gtsocial-umbx

cap.go (17742B)

---

 // Copyright 2014 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/s1"
 )
 
 var (
 	// centerPoint is the default center for Caps
 	centerPoint = PointFromCoords(1.0, 0, 0)
 )
 
 // Cap represents a disc-shaped region defined by a center and radius.
 // Technically this shape is called a "spherical cap" (rather than disc)
 // because it is not planar; the cap represents a portion of the sphere that
 // has been cut off by a plane. The boundary of the cap is the circle defined
 // by the intersection of the sphere and the plane. For containment purposes,
 // the cap is a closed set, i.e. it contains its boundary.
 //
 // For the most part, you can use a spherical cap wherever you would use a
 // disc in planar geometry. The radius of the cap is measured along the
 // surface of the sphere (rather than the straight-line distance through the
 // interior). Thus a cap of radius π/2 is a hemisphere, and a cap of radius
 // π covers the entire sphere.
 //
 // The center is a point on the surface of the unit sphere. (Hence the need for
 // it to be of unit length.)
 //
 // A cap can also be defined by its center point and height. The height is the
 // distance from the center point to the cutoff plane. There is also support for
 // "empty" and "full" caps, which contain no points and all points respectively.
 //
 // Here are some useful relationships between the cap height (h), the cap
 // radius (r), the maximum chord length from the cap's center (d), and the
 // radius of cap's base (a).
 //
 //     h = 1 - cos(r)
 //       = 2 * sin^2(r/2)
 //   d^2 = 2 * h
 //       = a^2 + h^2
 //
 // The zero value of Cap is an invalid cap. Use EmptyCap to get a valid empty cap.
 type Cap struct {
 	center Point
 	radius s1.ChordAngle
 }
 
 // CapFromPoint constructs a cap containing a single point.
 func CapFromPoint(p Point) Cap {
 	return CapFromCenterChordAngle(p, 0)
 }
 
 // CapFromCenterAngle constructs a cap with the given center and angle.
 func CapFromCenterAngle(center Point, angle s1.Angle) Cap {
 	return CapFromCenterChordAngle(center, s1.ChordAngleFromAngle(angle))
 }
 
 // CapFromCenterChordAngle constructs a cap where the angle is expressed as an
 // s1.ChordAngle. This constructor is more efficient than using an s1.Angle.
 func CapFromCenterChordAngle(center Point, radius s1.ChordAngle) Cap {
 	return Cap{
 		center: center,
 		radius: radius,
 	}
 }
 
 // CapFromCenterHeight constructs a cap with the given center and height. A
 // negative height yields an empty cap; a height of 2 or more yields a full cap.
 // The center should be unit length.
 func CapFromCenterHeight(center Point, height float64) Cap {
 	return CapFromCenterChordAngle(center, s1.ChordAngleFromSquaredLength(2*height))
 }
 
 // CapFromCenterArea constructs a cap with the given center and surface area.
 // Note that the area can also be interpreted as the solid angle subtended by the
 // cap (because the sphere has unit radius). A negative area yields an empty cap;
 // an area of 4*π or more yields a full cap.
 func CapFromCenterArea(center Point, area float64) Cap {
 	return CapFromCenterChordAngle(center, s1.ChordAngleFromSquaredLength(area/math.Pi))
 }
 
 // EmptyCap returns a cap that contains no points.
 func EmptyCap() Cap {
 	return CapFromCenterChordAngle(centerPoint, s1.NegativeChordAngle)
 }
 
 // FullCap returns a cap that contains all points.
 func FullCap() Cap {
 	return CapFromCenterChordAngle(centerPoint, s1.StraightChordAngle)
 }
 
 // IsValid reports whether the Cap is considered valid.
 func (c Cap) IsValid() bool {
 	return c.center.Vector.IsUnit() && c.radius <= s1.StraightChordAngle
 }
 
 // IsEmpty reports whether the cap is empty, i.e. it contains no points.
 func (c Cap) IsEmpty() bool {
 	return c.radius < 0
 }
 
 // IsFull reports whether the cap is full, i.e. it contains all points.
 func (c Cap) IsFull() bool {
 	return c.radius == s1.StraightChordAngle
 }
 
 // Center returns the cap's center point.
 func (c Cap) Center() Point {
 	return c.center
 }
 
 // Height returns the height of the cap. This is the distance from the center
 // point to the cutoff plane.
 func (c Cap) Height() float64 {
 	return float64(0.5 * c.radius)
 }
 
 // Radius returns the cap radius as an s1.Angle. (Note that the cap angle
 // is stored internally as a ChordAngle, so this method requires a trigonometric
 // operation and may yield a slightly different result than the value passed
 // to CapFromCenterAngle).
 func (c Cap) Radius() s1.Angle {
 	return c.radius.Angle()
 }
 
 // Area returns the surface area of the Cap on the unit sphere.
 func (c Cap) Area() float64 {
 	return 2.0 * math.Pi * math.Max(0, c.Height())
 }
 
 // Contains reports whether this cap contains the other.
 func (c Cap) Contains(other Cap) bool {
 	// In a set containment sense, every cap contains the empty cap.
 	if c.IsFull() || other.IsEmpty() {
 		return true
 	}
 	return c.radius >= ChordAngleBetweenPoints(c.center, other.center).Add(other.radius)
 }
 
 // Intersects reports whether this cap intersects the other cap.
 // i.e. whether they have any points in common.
 func (c Cap) Intersects(other Cap) bool {
 	if c.IsEmpty() || other.IsEmpty() {
 		return false
 	}
 
 	return c.radius.Add(other.radius) >= ChordAngleBetweenPoints(c.center, other.center)
 }
 
 // InteriorIntersects reports whether this caps interior intersects the other cap.
 func (c Cap) InteriorIntersects(other Cap) bool {
 	// Make sure this cap has an interior and the other cap is non-empty.
 	if c.radius <= 0 || other.IsEmpty() {
 		return false
 	}
 
 	return c.radius.Add(other.radius) > ChordAngleBetweenPoints(c.center, other.center)
 }
 
 // ContainsPoint reports whether this cap contains the point.
 func (c Cap) ContainsPoint(p Point) bool {
 	return ChordAngleBetweenPoints(c.center, p) <= c.radius
 }
 
 // InteriorContainsPoint reports whether the point is within the interior of this cap.
 func (c Cap) InteriorContainsPoint(p Point) bool {
 	return c.IsFull() || ChordAngleBetweenPoints(c.center, p) < c.radius
 }
 
 // Complement returns the complement of the interior of the cap. A cap and its
 // complement have the same boundary but do not share any interior points.
 // The complement operator is not a bijection because the complement of a
 // singleton cap (containing a single point) is the same as the complement
 // of an empty cap.
 func (c Cap) Complement() Cap {
 	if c.IsFull() {
 		return EmptyCap()
 	}
 	if c.IsEmpty() {
 		return FullCap()
 	}
 
 	return CapFromCenterChordAngle(Point{c.center.Mul(-1)}, s1.StraightChordAngle.Sub(c.radius))
 }
 
 // CapBound returns a bounding spherical cap. This is not guaranteed to be exact.
 func (c Cap) CapBound() Cap {
 	return c
 }
 
 // RectBound returns a bounding latitude-longitude rectangle.
 // The bounds are not guaranteed to be tight.
 func (c Cap) RectBound() Rect {
 	if c.IsEmpty() {
 		return EmptyRect()
 	}
 
 	capAngle := c.Radius().Radians()
 	allLongitudes := false
 	lat := r1.Interval{
 		Lo: latitude(c.center).Radians() - capAngle,
 		Hi: latitude(c.center).Radians() + capAngle,
 	}
 	lng := s1.FullInterval()
 
 	// Check whether cap includes the south pole.
 	if lat.Lo <= -math.Pi/2 {
 		lat.Lo = -math.Pi / 2
 		allLongitudes = true
 	}
 
 	// Check whether cap includes the north pole.
 	if lat.Hi >= math.Pi/2 {
 		lat.Hi = math.Pi / 2
 		allLongitudes = true
 	}
 
 	if !allLongitudes {
 		// Compute the range of longitudes covered by the cap. We use the law
 		// of sines for spherical triangles. Consider the triangle ABC where
 		// A is the north pole, B is the center of the cap, and C is the point
 		// of tangency between the cap boundary and a line of longitude. Then
 		// C is a right angle, and letting a,b,c denote the sides opposite A,B,C,
 		// we have sin(a)/sin(A) = sin(c)/sin(C), or sin(A) = sin(a)/sin(c).
 		// Here "a" is the cap angle, and "c" is the colatitude (90 degrees
 		// minus the latitude). This formula also works for negative latitudes.
 		//
 		// The formula for sin(a) follows from the relationship h = 1 - cos(a).
 		sinA := c.radius.Sin()
 		sinC := math.Cos(latitude(c.center).Radians())
 		if sinA <= sinC {
 			angleA := math.Asin(sinA / sinC)
 			lng.Lo = math.Remainder(longitude(c.center).Radians()-angleA, math.Pi*2)
 			lng.Hi = math.Remainder(longitude(c.center).Radians()+angleA, math.Pi*2)
 		}
 	}
 	return Rect{lat, lng}
 }
 
 // Equal reports whether this cap is equal to the other cap.
 func (c Cap) Equal(other Cap) bool {
 	return (c.radius == other.radius && c.center == other.center) ||
 		(c.IsEmpty() && other.IsEmpty()) ||
 		(c.IsFull() && other.IsFull())
 }
 
 // ApproxEqual reports whether this cap is equal to the other cap within the given tolerance.
 func (c Cap) ApproxEqual(other Cap) bool {
 	const epsilon = 1e-14
 	r2 := float64(c.radius)
 	otherR2 := float64(other.radius)
 	return c.center.ApproxEqual(other.center) &&
 		math.Abs(r2-otherR2) <= epsilon ||
 		c.IsEmpty() && otherR2 <= epsilon ||
 		other.IsEmpty() && r2 <= epsilon ||
 		c.IsFull() && otherR2 >= 2-epsilon ||
 		other.IsFull() && r2 >= 2-epsilon
 }
 
 // AddPoint increases the cap if necessary to include the given point. If this cap is empty,
 // then the center is set to the point with a zero height. p must be unit-length.
 func (c Cap) AddPoint(p Point) Cap {
 	if c.IsEmpty() {
 		c.center = p
 		c.radius = 0
 		return c
 	}
 
 	// After calling cap.AddPoint(p), cap.Contains(p) must be true. However
 	// we don't need to do anything special to achieve this because Contains()
 	// does exactly the same distance calculation that we do here.
 	if newRad := ChordAngleBetweenPoints(c.center, p); newRad > c.radius {
 		c.radius = newRad
 	}
 	return c
 }
 
 // AddCap increases the cap height if necessary to include the other cap. If this cap is empty,
 // it is set to the other cap.
 func (c Cap) AddCap(other Cap) Cap {
 	if c.IsEmpty() {
 		return other
 	}
 	if other.IsEmpty() {
 		return c
 	}
 
 	// We round up the distance to ensure that the cap is actually contained.
 	// TODO(roberts): Do some error analysis in order to guarantee this.
 	dist := ChordAngleBetweenPoints(c.center, other.center).Add(other.radius)
 	if newRad := dist.Expanded(dblEpsilon * float64(dist)); newRad > c.radius {
 		c.radius = newRad
 	}
 	return c
 }
 
 // Expanded returns a new cap expanded by the given angle. If the cap is empty,
 // it returns an empty cap.
 func (c Cap) Expanded(distance s1.Angle) Cap {
 	if c.IsEmpty() {
 		return EmptyCap()
 	}
 	return CapFromCenterChordAngle(c.center, c.radius.Add(s1.ChordAngleFromAngle(distance)))
 }
 
 func (c Cap) String() string {
 	return fmt.Sprintf("[Center=%v, Radius=%f]", c.center.Vector, c.Radius().Degrees())
 }
 
 // radiusToHeight converts an s1.Angle into the height of the cap.
 func radiusToHeight(r s1.Angle) float64 {
 	if r.Radians() < 0 {
 		return float64(s1.NegativeChordAngle)
 	}
 	if r.Radians() >= math.Pi {
 		return float64(s1.RightChordAngle)
 	}
 	return float64(0.5 * s1.ChordAngleFromAngle(r))
 
 }
 
 // ContainsCell reports whether the cap contains the given cell.
 func (c Cap) ContainsCell(cell Cell) bool {
 	// If the cap does not contain all cell vertices, return false.
 	var vertices [4]Point
 	for k := 0; k < 4; k++ {
 		vertices[k] = cell.Vertex(k)
 		if !c.ContainsPoint(vertices[k]) {
 			return false
 		}
 	}
 	// Otherwise, return true if the complement of the cap does not intersect the cell.
 	return !c.Complement().intersects(cell, vertices)
 }
 
 // IntersectsCell reports whether the cap intersects the cell.
 func (c Cap) IntersectsCell(cell Cell) bool {
 	// If the cap contains any cell vertex, return true.
 	var vertices [4]Point
 	for k := 0; k < 4; k++ {
 		vertices[k] = cell.Vertex(k)
 		if c.ContainsPoint(vertices[k]) {
 			return true
 		}
 	}
 	return c.intersects(cell, vertices)
 }
 
 // intersects reports whether the cap intersects any point of the cell excluding
 // its vertices (which are assumed to already have been checked).
 func (c Cap) intersects(cell Cell, vertices [4]Point) bool {
 	// If the cap is a hemisphere or larger, the cell and the complement of the cap
 	// are both convex. Therefore since no vertex of the cell is contained, no other
 	// interior point of the cell is contained either.
 	if c.radius >= s1.RightChordAngle {
 		return false
 	}
 
 	// We need to check for empty caps due to the center check just below.
 	if c.IsEmpty() {
 		return false
 	}
 
 	// Optimization: return true if the cell contains the cap center. This allows half
 	// of the edge checks below to be skipped.
 	if cell.ContainsPoint(c.center) {
 		return true
 	}
 
 	// At this point we know that the cell does not contain the cap center, and the cap
 	// does not contain any cell vertex. The only way that they can intersect is if the
 	// cap intersects the interior of some edge.
 	sin2Angle := c.radius.Sin2()
 	for k := 0; k < 4; k++ {
 		edge := cell.Edge(k).Vector
 		dot := c.center.Vector.Dot(edge)
 		if dot > 0 {
 			// The center is in the interior half-space defined by the edge. We do not need
 			// to consider these edges, since if the cap intersects this edge then it also
 			// intersects the edge on the opposite side of the cell, because the center is
 			// not contained with the cell.
 			continue
 		}
 
 		// The Norm2() factor is necessary because "edge" is not normalized.
 		if dot*dot > sin2Angle*edge.Norm2() {
 			return false
 		}
 
 		// Otherwise, the great circle containing this edge intersects the interior of the cap. We just
 		// need to check whether the point of closest approach occurs between the two edge endpoints.
 		dir := edge.Cross(c.center.Vector)
 		if dir.Dot(vertices[k].Vector) < 0 && dir.Dot(vertices[(k+1)&3].Vector) > 0 {
 			return true
 		}
 	}
 	return false
 }
 
 // CellUnionBound computes a covering of the Cap. In general the covering
 // consists of at most 4 cells except for very large caps, which may need
 // up to 6 cells. The output is not sorted.
 func (c Cap) CellUnionBound() []CellID {
 	// TODO(roberts): The covering could be made quite a bit tighter by mapping
 	// the cap to a rectangle in (i,j)-space and finding a covering for that.
 
 	// Find the maximum level such that the cap contains at most one cell vertex
 	// and such that CellID.AppendVertexNeighbors() can be called.
 	level := MinWidthMetric.MaxLevel(c.Radius().Radians()) - 1
 
 	// If level < 0, more than three face cells are required.
 	if level < 0 {
 		cellIDs := make([]CellID, 6)
 		for face := 0; face < 6; face++ {
 			cellIDs[face] = CellIDFromFace(face)
 		}
 		return cellIDs
 	}
 	// The covering consists of the 4 cells at the given level that share the
 	// cell vertex that is closest to the cap center.
 	return cellIDFromPoint(c.center).VertexNeighbors(level)
 }
 
 // Centroid returns the true centroid of the cap multiplied by its surface area
 // The result lies on the ray from the origin through the cap's center, but it
 // is not unit length. Note that if you just want the "surface centroid", i.e.
 // the normalized result, then it is simpler to call Center.
 //
 // The reason for multiplying the result by the cap area is to make it
 // easier to compute the centroid of more complicated shapes. The centroid
 // of a union of disjoint regions can be computed simply by adding their
 // Centroid() results. Caveat: for caps that contain a single point
 // (i.e., zero radius), this method always returns the origin (0, 0, 0).
 // This is because shapes with no area don't affect the centroid of a
 // union whose total area is positive.
 func (c Cap) Centroid() Point {
 	// From symmetry, the centroid of the cap must be somewhere on the line
 	// from the origin to the center of the cap on the surface of the sphere.
 	// When a sphere is divided into slices of constant thickness by a set of
 	// parallel planes, all slices have the same surface area. This implies
 	// that the radial component of the centroid is simply the midpoint of the
 	// range of radial distances spanned by the cap. That is easily computed
 	// from the cap height.
 	if c.IsEmpty() {
 		return Point{}
 	}
 	r := 1 - 0.5*c.Height()
 	return Point{c.center.Mul(r * c.Area())}
 }
 
 // Union returns the smallest cap which encloses this cap and other.
 func (c Cap) Union(other Cap) Cap {
 	// If the other cap is larger, swap c and other for the rest of the computations.
 	if c.radius < other.radius {
 		c, other = other, c
 	}
 
 	if c.IsFull() || other.IsEmpty() {
 		return c
 	}
 
 	// TODO: This calculation would be more efficient using s1.ChordAngles.
 	cRadius := c.Radius()
 	otherRadius := other.Radius()
 	distance := c.center.Distance(other.center)
 	if cRadius >= distance+otherRadius {
 		return c
 	}
 
 	resRadius := 0.5 * (distance + cRadius + otherRadius)
 	resCenter := InterpolateAtDistance(0.5*(distance-cRadius+otherRadius), c.center, other.center)
 	return CapFromCenterAngle(resCenter, resRadius)
 }
 
 // Encode encodes the Cap.
 func (c Cap) Encode(w io.Writer) error {
 	e := &encoder{w: w}
 	c.encode(e)
 	return e.err
 }
 
 func (c Cap) encode(e *encoder) {
 	e.writeFloat64(c.center.X)
 	e.writeFloat64(c.center.Y)
 	e.writeFloat64(c.center.Z)
 	e.writeFloat64(float64(c.radius))
 }
 
 // Decode decodes the Cap.
 func (c *Cap) Decode(r io.Reader) error {
 	d := &decoder{r: asByteReader(r)}
 	c.decode(d)
 	return d.err
 }
 
 func (c *Cap) decode(d *decoder) {
 	c.center.X = d.readFloat64()
 	c.center.Y = d.readFloat64()
 	c.center.Z = d.readFloat64()
 	c.radius = s1.ChordAngle(d.readFloat64())
 }