Grand test fixup (#138)

* start fixing up tests

* fix up tests + automate with drone

* fiddle with linting

* messing about with drone.yml

* some more fiddling

* hmmm

* add cache

* add vendor directory

* verbose

* ci updates

* update some little things

* update sig

vendor/github.com/golang/geo/s2/point_measures.go

@@ -0,0 +1,149 @@
+// 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
+}