeRadius = 2 * intersectionError
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)
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// A Crossing indicates how edges cross.
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type Crossing int
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const (
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// Cross means the edges cross.
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Cross Crossing = iota
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// MaybeCross means two vertices from different edges are the same.
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MaybeCross
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// DoNotCross means the edges do not cross.
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DoNotCross
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)
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func (c Crossing) String() string {
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switch c {
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case Cross:
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return "Cross"
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case MaybeCross:
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return "MaybeCross"
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case DoNotCross:
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return "DoNotCross"
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default:
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return fmt.Sprintf("(BAD CROSSING %d)", c)
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}
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}
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// CrossingSign reports whether the edge AB intersects the edge CD.
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// If AB crosses CD at a point that is interior to both edges, Cross is returned.
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// If any two vertices from different edges are the same it returns MaybeCross.
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// Otherwise it returns DoNotCross.
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// If either edge is degenerate (A == B or C == D), the return value is MaybeCross
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// if two vertices from different edges are the same and DoNotCross otherwise.
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//
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// Properties of CrossingSign:
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//
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// (1) CrossingSign(b,a,c,d) == CrossingSign(a,b,c,d)
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// (2) CrossingSign(c,d,a,b) == CrossingSign(a,b,c,d)
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// (3) CrossingSign(a,b,c,d) == MaybeCross if a==c, a==d, b==c, b==d
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// (3) CrossingSign(a,b,c,d) == DoNotCross or MaybeCross if a==b or c==d
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//
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// This method implements an exact, consistent perturbation model such
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// that no three points are ever considered to be collinear. This means
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// that even if you have 4 points A, B, C, D that lie exactly in a line
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// (say, around the equator), C and D will be treated as being slightly to
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// one side or the other of AB. This is done in a way such that the
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// results are always consistent (see RobustSign).
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func CrossingSign(a, b, c, d Point) Crossing {
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crosser := NewChainEdgeCrosser(a, b, c)
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return crosser.ChainCrossingSign(d)
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}
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// VertexCrossing reports whether two edges "cross" in such a way that point-in-polygon
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// containment tests can be implemented by counting the number of edge crossings.
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//
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// Given two edges AB and CD where at least two vertices are identical
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// (i.e. CrossingSign(a,b,c,d) == 0), the basic rule is that a "crossing"
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// occurs if AB is encountered after CD during a CCW sweep around the shared
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// vertex starting from a fixed reference point.
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//
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// Note that according to this rule, if AB crosses CD then in general CD
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// does not cross AB. However, this leads to the correct result when
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// counting polygon edge crossings. For example, suppose that A,B,C are
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// three consecutive vertices of a CCW polygon. If we now consider the edge
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// crossings of a segment BP as P sweeps around B, the crossing number
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// changes parity exactly when BP crosses BA or BC.
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//
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// Useful properties of VertexCrossing (VC):
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//
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// (1) VC(a,a,c,d) == VC(a,b,c,c) == false
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// (2) VC(a,b,a,b) == VC(a,b,b,a) == true
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// (3) VC(a,b,c,d) == VC(a,b,d,c) == VC(b,a,c,d) == VC(b,a,d,c)
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// (3) If exactly one of a,b equals one of c,d, then exactly one of
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// VC(a,b,c,d) and VC(c,d,a,b) is true
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//
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// It is an error to call this method with 4 distinct vertices.
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func VertexCrossing(a, b, c, d Point) bool {
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// If A == B or C == D there is no intersection. We need to check this
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// case first in case 3 or more input points are identical.
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if a == b || c == d {
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return false
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}
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// If any other pair of vertices is equal, there is a crossing if and only
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// if OrderedCCW indicates that the edge AB is further CCW around the
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// shared vertex O (either A or B) than the edge CD, starting from an
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// arbitrary fixed reference point.
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// Optimization: if AB=CD or AB=DC, we can avoid most of the calculations.
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switch {
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case a == c:
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return (b == d) || OrderedCCW(Point{a.Ortho()}, d, b, a)
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case b == d:
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return OrderedCCW(Point{b.Ortho()}, c, a, b)
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case a == d:
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return (b == c) || OrderedCCW(Point{a.Ortho()}, c, b, a)
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case b == c:
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return OrderedCCW(Point{b.Ortho()}, d, a, b)
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}
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return false
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}
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// EdgeOrVertexCrossing is a convenience function that calls CrossingSign to
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// handle cases where all four vertices are distinct, and VertexCrossing to
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// handle cases where two or more vertices are the same. This defines a crossing
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// function such that point-in-polygon containment tests can be implemented
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// by simply counting edge crossings.
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func EdgeOrVertexCrossing(a, b, c, d Point) bool {
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switch CrossingSign(a, b, c, d) {
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case DoNotCross:
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return false
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case Cross:
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return true
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default:
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return VertexCrossing(a, b, c, d)
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}
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}
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// Intersection returns the intersection point of two edges AB and CD that cross
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// (CrossingSign(a,b,c,d) == Crossing).
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//
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// Useful properties of Intersection:
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//
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// (1) Intersection(b,a,c,d) == Intersection(a,b,d,c) == Intersection(a,b,c,d)
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// (2) Intersection(c,d,a,b) == Intersection(a,b,c,d)
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//
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// The returned intersection point X is guaranteed to be very close to the
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// true intersection point of AB and CD, even if the edges intersect at a
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// very small angle.
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func Intersection(a0, a1, b0, b1 Point) Point {
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// It is difficult to compute the intersection point of two edges accurately
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// when the angle between the edges is very small. Previously we handled
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// this by only guaranteeing that the returned intersection point is within
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// intersectionError of each edge. However, this means that when the edges
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// cross at a very small angle, the computed result may be very far from the
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// true intersection point.
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//
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// Instead this function now guarantees that the result is always within
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// intersectionError of the true intersection. This requires using more
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// sophisticated techniques and in some cases extended precision.
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//
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// - intersectionStable computes the intersection point using
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// projection and interpolation, taking care to minimize cancellation
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// error.
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//
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// - intersectionExact computes the intersection point using precision
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// arithmetic and converts the final result back to an Point.
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pt, ok := intersectionStable(a0, a1, b0, b1)
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if !ok {
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pt = intersectionExact(a0, a1, b0, b1)
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}
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// Make sure the intersection point is on the correct side of the sphere.
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// Since all vertices are unit length, and edges are less than 180 degrees,
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// (a0 + a1) and (b0 + b1) both have positive dot product with the
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// intersection point. We use the sum of all vertices to make sure that the
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// result is unchanged when the edges are swapped or reversed.
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if pt.Dot((a0.Add(a1.Vector)).Add(b0.Add(b1.Vector))) < 0 {
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pt = Point{pt.Mul(-1)}
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}
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return pt
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}
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// Computes the cross product of two vectors, normalized to be unit length.
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// Also returns the length of the cross
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// product before normalization, which is useful for estimating the amount of
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// error in the result. For numerical stability, the vectors should both be
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// approximately unit length.
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func robustNormalWithLength(x, y r3.Vector) (r3.Vector, float64) {
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var pt r3.Vector
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// This computes 2 * (x.Cross(y)), but has much better numerical
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// stability when x and y are unit length.
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tmp := x.Sub(y).Cross(x.Add(y))
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length := tmp.Norm()
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if length != 0 {
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pt = tmp.Mul(1 / length)
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}
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return pt, 0.5 * length // Since tmp == 2 * (x.Cross(y))
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}
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/*
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// intersectionSimple is not used by the C++ so it is skipped here.
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*/
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// projection returns the projection of aNorm onto X (x.Dot(aNorm)), and a bound
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// on the error in the result. aNorm is not necessarily unit length.
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//
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// The remaining parameters (the length of aNorm (aNormLen) and the edge endpoints
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// a0 and a1) allow this dot product to be computed more accurately and efficiently.
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func projection(x, aNorm r3.Vector, aNormLen float64, a0, a1 Point) (proj, bound float64) {
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// The error in the dot product is proportional to the lengths of the input
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// vectors, so rather than using x itself (a unit-length vector) we use
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// the vectors from x to the closer of the two edge endpoints. This
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// typically reduces the error by a huge factor.
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x0 := x.Sub(a0.Vector)
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x1 := x.Sub(a1.Vector)
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x0Dist2 := x0.Norm2()
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x1Dist2 := x1.Norm2()
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// If both distances are the same, we need to be careful to choose one
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// endpoint deterministically so that the result does not change if the
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// order of the endpoints is reversed.
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var dist float64
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if x0Dist2 < x1Dist2 || (x0Dist2 == x1Dist2 && x0.Cmp(x1) == -1) {
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dist = math.Sqrt(x0Dist2)
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proj = x0.Dot(aNorm)
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} else {
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dist = math.Sqrt(x1Dist2)
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proj = x1.Dot(aNorm)
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}
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// This calculation bounds the error from all sources: the computation of
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// the normal, the subtraction of one endpoint, and the dot product itself.
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// dblError appears because the input points are assumed to be
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// normalized in double precision.
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//
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// For reference, the bounds that went into this calculation are:
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// ||N'-N|| <= ((1 + 2 * sqrt(3))||N|| + 32 * sqrt(3) * dblError) * epsilon
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// |(A.B)'-(A.B)| <= (1.5 * (A.B) + 1.5 * ||A|| * ||B||) * epsilon
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// ||(X-Y)'-(X-Y)|| <= ||X-Y|| * epsilon
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bound = (((3.5+2*math.Sqrt(3))*aNormLen+32*math.Sqrt(3)*dblError)*dist + 1.5*math.Abs(proj)) * epsilon
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return proj, bound
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}
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// compareEdges reports whether (a0,a1) is less than (b0,b1) with respect to a total
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// ordering on edges that is invariant under edge reversals.
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func compareEdges(a0, a1, b0, b1 Point) bool {
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if a0.Cmp(a1.Vector) != -1 {
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a0, a1 = a1, a0
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}
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if b0.Cmp(b1.Vector) != -1 {
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b0, b1 = b1, b0
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}
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return a0.Cmp(b0.Vector) == -1 || (a0 == b0 && b0.Cmp(b1.Vector) == -1)
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}
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// intersectionStable returns the intersection point of the edges (a0,a1) and
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// (b0,b1) if it can be computed to within an error of at most intersectionError
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// by this function.
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//
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// The intersection point is not guaranteed to have the correct sign because we
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// choose to use the longest of the two edges first. The sign is corrected by
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// Intersection.
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func intersectionStable(a0, a1, b0, b1 Point) (Point, bool) {
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// Sort the two edges so that (a0,a1) is longer, breaking ties in a
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// deterministic way that does not depend on the ordering of the endpoints.
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// This is desirable for two reasons:
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// - So that the result doesn't change when edges are swapped or reversed.
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// - It reduces error, since the first edge is used to compute the edge
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// normal (where a longer edge means less error), and the second edge
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// is used for interpolation (where a shorter edge means less error).
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aLen2 := a1.Sub(a0.Vector).Norm2()
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bLen2 := b1.Sub(b0.Vector).Norm2()
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if aLen2 < bLen2 || (aLen2 == bLen2 && compareEdges(a0, a1, b0, b1)) {
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return intersectionStableSorted(b0, b1, a0, a1)
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}
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return intersectionStableSorted(a0, a1, b0, b1)
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}
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// intersectionStableSorted is a helper function for intersectionStable.
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// It expects that the edges (a0,a1) and (b0,b1) have been sorted so that
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// the first edge passed in is longer.
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func intersectionStableSorted(a0, a1, b0, b1 Point) (Point, bool) {
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var pt Point
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// Compute the normal of the plane through (a0, a1) in a stable way.
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aNorm := a0.Sub(a1.Vector).Cross(a0.Add(a1.Vector))
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aNormLen := aNorm.Norm()
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bLen := b1.Sub(b0.Vector).Norm()
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// Compute the projection (i.e., signed distance) of b0 and b1 onto the
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// plane through (a0, a1). Distances are scaled by the length of aNorm.
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b0Dist, b0Error := projection(b0.Vector, aNorm, aNormLen, a0, a1)
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b1Dist, b1Error := projection(b1.Vector, aNorm, aNormLen, a0, a1)
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// The total distance from b0 to b1 measured perpendicularly to (a0,a1) is
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// |b0Dist - b1Dist|. Note that b0Dist and b1Dist generally have
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// opposite signs because b0 and b1 are on opposite sides of (a0, a1). The
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// code below finds the intersection point by interpolating along the edge
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// (b0, b1) to a fractional distance of b0Dist / (b0Dist - b1Dist).
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//
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// It can be shown that the maximum error in the interpolation fraction is
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//
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// (b0Dist * b1Error - b1Dist * b0Error) / (distSum * (distSum - errorSum))
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//
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// We save ourselves some work by scaling the result and the error bound by
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// "distSum", since the result is normalized to be unit length anyway.
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distSum := math.Abs(b0Dist - b1Dist)
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errorSum := b0Error + b1Error
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if distSum <= errorSum {
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return pt, false // Error is unbounded in this case.
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}
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x := b1.Mul(b0Dist).Sub(b0.Mul(b1Dist))
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err := bLen*math.Abs(b0Dist*b1Error-b1Dist*b0Error)/
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(distSum-errorSum) + 2*distSum*epsilon
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// Finally we normalize the result, compute the corresponding error, and
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// check whether the total error is acceptable.
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xLen := x.Norm()
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maxError := intersectionError
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if err > (float64(maxError)-epsilon)*xLen {
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return pt, false
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}
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return Point{x.Mul(1 / xLen)}, true
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}
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// intersectionExact returns the intersection point of (a0, a1) and (b0, b1)
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// using precise arithmetic. Note that the result is not exact because it is
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// rounded down to double precision at the end. Also, the intersection point
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// is not guaranteed to have the correct sign (i.e., the return value may need
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// to be negated).
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func intersectionExact(a0, a1, b0, b1 Point) Point {
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// Since we are using presice arithmetic, we don't need to worry about
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// numerical stability.
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a0P := r3.PreciseVectorFromVector(a0.Vector)
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a1P := r3.PreciseVectorFromVector(a1.Vector)
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b0P := r3.PreciseVectorFromVector(b0.Vector)
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b1P := r3.PreciseVectorFromVector(b1.Vector)
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aNormP := a0P.Cross(a1P)
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bNormP := b0P.Cross(b1P)
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xP := aNormP.Cross(bNormP)
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// The final Normalize() call is done in double precision, which creates a
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// directional error of up to 2*dblError. (Precise conversion and Normalize()
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// each contribute up to dblError of directional error.)
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x := xP.Vector()
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if x == (r3.Vector{}) {
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// The two edges are exactly collinear, but we still consider them to be
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// "crossing" because of simulation of simplicity. Out of the four
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// endpoints, exactly two lie in the interior of the other edge. Of
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// those two we return the one that is lexicographically smallest.
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x = r3.Vector{10, 10, 10} // Greater than any valid S2Point
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aNorm := Point{aNormP.Vector()}
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bNorm := Point{bNormP.Vector()}
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if OrderedCCW(b0, a0, b1, bNorm) && a0.Cmp(x) == -1 {
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return a0
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}
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if OrderedCCW(b0, a1, b1, bNorm) && a1.Cmp(x) == -1 {
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return a1
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}
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if OrderedCCW(a0, b0, a1, aNorm) && b0.Cmp(x) == -1 {
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return b0
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}
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if OrderedCCW(a0, b1, a1, aNorm) && b1.Cmp(x) == -1 {
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return b1
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}
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}
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return Point{x}
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}
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