// 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 s1
import (
"math"
)
// ChordAngle represents the angle subtended by a chord (i.e., the straight
// line segment connecting two points on the sphere). Its representation
// makes it very efficient for computing and comparing distances, but unlike
// Angle it is only capable of representing angles between 0 and π radians.
// Generally, ChordAngle should only be used in loops where many angles need
// to be calculated and compared. Otherwise it is simpler to use Angle.
//
// ChordAngle loses some accuracy as the angle approaches π radians.
// There are several different ways to measure this error, including the
// representational error (i.e., how accurately ChordAngle can represent
// angles near π radians), the conversion error (i.e., how much precision is
// lost when an Angle is converted to an ChordAngle), and the measurement
// error (i.e., how accurate the ChordAngle(a, b) constructor is when the
// points A and B are separated by angles close to π radians). All of these
// errors differ by a small constant factor.
//
// For the measurement error (which is the largest of these errors and also
// the most important in practice), let the angle between A and B be (π - x)
// radians, i.e. A and B are within "x" radians of being antipodal. The
// corresponding chord length is
//
// r = 2 * sin((π - x) / 2) = 2 * cos(x / 2)
//
// For values of x not close to π the relative error in the squared chord
// length is at most 4.5 * dblEpsilon (see MaxPointError below).
// The relative error in "r" is thus at most 2.25 * dblEpsilon ~= 5e-16. To
// convert this error into an equivalent angle, we have
//
// |dr / dx| = sin(x / 2)
//
// and therefore
//
// |dx| = dr / sin(x / 2)
// = 5e-16 * (2 * cos(x / 2)) / sin(x / 2)
// = 1e-15 / tan(x / 2)
//
// The maximum error is attained when
//
// x = |dx|
// = 1e-15 / tan(x / 2)
// ~= 1e-15 / (x / 2)
// ~= sqrt(2e-15)
//
// In summary, the measurement error for an angle (π - x) is at most
//
// dx = min(1e-15 / tan(x / 2), sqrt(2e-15))
// (~= min(2e-15 / x, sqrt(2e-15)) when x is small)
//
// On the Earth's surface (assuming a radius of 6371km), this corresponds to
// the following worst-case measurement errors:
//
// Accuracy: Unless antipodal to within:
// --------- ---------------------------
// 6.4 nanometers 10,000 km (90 degrees)
// 1 micrometer 81.2 kilometers
// 1 millimeter 81.2 meters
// 1 centimeter 8.12 meters
// 28.5 centimeters 28.5 centimeters
//
// The representational and conversion errors referred to earlier are somewhat
// smaller than this. For example, maximum distance between adjacent
// representable ChordAngle values is only 13.5 cm rather than 28.5 cm. To
// see this, observe that the closest representable value to r^2 = 4 is
// r^2 = 4 * (1 - dblEpsilon / 2). Thus r = 2 * (1 - dblEpsilon / 4) and
// the angle between these two representable values is
//
// x = 2 * acos(r / 2)
// = 2 * acos(1 - dblEpsilon / 4)
// ~= 2 * asin(sqrt(dblEpsilon / 2)
// ~= sqrt(2 * dblEpsilon)
// ~= 2.1e-8
//
// which is 13.5 cm on the Earth's surface.
//
// The worst case rounding error occurs when the value halfway between these
// two representable values is rounded up to 4. This halfway value is
// r^2 = (4 * (1 - dblEpsilon / 4)), thus r = 2 * (1 - dblEpsilon / 8) and
// the worst case rounding error is
//
// x = 2 * acos(r / 2)
// = 2 * acos(1 - dblEpsilon / 8)
// ~= 2 * asin(sqrt(dblEpsilon / 4)
// ~= sqrt(dblEpsilon)
// ~= 1.5e-8
//
// which is 9.5 cm on the Earth's surface.
type ChordAngle float64
const (
// NegativeChordAngle represents a chord angle smaller than the zero angle.
// The only valid operations on a NegativeChordAngle are comparisons,
// Angle conversions, and Successor/Predecessor.
NegativeChordAngle = ChordAngle(-1)
// RightChordAngle represents a chord angle of 90 degrees (a "right angle").
RightChordAngle = ChordAngle(2)
// StraightChordAngle represents a chord angle of 180 degrees (a "straight angle").
// This is the maximum finite chord angle.
StraightChordAngle = ChordAngle(4)
// maxLength2 is the square of the maximum length allowed in a ChordAngle.
maxLength2 = 4.0
)
// ChordAngleFromAngle returns a ChordAngle from the given Angle.
func ChordAngleFromAngle(a Angle) ChordAngle {
if a < 0 {
return NegativeChordAngle
}
if a.isInf() {
return InfChordAngle()
}
l := 2 * math.Sin(0.5*math.Min(math.Pi, a.Radians()))
return ChordAngle(l * l)
}
// ChordAngleFromSquaredLength returns a ChordAngle from the squared chord length.
// Note that the argument is automatically clamped to a maximum of 4 to
// handle possible roundoff errors. The argument must be non-negative.
func ChordAngleFromSquaredLength(length2 float64) ChordAngle {
if length2 > maxLength2 {
return StraightChordAngle
}
return ChordAngle(length2)
}
// Expanded returns a new ChordAngle that has been adjusted by the given error
// bound (which can be positive or negative). Error should be the value
// returned by either MaxPointError or MaxAngleError. For example:
// a := ChordAngleFromPoints(x, y)
// a1 := a.Expanded(a.MaxPointError())
func (c ChordAngle) Expanded(e float64) ChordAngle {
// If the angle is special, don't change it. Otherwise clamp it to the valid range.
if c.isSpecial() {
return c
}
return ChordAngle(math.Max(0.0, math.Min(maxLength2, float64(c)+e)))
}
// Angle converts this ChordAngle to an Angle.
func (c ChordAngle) Angle() Angle {
if c < 0 {
return -1 * Radian
}
if c.isInf() {
return InfAngle()
}
return Angle(2 * math.Asin(0.5*math.Sqrt(float64(c))))
}
// InfChordAngle returns a chord angle larger than any finite chord angle.
// The only valid operations on an InfChordAngle are comparisons, Angle
// conversions, and Successor/Predecessor.
func InfChordAngle() ChordAngle {
return ChordAngle(math.Inf(1))
}
// isInf reports whether this ChordAngle is infinite.
func (c ChordAngle) isInf() bool {
return math.IsInf(float64(c), 1)
}
// isSpecial reports whether this ChordAngle is one of the special cases.
func (c ChordAngle) isSpecial() bool {
return c < 0 || c.isInf()
}
// isValid reports whether this ChordAngle is valid or not.
func (c ChordAngle) isValid() bool {
return (c >= 0 && c <= maxLength2) || c.isSpecial()
}
// Successor returns the smallest representable ChordAngle larger than this one.
// This can be used to convert a "<" comparison to a "<=" comparison.
//
// Note the following special cases:
// NegativeChordAngle.Successor == 0
// StraightChordAngle.Successor == InfChordAngle
// InfChordAngle.Successor == InfChordAngle
func (c ChordAngle) Successor() ChordAngle {
if c >= maxLength2 {
return InfChordAngle()
}
if c < 0 {
return 0
}
return ChordAngle(math.Nextafter(float64(c), 10.0))
}
// Predecessor returns the largest representable ChordAngle less than this one.
//
// Note the following special cases:
// InfChordAngle.Predecessor == StraightChordAngle
// ChordAngle(0).Predecessor == NegativeChordAngle
// NegativeChordAngle.Predecessor == NegativeChordAngle
func (c ChordAngle) Predecessor() ChordAngle {
if c <= 0 {
return NegativeChordAngle
}
if c > maxLength2 {
return StraightChordAngle
}
return ChordAngle(math.Nextafter(float64(c), -10.0))
}
// MaxPointError returns the maximum error size for a ChordAngle constructed
// from 2 Points x and y, assuming that x and y are normalized to within the
// bounds guaranteed by s2.Point.Normalize. The error is defined with respect to
// the true distance after the points are projected to lie exactly on the sphere.
func (c ChordAngle) MaxPointError() float64 {
// There is a relative error of (2.5*dblEpsilon) when computing the squared
// distance, plus a relative error of 2 * dblEpsilon, plus an absolute error
// of (16 * dblEpsilon**2) because the lengths of the input points may differ
// from 1 by up to (2*dblEpsilon) each. (This is the maximum error in Normalize).
return 4.5*dblEpsilon*float64(c) + 16*dblEpsilon*dblEpsilon
}
// MaxAngleError returns the maximum error for a ChordAngle constructed
// as an Angle distance.
func (c ChordAngle) MaxAngleError() float64 {
return dblEpsilon * float64(c)
}
// Add adds the other ChordAngle to this one and returns the resulting value.
// This method assumes the ChordAngles are not special.
func (c ChordAngle) Add(other ChordAngle) ChordAngle {
// Note that this method (and Sub) is much more efficient than converting
// the ChordAngle to an Angle and adding those and converting back. It
// requires only one square root plus a few additions and multiplications.
// Optimization for the common case where b is an error tolerance
// parameter that happens to be set to zero.
if other == 0 {
return c
}
// Clamp the angle sum to at most 180 degrees.
if c+other >= maxLength2 {
return StraightChordAngle
}
// Let a and b be the (non-squared) chord lengths, and let c = a+b.
// Let A, B, and C be the corresponding half-angles (a = 2*sin(A), etc).
// Then the formula below can be derived from c = 2 * sin(A+B) and the
// relationships sin(A+B) = sin(A)*cos(B) + sin(B)*cos(A)
// cos(X) = sqrt(1 - sin^2(X))
x := float64(c * (1 - 0.25*other))
y := float64(other * (1 - 0.25*c))
return ChordAngle(math.Min(maxLength2, x+y+2*math.Sqrt(x*y)))
}
// Sub subtracts the other ChordAngle from this one and returns the resulting
// value. This method assumes the ChordAngles are not special.
func (c ChordAngle) Sub(other ChordAngle) ChordAngle {
if other == 0 {
return c
}
if c <= other {
return 0
}
x := float64(c * (1 - 0.25*other))
y := float64(other * (1 - 0.25*c))
return ChordAngle(math.Max(0.0, x+y-2*math.Sqrt(x*y)))
}
// Sin returns the sine of this chord angle. This method is more efficient
// than converting to Angle and performing the computation.
func (c ChordAngle) Sin() float64 {
return math.Sqrt(c.Sin2())
}
// Sin2 returns the square of the sine of this chord angle.
// It is more efficient than Sin.
func (c ChordAngle) Sin2() float64 {
// Let a be the (non-squared) chord length, and let A be the corresponding
// half-angle (a = 2*sin(A)). The formula below can be derived from:
// sin(2*A) = 2 * sin(A) * cos(A)
// cos^2(A) = 1 - sin^2(A)
// This is much faster than converting to an angle and computing its sine.
return float64(c * (1 - 0.25*c))
}
// Cos returns the cosine of this chord angle. This method is more efficient
// than converting to Angle and performing the computation.
func (c ChordAngle) Cos() float64 {
// cos(2*A) = cos^2(A) - sin^2(A) = 1 - 2*sin^2(A)
return float64(1 - 0.5*c)
}
// Tan returns the tangent of this chord angle.
func (c ChordAngle) Tan() float64 {
return c.Sin() / c.Cos()
}
// TODO(roberts): Differences from C++:
// Helpers to/from E5/E6/E7
// Helpers to/from degrees and radians directly.
// FastUpperBoundFrom(angle Angle)