gtsocial-umbx

point_measures.go (6420B)

---

 // Copyright 2018 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"
 
 	"github.com/golang/geo/s1"
 )
 
 // PointArea returns the area of triangle ABC. This method combines two different
 // algorithms to get accurate results for both large and small triangles.
 // The maximum error is about 5e-15 (about 0.25 square meters on the Earth's
 // surface), the same as GirardArea below, but unlike that method it is
 // also accurate for small triangles. Example: when the true area is 100
 // square meters, PointArea yields an error about 1 trillion times smaller than
 // GirardArea.
 //
 // All points should be unit length, and no two points should be antipodal.
 // The area is always positive.
 func PointArea(a, b, c Point) float64 {
 	// This method is based on l'Huilier's theorem,
 	//
 	//   tan(E/4) = sqrt(tan(s/2) tan((s-a)/2) tan((s-b)/2) tan((s-c)/2))
 	//
 	// where E is the spherical excess of the triangle (i.e. its area),
 	//       a, b, c are the side lengths, and
 	//       s is the semiperimeter (a + b + c) / 2.
 	//
 	// The only significant source of error using l'Huilier's method is the
 	// cancellation error of the terms (s-a), (s-b), (s-c). This leads to a
 	// *relative* error of about 1e-16 * s / min(s-a, s-b, s-c). This compares
 	// to a relative error of about 1e-15 / E using Girard's formula, where E is
 	// the true area of the triangle. Girard's formula can be even worse than
 	// this for very small triangles, e.g. a triangle with a true area of 1e-30
 	// might evaluate to 1e-5.
 	//
 	// So, we prefer l'Huilier's formula unless dmin < s * (0.1 * E), where
 	// dmin = min(s-a, s-b, s-c). This basically includes all triangles
 	// except for extremely long and skinny ones.
 	//
 	// Since we don't know E, we would like a conservative upper bound on
 	// the triangle area in terms of s and dmin. It's possible to show that
 	// E <= k1 * s * sqrt(s * dmin), where k1 = 2*sqrt(3)/Pi (about 1).
 	// Using this, it's easy to show that we should always use l'Huilier's
 	// method if dmin >= k2 * s^5, where k2 is about 1e-2. Furthermore,
 	// if dmin < k2 * s^5, the triangle area is at most k3 * s^4, where
 	// k3 is about 0.1. Since the best case error using Girard's formula
 	// is about 1e-15, this means that we shouldn't even consider it unless
 	// s >= 3e-4 or so.
 	sa := float64(b.Angle(c.Vector))
 	sb := float64(c.Angle(a.Vector))
 	sc := float64(a.Angle(b.Vector))
 	s := 0.5 * (sa + sb + sc)
 	if s >= 3e-4 {
 		// Consider whether Girard's formula might be more accurate.
 		dmin := s - math.Max(sa, math.Max(sb, sc))
 		if dmin < 1e-2*s*s*s*s*s {
 			// This triangle is skinny enough to use Girard's formula.
 			area := GirardArea(a, b, c)
 			if dmin < s*0.1*area {
 				return area
 			}
 		}
 	}
 
 	// Use l'Huilier's formula.
 	return 4 * math.Atan(math.Sqrt(math.Max(0.0, math.Tan(0.5*s)*math.Tan(0.5*(s-sa))*
 		math.Tan(0.5*(s-sb))*math.Tan(0.5*(s-sc)))))
 }
 
 // GirardArea returns the area of the triangle computed using Girard's formula.
 // All points should be unit length, and no two points should be antipodal.
 //
 // This method is about twice as fast as PointArea() but has poor relative
 // accuracy for small triangles. The maximum error is about 5e-15 (about
 // 0.25 square meters on the Earth's surface) and the average error is about
 // 1e-15. These bounds apply to triangles of any size, even as the maximum
 // edge length of the triangle approaches 180 degrees. But note that for
 // such triangles, tiny perturbations of the input points can change the
 // true mathematical area dramatically.
 func GirardArea(a, b, c Point) float64 {
 	// This is equivalent to the usual Girard's formula but is slightly more
 	// accurate, faster to compute, and handles a == b == c without a special
 	// case. PointCross is necessary to get good accuracy when two of
 	// the input points are very close together.
 	ab := a.PointCross(b)
 	bc := b.PointCross(c)
 	ac := a.PointCross(c)
 
 	area := float64(ab.Angle(ac.Vector) - ab.Angle(bc.Vector) + bc.Angle(ac.Vector))
 	if area < 0 {
 		area = 0
 	}
 	return area
 }
 
 // SignedArea returns a positive value for counterclockwise triangles and a negative
 // value otherwise (similar to PointArea).
 func SignedArea(a, b, c Point) float64 {
 	return float64(RobustSign(a, b, c)) * PointArea(a, b, c)
 }
 
 // Angle returns the interior angle at the vertex B in the triangle ABC. The
 // return value is always in the range [0, pi]. All points should be
 // normalized. Ensures that Angle(a,b,c) == Angle(c,b,a) for all a,b,c.
 //
 // The angle is undefined if A or C is diametrically opposite from B, and
 // becomes numerically unstable as the length of edge AB or BC approaches
 // 180 degrees.
 func Angle(a, b, c Point) s1.Angle {
 	// PointCross is necessary to get good accuracy when two of the input
 	// points are very close together.
 	return a.PointCross(b).Angle(c.PointCross(b).Vector)
 }
 
 // TurnAngle returns the exterior angle at vertex B in the triangle ABC. The
 // return value is positive if ABC is counterclockwise and negative otherwise.
 // If you imagine an ant walking from A to B to C, this is the angle that the
 // ant turns at vertex B (positive = left = CCW, negative = right = CW).
 // This quantity is also known as the "geodesic curvature" at B.
 //
 // Ensures that TurnAngle(a,b,c) == -TurnAngle(c,b,a) for all distinct
 // a,b,c. The result is undefined if (a == b || b == c), but is either
 // -Pi or Pi if (a == c). All points should be normalized.
 func TurnAngle(a, b, c Point) s1.Angle {
 	// We use PointCross to get good accuracy when two points are very
 	// close together, and RobustSign to ensure that the sign is correct for
 	// turns that are close to 180 degrees.
 	angle := a.PointCross(b).Angle(b.PointCross(c).Vector)
 
 	// Don't return RobustSign * angle because it is legal to have (a == c).
 	if RobustSign(a, b, c) == CounterClockwise {
 		return angle
 	}
 	return -angle
 }