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// 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
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