gtsocial-umbx

predicates.go (27138B)

---

 // 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