259 lines
9.2 KiB
Go
259 lines
9.2 KiB
Go
// Copyright 2014 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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"io"
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"math"
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"sort"
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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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// Point represents a point on the unit sphere as a normalized 3D vector.
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// Fields should be treated as read-only. Use one of the factory methods for creation.
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type Point struct {
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r3.Vector
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}
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// sortPoints sorts the slice of Points in place.
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func sortPoints(e []Point) {
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sort.Sort(points(e))
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}
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// points implements the Sort interface for slices of Point.
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type points []Point
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func (p points) Len() int { return len(p) }
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func (p points) Swap(i, j int) { p[i], p[j] = p[j], p[i] }
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func (p points) Less(i, j int) bool { return p[i].Cmp(p[j].Vector) == -1 }
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// PointFromCoords creates a new normalized point from coordinates.
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//
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// This always returns a valid point. If the given coordinates can not be normalized
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// the origin point will be returned.
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//
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// This behavior is different from the C++ construction of a S2Point from coordinates
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// (i.e. S2Point(x, y, z)) in that in C++ they do not Normalize.
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func PointFromCoords(x, y, z float64) Point {
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if x == 0 && y == 0 && z == 0 {
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return OriginPoint()
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}
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return Point{r3.Vector{x, y, z}.Normalize()}
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}
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// OriginPoint returns a unique "origin" on the sphere for operations that need a fixed
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// reference point. In particular, this is the "point at infinity" used for
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// point-in-polygon testing (by counting the number of edge crossings).
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//
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// It should *not* be a point that is commonly used in edge tests in order
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// to avoid triggering code to handle degenerate cases (this rules out the
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// north and south poles). It should also not be on the boundary of any
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// low-level S2Cell for the same reason.
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func OriginPoint() Point {
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return Point{r3.Vector{-0.0099994664350250197, 0.0025924542609324121, 0.99994664350250195}}
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}
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// PointCross returns a Point that is orthogonal to both p and op. This is similar to
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// p.Cross(op) (the true cross product) except that it does a better job of
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// ensuring orthogonality when the Point is nearly parallel to op, it returns
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// a non-zero result even when p == op or p == -op and the result is a Point.
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//
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// It satisfies the following properties (f == PointCross):
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//
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// (1) f(p, op) != 0 for all p, op
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// (2) f(op,p) == -f(p,op) unless p == op or p == -op
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// (3) f(-p,op) == -f(p,op) unless p == op or p == -op
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// (4) f(p,-op) == -f(p,op) unless p == op or p == -op
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func (p Point) PointCross(op Point) Point {
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// NOTE(dnadasi): In the C++ API the equivalent method here was known as "RobustCrossProd",
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// but PointCross more accurately describes how this method is used.
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x := p.Add(op.Vector).Cross(op.Sub(p.Vector))
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// Compare exactly to the 0 vector.
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if x == (r3.Vector{}) {
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// The only result that makes sense mathematically is to return zero, but
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// we find it more convenient to return an arbitrary orthogonal vector.
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return Point{p.Ortho()}
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}
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return Point{x}
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}
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// OrderedCCW returns true if the edges OA, OB, and OC are encountered in that
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// order while sweeping CCW around the point O.
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//
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// You can think of this as testing whether A <= B <= C with respect to the
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// CCW ordering around O that starts at A, or equivalently, whether B is
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// contained in the range of angles (inclusive) that starts at A and extends
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// CCW to C. Properties:
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//
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// (1) If OrderedCCW(a,b,c,o) && OrderedCCW(b,a,c,o), then a == b
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// (2) If OrderedCCW(a,b,c,o) && OrderedCCW(a,c,b,o), then b == c
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// (3) If OrderedCCW(a,b,c,o) && OrderedCCW(c,b,a,o), then a == b == c
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// (4) If a == b or b == c, then OrderedCCW(a,b,c,o) is true
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// (5) Otherwise if a == c, then OrderedCCW(a,b,c,o) is false
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func OrderedCCW(a, b, c, o Point) bool {
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sum := 0
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if RobustSign(b, o, a) != Clockwise {
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sum++
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}
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if RobustSign(c, o, b) != Clockwise {
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sum++
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}
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if RobustSign(a, o, c) == CounterClockwise {
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sum++
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}
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return sum >= 2
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}
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// Distance returns the angle between two points.
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func (p Point) Distance(b Point) s1.Angle {
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return p.Vector.Angle(b.Vector)
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}
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// ApproxEqual reports whether the two points are similar enough to be equal.
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func (p Point) ApproxEqual(other Point) bool {
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return p.approxEqual(other, s1.Angle(epsilon))
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}
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// approxEqual reports whether the two points are within the given epsilon.
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func (p Point) approxEqual(other Point, eps s1.Angle) bool {
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return p.Vector.Angle(other.Vector) <= eps
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}
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// ChordAngleBetweenPoints constructs a ChordAngle corresponding to the distance
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// between the two given points. The points must be unit length.
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func ChordAngleBetweenPoints(x, y Point) s1.ChordAngle {
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return s1.ChordAngle(math.Min(4.0, x.Sub(y.Vector).Norm2()))
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}
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// regularPoints generates a slice of points shaped as a regular polygon with
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// the numVertices vertices, all located on a circle of the specified angular radius
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// around the center. The radius is the actual distance from center to each vertex.
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func regularPoints(center Point, radius s1.Angle, numVertices int) []Point {
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return regularPointsForFrame(getFrame(center), radius, numVertices)
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}
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// regularPointsForFrame generates a slice of points shaped as a regular polygon
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// with numVertices vertices, all on a circle of the specified angular radius around
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// the center. The radius is the actual distance from the center to each vertex.
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func regularPointsForFrame(frame matrix3x3, radius s1.Angle, numVertices int) []Point {
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// We construct the loop in the given frame coordinates, with the center at
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// (0, 0, 1). For a loop of radius r, the loop vertices have the form
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// (x, y, z) where x^2 + y^2 = sin(r) and z = cos(r). The distance on the
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// sphere (arc length) from each vertex to the center is acos(cos(r)) = r.
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z := math.Cos(radius.Radians())
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r := math.Sin(radius.Radians())
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radianStep := 2 * math.Pi / float64(numVertices)
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var vertices []Point
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for i := 0; i < numVertices; i++ {
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angle := float64(i) * radianStep
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p := Point{r3.Vector{r * math.Cos(angle), r * math.Sin(angle), z}}
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vertices = append(vertices, Point{fromFrame(frame, p).Normalize()})
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}
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return vertices
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}
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// CapBound returns a bounding cap for this point.
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func (p Point) CapBound() Cap {
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return CapFromPoint(p)
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}
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// RectBound returns a bounding latitude-longitude rectangle from this point.
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func (p Point) RectBound() Rect {
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return RectFromLatLng(LatLngFromPoint(p))
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}
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// ContainsCell returns false as Points do not contain any other S2 types.
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func (p Point) ContainsCell(c Cell) bool { return false }
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// IntersectsCell reports whether this Point intersects the given cell.
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func (p Point) IntersectsCell(c Cell) bool {
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return c.ContainsPoint(p)
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}
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// ContainsPoint reports if this Point contains the other Point.
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// (This method is named to satisfy the Region interface.)
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func (p Point) ContainsPoint(other Point) bool {
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return p.Contains(other)
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}
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// CellUnionBound computes a covering of the Point.
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func (p Point) CellUnionBound() []CellID {
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return p.CapBound().CellUnionBound()
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}
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// Contains reports if this Point contains the other Point.
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// (This method matches all other s2 types where the reflexive Contains
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// method does not contain the type's name.)
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func (p Point) Contains(other Point) bool { return p == other }
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// Encode encodes the Point.
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func (p Point) Encode(w io.Writer) error {
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e := &encoder{w: w}
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p.encode(e)
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return e.err
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}
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func (p Point) encode(e *encoder) {
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e.writeInt8(encodingVersion)
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e.writeFloat64(p.X)
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e.writeFloat64(p.Y)
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e.writeFloat64(p.Z)
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}
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// Decode decodes the Point.
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func (p *Point) Decode(r io.Reader) error {
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d := &decoder{r: asByteReader(r)}
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p.decode(d)
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return d.err
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}
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func (p *Point) decode(d *decoder) {
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version := d.readInt8()
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if d.err != nil {
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return
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}
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if version != encodingVersion {
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d.err = fmt.Errorf("only version %d is supported", encodingVersion)
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return
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}
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p.X = d.readFloat64()
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p.Y = d.readFloat64()
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p.Z = d.readFloat64()
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}
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// Rotate the given point about the given axis by the given angle. p and
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// axis must be unit length; angle has no restrictions (e.g., it can be
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// positive, negative, greater than 360 degrees, etc).
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func Rotate(p, axis Point, angle s1.Angle) Point {
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// Let M be the plane through P that is perpendicular to axis, and let
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// center be the point where M intersects axis. We construct a
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// right-handed orthogonal frame (dx, dy, center) such that dx is the
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// vector from center to P, and dy has the same length as dx. The
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// result can then be expressed as (cos(angle)*dx + sin(angle)*dy + center).
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center := axis.Mul(p.Dot(axis.Vector))
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dx := p.Sub(center)
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dy := axis.Cross(p.Vector)
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// Mathematically the result is unit length, but normalization is necessary
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// to ensure that numerical errors don't accumulate.
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return Point{dx.Mul(math.Cos(angle.Radians())).Add(dy.Mul(math.Sin(angle.Radians()))).Add(center).Normalize()}
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}
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