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

1 files changed, 701 insertions, 0 deletions

diff --git a/vendor/github.com/golang/geo/s2/predicates.go b/vendor/github.com/golang/geo/s2/predicates.go
new file mode 100644
index 000000000..9fc5e1751
--- /dev/null
+++ b/vendor/github.com/golang/geo/s2/predicates.go
@@ -0,0 +1,701 @@
+// Copyright 2016 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
+
+// This file contains various predicates that are guaranteed to produce
+// correct, consistent results. They are also relatively efficient. This is
+// achieved by computing conservative error bounds and falling back to high
+// precision or even exact arithmetic when the result is uncertain. Such
+// predicates are useful in implementing robust algorithms.
+//
+// See also EdgeCrosser, which implements various exact
+// edge-crossing predicates more efficiently than can be done here.
+
+import (
+	"math"
+	"math/big"
+
+	"github.com/golang/geo/r3"
+	"github.com/golang/geo/s1"
+)
+
+const (
+	// If any other machine architectures need to be suppported, these next three
+	// values will need to be updated.
+
+	// epsilon is a small number that represents a reasonable level of noise between two
+	// values that can be considered to be equal.
+	epsilon = 1e-15
+	// dblEpsilon is a smaller number for values that require more precision.
+	// This is the C++ DBL_EPSILON equivalent.
+	dblEpsilon = 2.220446049250313e-16
+	// dblError is the C++ value for S2 rounding_epsilon().
+	dblError = 1.110223024625156e-16
+
+	// maxDeterminantError is the maximum error in computing (AxB).C where all vectors
+	// are unit length. Using standard inequalities, it can be shown that
+	//
+	//  fl(AxB) = AxB + D where |D| <= (|AxB| + (2/sqrt(3))*|A|*|B|) * e
+	//
+	// where "fl()" denotes a calculation done in floating-point arithmetic,
+	// |x| denotes either absolute value or the L2-norm as appropriate, and
+	// e is a reasonably small value near the noise level of floating point
+	// number accuracy. Similarly,
+	//
+	//  fl(B.C) = B.C + d where |d| <= (|B.C| + 2*|B|*|C|) * e .
+	//
+	// Applying these bounds to the unit-length vectors A,B,C and neglecting
+	// relative error (which does not affect the sign of the result), we get
+	//
+	//  fl((AxB).C) = (AxB).C + d where |d| <= (3 + 2/sqrt(3)) * e
+	maxDeterminantError = 1.8274 * dblEpsilon
+
+	// detErrorMultiplier is the factor to scale the magnitudes by when checking
+	// for the sign of set of points with certainty. Using a similar technique to
+	// the one used for maxDeterminantError, the error is at most:
+	//
+	//   |d| <= (3 + 6/sqrt(3)) * |A-C| * |B-C| * e
+	//
+	// If the determinant magnitude is larger than this value then we know
+	// its sign with certainty.
+	detErrorMultiplier = 3.2321 * dblEpsilon
+)
+
+// Direction is an indication of the ordering of a set of points.
+type Direction int
+
+// These are the three options for the direction of a set of points.
+const (
+	Clockwise        Direction = -1
+	Indeterminate    Direction = 0
+	CounterClockwise Direction = 1
+)
+
+// newBigFloat constructs a new big.Float with maximum precision.
+func newBigFloat() *big.Float { return new(big.Float).SetPrec(big.MaxPrec) }
+
+// Sign returns true if the points A, B, C are strictly counterclockwise,
+// and returns false if the points are clockwise or collinear (i.e. if they are all
+// contained on some great circle).
+//
+// Due to numerical errors, situations may arise that are mathematically
+// impossible, e.g. ABC may be considered strictly CCW while BCA is not.
+// However, the implementation guarantees the following:
+//
+// If Sign(a,b,c), then !Sign(c,b,a) for all a,b,c.
+func Sign(a, b, c Point) bool {
+	// NOTE(dnadasi): In the C++ API the equivalent method here was known as "SimpleSign".
+
+	// We compute the signed volume of the parallelepiped ABC. The usual
+	// formula for this is (A ⨯ B) · C, but we compute it here using (C ⨯ A) · B
+	// in order to ensure that ABC and CBA are not both CCW. This follows
+	// from the following identities (which are true numerically, not just
+	// mathematically):
+	//
+	//     (1) x ⨯ y == -(y ⨯ x)
+	//     (2) -x · y == -(x · y)
+	return c.Cross(a.Vector).Dot(b.Vector) > 0
+}
+
+// RobustSign returns a Direction representing the ordering of the points.
+// CounterClockwise is returned if the points are in counter-clockwise order,
+// Clockwise for clockwise, and Indeterminate if any two points are the same (collinear),
+// or the sign could not completely be determined.
+//
+// This function has additional logic to make sure that the above properties hold even
+// when the three points are coplanar, and to deal with the limitations of
+// floating-point arithmetic.
+//
+// RobustSign satisfies the following conditions:
+//
+//  (1) RobustSign(a,b,c) == Indeterminate if and only if a == b, b == c, or c == a
+//  (2) RobustSign(b,c,a) == RobustSign(a,b,c) for all a,b,c
+//  (3) RobustSign(c,b,a) == -RobustSign(a,b,c) for all a,b,c
+//
+// In other words:
+//
+//  (1) The result is Indeterminate if and only if two points are the same.
+//  (2) Rotating the order of the arguments does not affect the result.
+//  (3) Exchanging any two arguments inverts the result.
+//
+// On the other hand, note that it is not true in general that
+// RobustSign(-a,b,c) == -RobustSign(a,b,c), or any similar identities
+// involving antipodal points.
+func RobustSign(a, b, c Point) Direction {
+	sign := triageSign(a, b, c)
+	if sign == Indeterminate {
+		sign = expensiveSign(a, b, c)
+	}
+	return sign
+}
+
+// stableSign reports the direction sign of the points in a numerically stable way.
+// Unlike triageSign, this method can usually compute the correct determinant sign
+// even when all three points are as collinear as possible. For example if three
+// points are spaced 1km apart along a random line on the Earth's surface using
+// the nearest representable points, there is only a 0.4% chance that this method
+// will not be able to find the determinant sign. The probability of failure
+// decreases as the points get closer together; if the collinear points are 1 meter
+// apart, the failure rate drops to 0.0004%.
+//
+// This method could be extended to also handle nearly-antipodal points, but antipodal
+// points are rare in practice so it seems better to simply fall back to
+// exact arithmetic in that case.
+func stableSign(a, b, c Point) Direction {
+	ab := b.Sub(a.Vector)
+	ab2 := ab.Norm2()
+	bc := c.Sub(b.Vector)
+	bc2 := bc.Norm2()
+	ca := a.Sub(c.Vector)
+	ca2 := ca.Norm2()
+
+	// Now compute the determinant ((A-C)x(B-C)).C, where the vertices have been
+	// cyclically permuted if necessary so that AB is the longest edge. (This
+	// minimizes the magnitude of cross product.)  At the same time we also
+	// compute the maximum error in the determinant.
+
+	// The two shortest edges, pointing away from their common point.
+	var e1, e2, op r3.Vector
+	if ab2 >= bc2 && ab2 >= ca2 {
+		// AB is the longest edge.
+		e1, e2, op = ca, bc, c.Vector
+	} else if bc2 >= ca2 {
+		// BC is the longest edge.
+		e1, e2, op = ab, ca, a.Vector
+	} else {
+		// CA is the longest edge.
+		e1, e2, op = bc, ab, b.Vector
+	}
+
+	det := -e1.Cross(e2).Dot(op)
+	maxErr := detErrorMultiplier * math.Sqrt(e1.Norm2()*e2.Norm2())
+
+	// If the determinant isn't zero, within maxErr, we know definitively the point ordering.
+	if det > maxErr {
+		return CounterClockwise
+	}
+	if det < -maxErr {
+		return Clockwise
+	}
+	return Indeterminate
+}
+
+// triageSign returns the direction sign of the points. It returns Indeterminate if two
+// points are identical or the result is uncertain. Uncertain cases can be resolved, if
+// desired, by calling expensiveSign.
+//
+// The purpose of this method is to allow additional cheap tests to be done without
+// calling expensiveSign.
+func triageSign(a, b, c Point) Direction {
+	det := a.Cross(b.Vector).Dot(c.Vector)
+	if det > maxDeterminantError {
+		return CounterClockwise
+	}
+	if det < -maxDeterminantError {
+		return Clockwise
+	}
+	return Indeterminate
+}
+
+// expensiveSign reports the direction sign of the points. It returns Indeterminate
+// if two of the input points are the same. It uses multiple-precision arithmetic
+// to ensure that its results are always self-consistent.
+func expensiveSign(a, b, c Point) Direction {
+	// Return Indeterminate if and only if two points are the same.
+	// This ensures RobustSign(a,b,c) == Indeterminate if and only if a == b, b == c, or c == a.
+	// ie. Property 1 of RobustSign.
+	if a == b || b == c || c == a {
+		return Indeterminate
+	}
+
+	// Next we try recomputing the determinant still using floating-point
+	// arithmetic but in a more precise way. This is more expensive than the
+	// simple calculation done by triageSign, but it is still *much* cheaper
+	// than using arbitrary-precision arithmetic. This optimization is able to
+	// compute the correct determinant sign in virtually all cases except when
+	// the three points are truly collinear (e.g., three points on the equator).
+	detSign := stableSign(a, b, c)
+	if detSign != Indeterminate {
+		return detSign
+	}
+
+	// Otherwise fall back to exact arithmetic and symbolic permutations.
+	return exactSign(a, b, c, true)
+}
+
+// exactSign reports the direction sign of the points computed using high-precision
+// arithmetic and/or symbolic perturbations.
+func exactSign(a, b, c Point, perturb bool) Direction {
+	// Sort the three points in lexicographic order, keeping track of the sign
+	// of the permutation. (Each exchange inverts the sign of the determinant.)
+	permSign := CounterClockwise
+	pa := &a
+	pb := &b
+	pc := &c
+	if pa.Cmp(pb.Vector) > 0 {
+		pa, pb = pb, pa
+		permSign = -permSign
+	}
+	if pb.Cmp(pc.Vector) > 0 {
+		pb, pc = pc, pb
+		permSign = -permSign
+	}
+	if pa.Cmp(pb.Vector) > 0 {
+		pa, pb = pb, pa
+		permSign = -permSign
+	}
+
+	// Construct multiple-precision versions of the sorted points and compute
+	// their precise 3x3 determinant.
+	xa := r3.PreciseVectorFromVector(pa.Vector)
+	xb := r3.PreciseVectorFromVector(pb.Vector)
+	xc := r3.PreciseVectorFromVector(pc.Vector)
+	xbCrossXc := xb.Cross(xc)
+	det := xa.Dot(xbCrossXc)
+
+	// The precision of big.Float is high enough that the result should always
+	// be exact enough (no rounding was performed).
+
+	// If the exact determinant is non-zero, we're done.
+	detSign := Direction(det.Sign())
+	if detSign == Indeterminate && perturb {
+		// Otherwise, we need to resort to symbolic perturbations to resolve the
+		// sign of the determinant.
+		detSign = symbolicallyPerturbedSign(xa, xb, xc, xbCrossXc)
+	}
+	return permSign * detSign
+}
+
+// symbolicallyPerturbedSign reports the sign of the determinant of three points
+// A, B, C under a model where every possible Point is slightly perturbed by
+// a unique infinitesmal amount such that no three perturbed points are
+// collinear and no four points are coplanar. The perturbations are so small
+// that they do not change the sign of any determinant that was non-zero
+// before the perturbations, and therefore can be safely ignored unless the
+// determinant of three points is exactly zero (using multiple-precision
+// arithmetic). This returns CounterClockwise or Clockwise according to the
+// sign of the determinant after the symbolic perturbations are taken into account.
+//
+// Since the symbolic perturbation of a given point is fixed (i.e., the
+// perturbation is the same for all calls to this method and does not depend
+// on the other two arguments), the results of this method are always
+// self-consistent. It will never return results that would correspond to an
+// impossible configuration of non-degenerate points.
+//
+// This requires that the 3x3 determinant of A, B, C must be exactly zero.
+// And the points must be distinct, with A < B < C in lexicographic order.
+//
+// Reference:
+//   "Simulation of Simplicity" (Edelsbrunner and Muecke, ACM Transactions on
+//   Graphics, 1990).
+//
+func symbolicallyPerturbedSign(a, b, c, bCrossC r3.PreciseVector) Direction {
+	// This method requires that the points are sorted in lexicographically
+	// increasing order. This is because every possible Point has its own
+	// symbolic perturbation such that if A < B then the symbolic perturbation
+	// for A is much larger than the perturbation for B.
+	//
+	// Alternatively, we could sort the points in this method and keep track of
+	// the sign of the permutation, but it is more efficient to do this before
+	// converting the inputs to the multi-precision representation, and this
+	// also lets us re-use the result of the cross product B x C.
+	//
+	// Every input coordinate x[i] is assigned a symbolic perturbation dx[i].
+	// We then compute the sign of the determinant of the perturbed points,
+	// i.e.
+	//               | a.X+da.X  a.Y+da.Y  a.Z+da.Z |
+	//               | b.X+db.X  b.Y+db.Y  b.Z+db.Z |
+	//               | c.X+dc.X  c.Y+dc.Y  c.Z+dc.Z |
+	//
+	// The perturbations are chosen such that
+	//
+	//   da.Z > da.Y > da.X > db.Z > db.Y > db.X > dc.Z > dc.Y > dc.X
+	//
+	// where each perturbation is so much smaller than the previous one that we
+	// don't even need to consider it unless the coefficients of all previous
+	// perturbations are zero. In fact, it is so small that we don't need to
+	// consider it unless the coefficient of all products of the previous
+	// perturbations are zero. For example, we don't need to consider the
+	// coefficient of db.Y unless the coefficient of db.Z *da.X is zero.
+	//
+	// The follow code simply enumerates the coefficients of the perturbations
+	// (and products of perturbations) that appear in the determinant above, in
+	// order of decreasing perturbation magnitude. The first non-zero
+	// coefficient determines the sign of the result. The easiest way to
+	// enumerate the coefficients in the correct order is to pretend that each
+	// perturbation is some tiny value "eps" raised to a power of two:
+	//
+	// eps**     1      2      4      8     16     32     64    128    256
+	//        da.Z   da.Y   da.X   db.Z   db.Y   db.X   dc.Z   dc.Y   dc.X
+	//
+	// Essentially we can then just count in binary and test the corresponding
+	// subset of perturbations at each step. So for example, we must test the
+	// coefficient of db.Z*da.X before db.Y because eps**12 > eps**16.
+	//
+	// Of course, not all products of these perturbations appear in the
+	// determinant above, since the determinant only contains the products of
+	// elements in distinct rows and columns. Thus we don't need to consider
+	// da.Z*da.Y, db.Y *da.Y, etc. Furthermore, sometimes different pairs of
+	// perturbations have the same coefficient in the determinant; for example,
+	// da.Y*db.X and db.Y*da.X have the same coefficient (c.Z). Therefore
+	// we only need to test this coefficient the first time we encounter it in
+	// the binary order above (which will be db.Y*da.X).
+	//
+	// The sequence of tests below also appears in Table 4-ii of the paper
+	// referenced above, if you just want to look it up, with the following
+	// translations: [a,b,c] -> [i,j,k] and [0,1,2] -> [1,2,3]. Also note that
+	// some of the signs are different because the opposite cross product is
+	// used (e.g., B x C rather than C x B).
+
+	detSign := bCrossC.Z.Sign() // da.Z
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+	detSign = bCrossC.Y.Sign() // da.Y
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+	detSign = bCrossC.X.Sign() // da.X
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+
+	detSign = newBigFloat().Sub(newBigFloat().Mul(c.X, a.Y), newBigFloat().Mul(c.Y, a.X)).Sign() // db.Z
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+	detSign = c.X.Sign() // db.Z * da.Y
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+	detSign = -(c.Y.Sign()) // db.Z * da.X
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+
+	detSign = newBigFloat().Sub(newBigFloat().Mul(c.Z, a.X), newBigFloat().Mul(c.X, a.Z)).Sign() // db.Y
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+	detSign = c.Z.Sign() // db.Y * da.X
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+
+	// The following test is listed in the paper, but it is redundant because
+	// the previous tests guarantee that C == (0, 0, 0).
+	// (c.Y*a.Z - c.Z*a.Y).Sign() // db.X
+
+	detSign = newBigFloat().Sub(newBigFloat().Mul(a.X, b.Y), newBigFloat().Mul(a.Y, b.X)).Sign() // dc.Z
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+	detSign = -(b.X.Sign()) // dc.Z * da.Y
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+	detSign = b.Y.Sign() // dc.Z * da.X
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+	detSign = a.X.Sign() // dc.Z * db.Y
+	if detSign != 0 {
+		return Direction(detSign)
+	}
+	return CounterClockwise // dc.Z * db.Y * da.X
+}
+
+// CompareDistances returns -1, 0, or +1 according to whether AX < BX, A == B,
+// or AX > BX respectively. Distances are measured with respect to the positions
+// of X, A, and B as though they were reprojected to lie exactly on the surface of
+// the unit sphere. Furthermore, this method uses symbolic perturbations to
+// ensure that the result is non-zero whenever A != B, even when AX == BX
+// exactly, or even when A and B project to the same point on the sphere.
+// Such results are guaranteed to be self-consistent, i.e. if AB < BC and
+// BC < AC, then AB < AC.
+func CompareDistances(x, a, b Point) int {
+	// We start by comparing distances using dot products (i.e., cosine of the
+	// angle), because (1) this is the cheapest technique, and (2) it is valid
+	// over the entire range of possible angles. (We can only use the sin^2
+	// technique if both angles are less than 90 degrees or both angles are
+	// greater than 90 degrees.)
+	sign := triageCompareCosDistances(x, a, b)
+	if sign != 0 {
+		return sign
+	}
+
+	// Optimization for (a == b) to avoid falling back to exact arithmetic.
+	if a == b {
+		return 0
+	}
+
+	// It is much better numerically to compare distances using cos(angle) if
+	// the distances are near 90 degrees and sin^2(angle) if the distances are
+	// near 0 or 180 degrees. We only need to check one of the two angles when
+	// making this decision because the fact that the test above failed means
+	// that angles "a" and "b" are very close together.
+	cosAX := a.Dot(x.Vector)
+	if cosAX > 1/math.Sqrt2 {
+		// Angles < 45 degrees.
+		sign = triageCompareSin2Distances(x, a, b)
+	} else if cosAX < -1/math.Sqrt2 {
+		// Angles > 135 degrees. sin^2(angle) is decreasing in this range.
+		sign = -triageCompareSin2Distances(x, a, b)
+	}
+	// C++ adds an additional check here using 80-bit floats.
+	// This is skipped in Go because we only have 32 and 64 bit floats.
+
+	if sign != 0 {
+		return sign
+	}
+
+	sign = exactCompareDistances(r3.PreciseVectorFromVector(x.Vector), r3.PreciseVectorFromVector(a.Vector), r3.PreciseVectorFromVector(b.Vector))
+	if sign != 0 {
+		return sign
+	}
+	return symbolicCompareDistances(x, a, b)
+}
+
+// cosDistance returns cos(XY) where XY is the angle between X and Y, and the
+// maximum error amount in the result. This requires X and Y be normalized.
+func cosDistance(x, y Point) (cos, err float64) {
+	cos = x.Dot(y.Vector)
+	return cos, 9.5*dblError*math.Abs(cos) + 1.5*dblError
+}
+
+// sin2Distance returns sin**2(XY), where XY is the angle between X and Y,
+// and the maximum error amount in the result. This requires X and Y be normalized.
+func sin2Distance(x, y Point) (sin2, err float64) {
+	// The (x-y).Cross(x+y) trick eliminates almost all of error due to x
+	// and y being not quite unit length. This method is extremely accurate
+	// for small distances; the *relative* error in the result is O(dblError) for
+	// distances as small as dblError.
+	n := x.Sub(y.Vector).Cross(x.Add(y.Vector))
+	sin2 = 0.25 * n.Norm2()
+	err = ((21+4*math.Sqrt(3))*dblError*sin2 +
+		32*math.Sqrt(3)*dblError*dblError*math.Sqrt(sin2) +
+		768*dblError*dblError*dblError*dblError)
+	return sin2, err
+}
+
+// triageCompareCosDistances returns -1, 0, or +1 according to whether AX < BX,
+// A == B, or AX > BX by comparing the distances between them using cosDistance.
+func triageCompareCosDistances(x, a, b Point) int {
+	cosAX, cosAXerror := cosDistance(a, x)
+	cosBX, cosBXerror := cosDistance(b, x)
+	diff := cosAX - cosBX
+	err := cosAXerror + cosBXerror
+	if diff > err {
+		return -1
+	}
+	if diff < -err {
+		return 1
+	}
+	return 0
+}
+
+// triageCompareSin2Distances returns -1, 0, or +1 according to whether AX < BX,
+// A == B, or AX > BX by comparing the distances between them using sin2Distance.
+func triageCompareSin2Distances(x, a, b Point) int {
+	sin2AX, sin2AXerror := sin2Distance(a, x)
+	sin2BX, sin2BXerror := sin2Distance(b, x)
+	diff := sin2AX - sin2BX
+	err := sin2AXerror + sin2BXerror
+	if diff > err {
+		return 1
+	}
+	if diff < -err {
+		return -1
+	}
+	return 0
+}
+
+// exactCompareDistances returns -1, 0, or 1 after comparing using the values as
+// PreciseVectors.
+func exactCompareDistances(x, a, b r3.PreciseVector) int {
+	// This code produces the same result as though all points were reprojected
+	// to lie exactly on the surface of the unit sphere. It is based on testing
+	// whether x.Dot(a.Normalize()) < x.Dot(b.Normalize()), reformulated
+	// so that it can be evaluated using exact arithmetic.
+	cosAX := x.Dot(a)
+	cosBX := x.Dot(b)
+
+	// If the two values have different signs, we need to handle that case now
+	// before squaring them below.
+	aSign := cosAX.Sign()
+	bSign := cosBX.Sign()
+	if aSign != bSign {
+		// If cos(AX) > cos(BX), then AX < BX.
+		if aSign > bSign {
+			return -1
+		}
+		return 1
+	}
+	cosAX2 := newBigFloat().Mul(cosAX, cosAX)
+	cosBX2 := newBigFloat().Mul(cosBX, cosBX)
+	cmp := newBigFloat().Sub(cosBX2.Mul(cosBX2, a.Norm2()), cosAX2.Mul(cosAX2, b.Norm2()))
+	return aSign * cmp.Sign()
+}
+
+// symbolicCompareDistances returns -1, 0, or +1 given three points such that AX == BX
+// (exactly) according to whether AX < BX, AX == BX, or AX > BX after symbolic
+// perturbations are taken into account.
+func symbolicCompareDistances(x, a, b Point) int {
+	// Our symbolic perturbation strategy is based on the following model.
+	// Similar to "simulation of simplicity", we assign a perturbation to every
+	// point such that if A < B, then the symbolic perturbation for A is much,
+	// much larger than the symbolic perturbation for B. We imagine that
+	// rather than projecting every point to lie exactly on the unit sphere,
+	// instead each point is positioned on its own tiny pedestal that raises it
+	// just off the surface of the unit sphere. This means that the distance AX
+	// is actually the true distance AX plus the (symbolic) heights of the
+	// pedestals for A and X. The pedestals are infinitesmally thin, so they do
+	// not affect distance measurements except at the two endpoints. If several
+	// points project to exactly the same point on the unit sphere, we imagine
+	// that they are placed on separate pedestals placed close together, where
+	// the distance between pedestals is much, much less than the height of any
+	// pedestal. (There are a finite number of Points, and therefore a finite
+	// number of pedestals, so this is possible.)
+	//
+	// If A < B, then A is on a higher pedestal than B, and therefore AX > BX.
+	switch a.Cmp(b.Vector) {
+	case -1:
+		return 1
+	case 1:
+		return -1
+	default:
+		return 0
+	}
+}
+
+var (
+	// ca45Degrees is a predefined ChordAngle representing (approximately) 45 degrees.
+	ca45Degrees = s1.ChordAngleFromSquaredLength(2 - math.Sqrt2)
+)
+
+// CompareDistance returns -1, 0, or +1 according to whether the distance XY is
+// respectively less than, equal to, or greater than the provided chord angle. Distances are measured
+// with respect to the positions of all points as though they are projected to lie
+// exactly on the surface of the unit sphere.
+func CompareDistance(x, y Point, r s1.ChordAngle) int {
+	// As with CompareDistances, we start by comparing dot products because
+	// the sin^2 method is only valid when the distance XY and the limit "r" are
+	// both less than 90 degrees.
+	sign := triageCompareCosDistance(x, y, float64(r))
+	if sign != 0 {
+		return sign
+	}
+
+	// Unlike with CompareDistances, it's not worth using the sin^2 method
+	// when the distance limit is near 180 degrees because the ChordAngle
+	// representation itself has has a rounding error of up to 2e-8 radians for
+	// distances near 180 degrees.
+	if r < ca45Degrees {
+		sign = triageCompareSin2Distance(x, y, float64(r))
+		if sign != 0 {
+			return sign
+		}
+	}
+	return exactCompareDistance(r3.PreciseVectorFromVector(x.Vector), r3.PreciseVectorFromVector(y.Vector), big.NewFloat(float64(r)).SetPrec(big.MaxPrec))
+}
+
+// triageCompareCosDistance returns -1, 0, or +1 according to whether the distance XY is
+// less than, equal to, or greater than r2 respectively using cos distance.
+func triageCompareCosDistance(x, y Point, r2 float64) int {
+	cosXY, cosXYError := cosDistance(x, y)
+	cosR := 1.0 - 0.5*r2
+	cosRError := 2.0 * dblError * cosR
+	diff := cosXY - cosR
+	err := cosXYError + cosRError
+	if diff > err {
+		return -1
+	}
+	if diff < -err {
+		return 1
+	}
+	return 0
+}
+
+// triageCompareSin2Distance returns -1, 0, or +1 according to whether the distance XY is
+// less than, equal to, or greater than r2 respectively using sin^2 distance.
+func triageCompareSin2Distance(x, y Point, r2 float64) int {
+	// Only valid for distance limits < 90 degrees.
+	sin2XY, sin2XYError := sin2Distance(x, y)
+	sin2R := r2 * (1.0 - 0.25*r2)
+	sin2RError := 3.0 * dblError * sin2R
+	diff := sin2XY - sin2R
+	err := sin2XYError + sin2RError
+	if diff > err {
+		return 1
+	}
+	if diff < -err {
+		return -1
+	}
+	return 0
+}
+
+var (
+	bigOne  = big.NewFloat(1.0).SetPrec(big.MaxPrec)
+	bigHalf = big.NewFloat(0.5).SetPrec(big.MaxPrec)
+)
+
+// exactCompareDistance returns -1, 0, or +1 after comparing using PreciseVectors.
+func exactCompareDistance(x, y r3.PreciseVector, r2 *big.Float) int {
+	// This code produces the same result as though all points were reprojected
+	// to lie exactly on the surface of the unit sphere.  It is based on
+	// comparing the cosine of the angle XY (when both points are projected to
+	// lie exactly on the sphere) to the given threshold.
+	cosXY := x.Dot(y)
+	cosR := newBigFloat().Sub(bigOne, newBigFloat().Mul(bigHalf, r2))
+
+	// If the two values have different signs, we need to handle that case now
+	// before squaring them below.
+	xySign := cosXY.Sign()
+	rSign := cosR.Sign()
+	if xySign != rSign {
+		if xySign > rSign {
+			return -1
+		}
+		return 1 // If cos(XY) > cos(r), then XY < r.
+	}
+	cmp := newBigFloat().Sub(
+		newBigFloat().Mul(
+			newBigFloat().Mul(cosR, cosR), newBigFloat().Mul(x.Norm2(), y.Norm2())),
+		newBigFloat().Mul(cosXY, cosXY))
+	return xySign * cmp.Sign()
+}
+
+// TODO(roberts): Differences from C++
+// CompareEdgeDistance
+// CompareEdgeDirections
+// EdgeCircumcenterSign
+// GetVoronoiSiteExclusion
+// GetClosestVertex
+// TriageCompareLineSin2Distance
+// TriageCompareLineCos2Distance
+// TriageCompareLineDistance
+// TriageCompareEdgeDistance
+// ExactCompareLineDistance
+// ExactCompareEdgeDistance
+// TriageCompareEdgeDirections
+// ExactCompareEdgeDirections
+// ArePointsAntipodal
+// ArePointsLinearlyDependent
+// GetCircumcenter
+// TriageEdgeCircumcenterSign
+// ExactEdgeCircumcenterSign
+// UnperturbedSign
+// SymbolicEdgeCircumcenterSign
+// ExactVoronoiSiteExclusion