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, 710 insertions, 0 deletions

diff --git a/vendor/github.com/golang/geo/s2/rect.go b/vendor/github.com/golang/geo/s2/rect.go
new file mode 100644
index 000000000..f6b52a59e
--- /dev/null
+++ b/vendor/github.com/golang/geo/s2/rect.go
@@ -0,0 +1,710 @@
+// Copyright 2014 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 (
+	"fmt"
+	"io"
+	"math"
+
+	"github.com/golang/geo/r1"
+	"github.com/golang/geo/r3"
+	"github.com/golang/geo/s1"
+)
+
+// Rect represents a closed latitude-longitude rectangle.
+type Rect struct {
+	Lat r1.Interval
+	Lng s1.Interval
+}
+
+var (
+	validRectLatRange = r1.Interval{-math.Pi / 2, math.Pi / 2}
+	validRectLngRange = s1.FullInterval()
+)
+
+// EmptyRect returns the empty rectangle.
+func EmptyRect() Rect { return Rect{r1.EmptyInterval(), s1.EmptyInterval()} }
+
+// FullRect returns the full rectangle.
+func FullRect() Rect { return Rect{validRectLatRange, validRectLngRange} }
+
+// RectFromLatLng constructs a rectangle containing a single point p.
+func RectFromLatLng(p LatLng) Rect {
+	return Rect{
+		Lat: r1.Interval{p.Lat.Radians(), p.Lat.Radians()},
+		Lng: s1.Interval{p.Lng.Radians(), p.Lng.Radians()},
+	}
+}
+
+// RectFromCenterSize constructs a rectangle with the given size and center.
+// center needs to be normalized, but size does not. The latitude
+// interval of the result is clamped to [-90,90] degrees, and the longitude
+// interval of the result is FullRect() if and only if the longitude size is
+// 360 degrees or more.
+//
+// Examples of clamping (in degrees):
+//   center=(80,170),  size=(40,60)   -> lat=[60,90],   lng=[140,-160]
+//   center=(10,40),   size=(210,400) -> lat=[-90,90],  lng=[-180,180]
+//   center=(-90,180), size=(20,50)   -> lat=[-90,-80], lng=[155,-155]
+func RectFromCenterSize(center, size LatLng) Rect {
+	half := LatLng{size.Lat / 2, size.Lng / 2}
+	return RectFromLatLng(center).expanded(half)
+}
+
+// IsValid returns true iff the rectangle is valid.
+// This requires Lat ⊆ [-π/2,π/2] and Lng ⊆ [-π,π], and Lat = ∅ iff Lng = ∅
+func (r Rect) IsValid() bool {
+	return math.Abs(r.Lat.Lo) <= math.Pi/2 &&
+		math.Abs(r.Lat.Hi) <= math.Pi/2 &&
+		r.Lng.IsValid() &&
+		r.Lat.IsEmpty() == r.Lng.IsEmpty()
+}
+
+// IsEmpty reports whether the rectangle is empty.
+func (r Rect) IsEmpty() bool { return r.Lat.IsEmpty() }
+
+// IsFull reports whether the rectangle is full.
+func (r Rect) IsFull() bool { return r.Lat.Equal(validRectLatRange) && r.Lng.IsFull() }
+
+// IsPoint reports whether the rectangle is a single point.
+func (r Rect) IsPoint() bool { return r.Lat.Lo == r.Lat.Hi && r.Lng.Lo == r.Lng.Hi }
+
+// Vertex returns the i-th vertex of the rectangle (i = 0,1,2,3) in CCW order
+// (lower left, lower right, upper right, upper left).
+func (r Rect) Vertex(i int) LatLng {
+	var lat, lng float64
+
+	switch i {
+	case 0:
+		lat = r.Lat.Lo
+		lng = r.Lng.Lo
+	case 1:
+		lat = r.Lat.Lo
+		lng = r.Lng.Hi
+	case 2:
+		lat = r.Lat.Hi
+		lng = r.Lng.Hi
+	case 3:
+		lat = r.Lat.Hi
+		lng = r.Lng.Lo
+	}
+	return LatLng{s1.Angle(lat) * s1.Radian, s1.Angle(lng) * s1.Radian}
+}
+
+// Lo returns one corner of the rectangle.
+func (r Rect) Lo() LatLng {
+	return LatLng{s1.Angle(r.Lat.Lo) * s1.Radian, s1.Angle(r.Lng.Lo) * s1.Radian}
+}
+
+// Hi returns the other corner of the rectangle.
+func (r Rect) Hi() LatLng {
+	return LatLng{s1.Angle(r.Lat.Hi) * s1.Radian, s1.Angle(r.Lng.Hi) * s1.Radian}
+}
+
+// Center returns the center of the rectangle.
+func (r Rect) Center() LatLng {
+	return LatLng{s1.Angle(r.Lat.Center()) * s1.Radian, s1.Angle(r.Lng.Center()) * s1.Radian}
+}
+
+// Size returns the size of the Rect.
+func (r Rect) Size() LatLng {
+	return LatLng{s1.Angle(r.Lat.Length()) * s1.Radian, s1.Angle(r.Lng.Length()) * s1.Radian}
+}
+
+// Area returns the surface area of the Rect.
+func (r Rect) Area() float64 {
+	if r.IsEmpty() {
+		return 0
+	}
+	capDiff := math.Abs(math.Sin(r.Lat.Hi) - math.Sin(r.Lat.Lo))
+	return r.Lng.Length() * capDiff
+}
+
+// AddPoint increases the size of the rectangle to include the given point.
+func (r Rect) AddPoint(ll LatLng) Rect {
+	if !ll.IsValid() {
+		return r
+	}
+	return Rect{
+		Lat: r.Lat.AddPoint(ll.Lat.Radians()),
+		Lng: r.Lng.AddPoint(ll.Lng.Radians()),
+	}
+}
+
+// expanded returns a rectangle that has been expanded by margin.Lat on each side
+// in the latitude direction, and by margin.Lng on each side in the longitude
+// direction. If either margin is negative, then it shrinks the rectangle on
+// the corresponding sides instead. The resulting rectangle may be empty.
+//
+// The latitude-longitude space has the topology of a cylinder. Longitudes
+// "wrap around" at +/-180 degrees, while latitudes are clamped to range [-90, 90].
+// This means that any expansion (positive or negative) of the full longitude range
+// remains full (since the "rectangle" is actually a continuous band around the
+// cylinder), while expansion of the full latitude range remains full only if the
+// margin is positive.
+//
+// If either the latitude or longitude interval becomes empty after
+// expansion by a negative margin, the result is empty.
+//
+// Note that if an expanded rectangle contains a pole, it may not contain
+// all possible lat/lng representations of that pole, e.g., both points [π/2,0]
+// and [π/2,1] represent the same pole, but they might not be contained by the
+// same Rect.
+//
+// If you are trying to grow a rectangle by a certain distance on the
+// sphere (e.g. 5km), refer to the ExpandedByDistance() C++ method implementation
+// instead.
+func (r Rect) expanded(margin LatLng) Rect {
+	lat := r.Lat.Expanded(margin.Lat.Radians())
+	lng := r.Lng.Expanded(margin.Lng.Radians())
+
+	if lat.IsEmpty() || lng.IsEmpty() {
+		return EmptyRect()
+	}
+
+	return Rect{
+		Lat: lat.Intersection(validRectLatRange),
+		Lng: lng,
+	}
+}
+
+func (r Rect) String() string { return fmt.Sprintf("[Lo%v, Hi%v]", r.Lo(), r.Hi()) }
+
+// PolarClosure returns the rectangle unmodified if it does not include either pole.
+// If it includes either pole, PolarClosure returns an expansion of the rectangle along
+// the longitudinal range to include all possible representations of the contained poles.
+func (r Rect) PolarClosure() Rect {
+	if r.Lat.Lo == -math.Pi/2 || r.Lat.Hi == math.Pi/2 {
+		return Rect{r.Lat, s1.FullInterval()}
+	}
+	return r
+}
+
+// Union returns the smallest Rect containing the union of this rectangle and the given rectangle.
+func (r Rect) Union(other Rect) Rect {
+	return Rect{
+		Lat: r.Lat.Union(other.Lat),
+		Lng: r.Lng.Union(other.Lng),
+	}
+}
+
+// Intersection returns the smallest rectangle containing the intersection of
+// this rectangle and the given rectangle. Note that the region of intersection
+// may consist of two disjoint rectangles, in which case a single rectangle
+// spanning both of them is returned.
+func (r Rect) Intersection(other Rect) Rect {
+	lat := r.Lat.Intersection(other.Lat)
+	lng := r.Lng.Intersection(other.Lng)
+
+	if lat.IsEmpty() || lng.IsEmpty() {
+		return EmptyRect()
+	}
+	return Rect{lat, lng}
+}
+
+// Intersects reports whether this rectangle and the other have any points in common.
+func (r Rect) Intersects(other Rect) bool {
+	return r.Lat.Intersects(other.Lat) && r.Lng.Intersects(other.Lng)
+}
+
+// CapBound returns a cap that contains Rect.
+func (r Rect) CapBound() Cap {
+	// We consider two possible bounding caps, one whose axis passes
+	// through the center of the lat-long rectangle and one whose axis
+	// is the north or south pole.  We return the smaller of the two caps.
+
+	if r.IsEmpty() {
+		return EmptyCap()
+	}
+
+	var poleZ, poleAngle float64
+	if r.Lat.Hi+r.Lat.Lo < 0 {
+		// South pole axis yields smaller cap.
+		poleZ = -1
+		poleAngle = math.Pi/2 + r.Lat.Hi
+	} else {
+		poleZ = 1
+		poleAngle = math.Pi/2 - r.Lat.Lo
+	}
+	poleCap := CapFromCenterAngle(Point{r3.Vector{0, 0, poleZ}}, s1.Angle(poleAngle)*s1.Radian)
+
+	// For bounding rectangles that span 180 degrees or less in longitude, the
+	// maximum cap size is achieved at one of the rectangle vertices.  For
+	// rectangles that are larger than 180 degrees, we punt and always return a
+	// bounding cap centered at one of the two poles.
+	if math.Remainder(r.Lng.Hi-r.Lng.Lo, 2*math.Pi) >= 0 && r.Lng.Hi-r.Lng.Lo < 2*math.Pi {
+		midCap := CapFromPoint(PointFromLatLng(r.Center())).AddPoint(PointFromLatLng(r.Lo())).AddPoint(PointFromLatLng(r.Hi()))
+		if midCap.Height() < poleCap.Height() {
+			return midCap
+		}
+	}
+	return poleCap
+}
+
+// RectBound returns itself.
+func (r Rect) RectBound() Rect {
+	return r
+}
+
+// Contains reports whether this Rect contains the other Rect.
+func (r Rect) Contains(other Rect) bool {
+	return r.Lat.ContainsInterval(other.Lat) && r.Lng.ContainsInterval(other.Lng)
+}
+
+// ContainsCell reports whether the given Cell is contained by this Rect.
+func (r Rect) ContainsCell(c Cell) bool {
+	// A latitude-longitude rectangle contains a cell if and only if it contains
+	// the cell's bounding rectangle. This test is exact from a mathematical
+	// point of view, assuming that the bounds returned by Cell.RectBound()
+	// are tight. However, note that there can be a loss of precision when
+	// converting between representations -- for example, if an s2.Cell is
+	// converted to a polygon, the polygon's bounding rectangle may not contain
+	// the cell's bounding rectangle. This has some slightly unexpected side
+	// effects; for instance, if one creates an s2.Polygon from an s2.Cell, the
+	// polygon will contain the cell, but the polygon's bounding box will not.
+	return r.Contains(c.RectBound())
+}
+
+// ContainsLatLng reports whether the given LatLng is within the Rect.
+func (r Rect) ContainsLatLng(ll LatLng) bool {
+	if !ll.IsValid() {
+		return false
+	}
+	return r.Lat.Contains(ll.Lat.Radians()) && r.Lng.Contains(ll.Lng.Radians())
+}
+
+// ContainsPoint reports whether the given Point is within the Rect.
+func (r Rect) ContainsPoint(p Point) bool {
+	return r.ContainsLatLng(LatLngFromPoint(p))
+}
+
+// CellUnionBound computes a covering of the Rect.
+func (r Rect) CellUnionBound() []CellID {
+	return r.CapBound().CellUnionBound()
+}
+
+// intersectsLatEdge reports whether the edge AB intersects the given edge of constant
+// latitude. Requires the points to have unit length.
+func intersectsLatEdge(a, b Point, lat s1.Angle, lng s1.Interval) bool {
+	// Unfortunately, lines of constant latitude are curves on
+	// the sphere. They can intersect a straight edge in 0, 1, or 2 points.
+
+	// First, compute the normal to the plane AB that points vaguely north.
+	z := Point{a.PointCross(b).Normalize()}
+	if z.Z < 0 {
+		z = Point{z.Mul(-1)}
+	}
+
+	// Extend this to an orthonormal frame (x,y,z) where x is the direction
+	// where the great circle through AB achieves its maximium latitude.
+	y := Point{z.PointCross(PointFromCoords(0, 0, 1)).Normalize()}
+	x := y.Cross(z.Vector)
+
+	// Compute the angle "theta" from the x-axis (in the x-y plane defined
+	// above) where the great circle intersects the given line of latitude.
+	sinLat := math.Sin(float64(lat))
+	if math.Abs(sinLat) >= x.Z {
+		// The great circle does not reach the given latitude.
+		return false
+	}
+
+	cosTheta := sinLat / x.Z
+	sinTheta := math.Sqrt(1 - cosTheta*cosTheta)
+	theta := math.Atan2(sinTheta, cosTheta)
+
+	// The candidate intersection points are located +/- theta in the x-y
+	// plane. For an intersection to be valid, we need to check that the
+	// intersection point is contained in the interior of the edge AB and
+	// also that it is contained within the given longitude interval "lng".
+
+	// Compute the range of theta values spanned by the edge AB.
+	abTheta := s1.IntervalFromPointPair(
+		math.Atan2(a.Dot(y.Vector), a.Dot(x)),
+		math.Atan2(b.Dot(y.Vector), b.Dot(x)))
+
+	if abTheta.Contains(theta) {
+		// Check if the intersection point is also in the given lng interval.
+		isect := x.Mul(cosTheta).Add(y.Mul(sinTheta))
+		if lng.Contains(math.Atan2(isect.Y, isect.X)) {
+			return true
+		}
+	}
+
+	if abTheta.Contains(-theta) {
+		// Check if the other intersection point is also in the given lng interval.
+		isect := x.Mul(cosTheta).Sub(y.Mul(sinTheta))
+		if lng.Contains(math.Atan2(isect.Y, isect.X)) {
+			return true
+		}
+	}
+	return false
+}
+
+// intersectsLngEdge reports whether the edge AB intersects the given edge of constant
+// longitude. Requires the points to have unit length.
+func intersectsLngEdge(a, b Point, lat r1.Interval, lng s1.Angle) bool {
+	// The nice thing about edges of constant longitude is that
+	// they are straight lines on the sphere (geodesics).
+	return CrossingSign(a, b, PointFromLatLng(LatLng{s1.Angle(lat.Lo), lng}),
+		PointFromLatLng(LatLng{s1.Angle(lat.Hi), lng})) == Cross
+}
+
+// IntersectsCell reports whether this rectangle intersects the given cell. This is an
+// exact test and may be fairly expensive.
+func (r Rect) IntersectsCell(c Cell) bool {
+	// First we eliminate the cases where one region completely contains the
+	// other. Once these are disposed of, then the regions will intersect
+	// if and only if their boundaries intersect.
+	if r.IsEmpty() {
+		return false
+	}
+	if r.ContainsPoint(Point{c.id.rawPoint()}) {
+		return true
+	}
+	if c.ContainsPoint(PointFromLatLng(r.Center())) {
+		return true
+	}
+
+	// Quick rejection test (not required for correctness).
+	if !r.Intersects(c.RectBound()) {
+		return false
+	}
+
+	// Precompute the cell vertices as points and latitude-longitudes. We also
+	// check whether the Cell contains any corner of the rectangle, or
+	// vice-versa, since the edge-crossing tests only check the edge interiors.
+	vertices := [4]Point{}
+	latlngs := [4]LatLng{}
+
+	for i := range vertices {
+		vertices[i] = c.Vertex(i)
+		latlngs[i] = LatLngFromPoint(vertices[i])
+		if r.ContainsLatLng(latlngs[i]) {
+			return true
+		}
+		if c.ContainsPoint(PointFromLatLng(r.Vertex(i))) {
+			return true
+		}
+	}
+
+	// Now check whether the boundaries intersect. Unfortunately, a
+	// latitude-longitude rectangle does not have straight edges: two edges
+	// are curved, and at least one of them is concave.
+	for i := range vertices {
+		edgeLng := s1.IntervalFromEndpoints(latlngs[i].Lng.Radians(), latlngs[(i+1)&3].Lng.Radians())
+		if !r.Lng.Intersects(edgeLng) {
+			continue
+		}
+
+		a := vertices[i]
+		b := vertices[(i+1)&3]
+		if edgeLng.Contains(r.Lng.Lo) && intersectsLngEdge(a, b, r.Lat, s1.Angle(r.Lng.Lo)) {
+			return true
+		}
+		if edgeLng.Contains(r.Lng.Hi) && intersectsLngEdge(a, b, r.Lat, s1.Angle(r.Lng.Hi)) {
+			return true
+		}
+		if intersectsLatEdge(a, b, s1.Angle(r.Lat.Lo), r.Lng) {
+			return true
+		}
+		if intersectsLatEdge(a, b, s1.Angle(r.Lat.Hi), r.Lng) {
+			return true
+		}
+	}
+	return false
+}
+
+// Encode encodes the Rect.
+func (r Rect) Encode(w io.Writer) error {
+	e := &encoder{w: w}
+	r.encode(e)
+	return e.err
+}
+
+func (r Rect) encode(e *encoder) {
+	e.writeInt8(encodingVersion)
+	e.writeFloat64(r.Lat.Lo)
+	e.writeFloat64(r.Lat.Hi)
+	e.writeFloat64(r.Lng.Lo)
+	e.writeFloat64(r.Lng.Hi)
+}
+
+// Decode decodes a rectangle.
+func (r *Rect) Decode(rd io.Reader) error {
+	d := &decoder{r: asByteReader(rd)}
+	r.decode(d)
+	return d.err
+}
+
+func (r *Rect) decode(d *decoder) {
+	if version := d.readUint8(); int8(version) != encodingVersion && d.err == nil {
+		d.err = fmt.Errorf("can't decode version %d; my version: %d", version, encodingVersion)
+		return
+	}
+	r.Lat.Lo = d.readFloat64()
+	r.Lat.Hi = d.readFloat64()
+	r.Lng.Lo = d.readFloat64()
+	r.Lng.Hi = d.readFloat64()
+	return
+}
+
+// DistanceToLatLng returns the minimum distance (measured along the surface of the sphere)
+// from a given point to the rectangle (both its boundary and its interior).
+// If r is empty, the result is meaningless.
+// The latlng must be valid.
+func (r Rect) DistanceToLatLng(ll LatLng) s1.Angle {
+	if r.Lng.Contains(float64(ll.Lng)) {
+		return maxAngle(0, ll.Lat-s1.Angle(r.Lat.Hi), s1.Angle(r.Lat.Lo)-ll.Lat)
+	}
+
+	i := s1.IntervalFromEndpoints(r.Lng.Hi, r.Lng.ComplementCenter())
+	rectLng := r.Lng.Lo
+	if i.Contains(float64(ll.Lng)) {
+		rectLng = r.Lng.Hi
+	}
+
+	lo := LatLng{s1.Angle(r.Lat.Lo) * s1.Radian, s1.Angle(rectLng) * s1.Radian}
+	hi := LatLng{s1.Angle(r.Lat.Hi) * s1.Radian, s1.Angle(rectLng) * s1.Radian}
+	return DistanceFromSegment(PointFromLatLng(ll), PointFromLatLng(lo), PointFromLatLng(hi))
+}
+
+// DirectedHausdorffDistance returns the directed Hausdorff distance (measured along the
+// surface of the sphere) to the given Rect. The directed Hausdorff
+// distance from rectangle A to rectangle B is given by
+//     h(A, B) = max_{p in A} min_{q in B} d(p, q).
+func (r Rect) DirectedHausdorffDistance(other Rect) s1.Angle {
+	if r.IsEmpty() {
+		return 0 * s1.Radian
+	}
+	if other.IsEmpty() {
+		return math.Pi * s1.Radian
+	}
+
+	lng := r.Lng.DirectedHausdorffDistance(other.Lng)
+	return directedHausdorffDistance(lng, r.Lat, other.Lat)
+}
+
+// HausdorffDistance returns the undirected Hausdorff distance (measured along the
+// surface of the sphere) to the given Rect.
+// The Hausdorff distance between rectangle A and rectangle B is given by
+//     H(A, B) = max{h(A, B), h(B, A)}.
+func (r Rect) HausdorffDistance(other Rect) s1.Angle {
+	return maxAngle(r.DirectedHausdorffDistance(other),
+		other.DirectedHausdorffDistance(r))
+}
+
+// ApproxEqual reports whether the latitude and longitude intervals of the two rectangles
+// are the same up to a small tolerance.
+func (r Rect) ApproxEqual(other Rect) bool {
+	return r.Lat.ApproxEqual(other.Lat) && r.Lng.ApproxEqual(other.Lng)
+}
+
+// directedHausdorffDistance returns the directed Hausdorff distance
+// from one longitudinal edge spanning latitude range 'a' to the other
+// longitudinal edge spanning latitude range 'b', with their longitudinal
+// difference given by 'lngDiff'.
+func directedHausdorffDistance(lngDiff s1.Angle, a, b r1.Interval) s1.Angle {
+	// By symmetry, we can assume a's longitude is 0 and b's longitude is
+	// lngDiff. Call b's two endpoints bLo and bHi. Let H be the hemisphere
+	// containing a and delimited by the longitude line of b. The Voronoi diagram
+	// of b on H has three edges (portions of great circles) all orthogonal to b
+	// and meeting at bLo cross bHi.
+	// E1: (bLo, bLo cross bHi)
+	// E2: (bHi, bLo cross bHi)
+	// E3: (-bMid, bLo cross bHi), where bMid is the midpoint of b
+	//
+	// They subdivide H into three Voronoi regions. Depending on how longitude 0
+	// (which contains edge a) intersects these regions, we distinguish two cases:
+	// Case 1: it intersects three regions. This occurs when lngDiff <= π/2.
+	// Case 2: it intersects only two regions. This occurs when lngDiff > π/2.
+	//
+	// In the first case, the directed Hausdorff distance to edge b can only be
+	// realized by the following points on a:
+	// A1: two endpoints of a.
+	// A2: intersection of a with the equator, if b also intersects the equator.
+	//
+	// In the second case, the directed Hausdorff distance to edge b can only be
+	// realized by the following points on a:
+	// B1: two endpoints of a.
+	// B2: intersection of a with E3
+	// B3: farthest point from bLo to the interior of D, and farthest point from
+	//     bHi to the interior of U, if any, where D (resp. U) is the portion
+	//     of edge a below (resp. above) the intersection point from B2.
+
+	if lngDiff < 0 {
+		panic("impossible: negative lngDiff")
+	}
+	if lngDiff > math.Pi {
+		panic("impossible: lngDiff > Pi")
+	}
+
+	if lngDiff == 0 {
+		return s1.Angle(a.DirectedHausdorffDistance(b))
+	}
+
+	// Assumed longitude of b.
+	bLng := lngDiff
+	// Two endpoints of b.
+	bLo := PointFromLatLng(LatLng{s1.Angle(b.Lo), bLng})
+	bHi := PointFromLatLng(LatLng{s1.Angle(b.Hi), bLng})
+
+	// Cases A1 and B1.
+	aLo := PointFromLatLng(LatLng{s1.Angle(a.Lo), 0})
+	aHi := PointFromLatLng(LatLng{s1.Angle(a.Hi), 0})
+	maxDistance := maxAngle(
+		DistanceFromSegment(aLo, bLo, bHi),
+		DistanceFromSegment(aHi, bLo, bHi))
+
+	if lngDiff <= math.Pi/2 {
+		// Case A2.
+		if a.Contains(0) && b.Contains(0) {
+			maxDistance = maxAngle(maxDistance, lngDiff)
+		}
+		return maxDistance
+	}
+
+	// Case B2.
+	p := bisectorIntersection(b, bLng)
+	pLat := LatLngFromPoint(p).Lat
+	if a.Contains(float64(pLat)) {
+		maxDistance = maxAngle(maxDistance, p.Angle(bLo.Vector))
+	}
+
+	// Case B3.
+	if pLat > s1.Angle(a.Lo) {
+		intDist, ok := interiorMaxDistance(r1.Interval{a.Lo, math.Min(float64(pLat), a.Hi)}, bLo)
+		if ok {
+			maxDistance = maxAngle(maxDistance, intDist)
+		}
+	}
+	if pLat < s1.Angle(a.Hi) {
+		intDist, ok := interiorMaxDistance(r1.Interval{math.Max(float64(pLat), a.Lo), a.Hi}, bHi)
+		if ok {
+			maxDistance = maxAngle(maxDistance, intDist)
+		}
+	}
+
+	return maxDistance
+}
+
+// interiorMaxDistance returns the max distance from a point b to the segment spanning latitude range
+// aLat on longitude 0 if the max occurs in the interior of aLat. Otherwise, returns (0, false).
+func interiorMaxDistance(aLat r1.Interval, b Point) (a s1.Angle, ok bool) {
+	// Longitude 0 is in the y=0 plane. b.X >= 0 implies that the maximum
+	// does not occur in the interior of aLat.
+	if aLat.IsEmpty() || b.X >= 0 {
+		return 0, false
+	}
+
+	// Project b to the y=0 plane. The antipodal of the normalized projection is
+	// the point at which the maxium distance from b occurs, if it is contained
+	// in aLat.
+	intersectionPoint := PointFromCoords(-b.X, 0, -b.Z)
+	if !aLat.InteriorContains(float64(LatLngFromPoint(intersectionPoint).Lat)) {
+		return 0, false
+	}
+	return b.Angle(intersectionPoint.Vector), true
+}
+
+// bisectorIntersection return the intersection of longitude 0 with the bisector of an edge
+// on longitude 'lng' and spanning latitude range 'lat'.
+func bisectorIntersection(lat r1.Interval, lng s1.Angle) Point {
+	lng = s1.Angle(math.Abs(float64(lng)))
+	latCenter := s1.Angle(lat.Center())
+
+	// A vector orthogonal to the bisector of the given longitudinal edge.
+	orthoBisector := LatLng{latCenter - math.Pi/2, lng}
+	if latCenter < 0 {
+		orthoBisector = LatLng{-latCenter - math.Pi/2, lng - math.Pi}
+	}
+
+	// A vector orthogonal to longitude 0.
+	orthoLng := Point{r3.Vector{0, -1, 0}}
+
+	return orthoLng.PointCross(PointFromLatLng(orthoBisector))
+}
+
+// Centroid returns the true centroid of the given Rect multiplied by its
+// surface area. The result is not unit length, so you may want to normalize it.
+// Note that in general the centroid is *not* at the center of the rectangle, and
+// in fact it may not even be contained by the rectangle. (It is the "center of
+// mass" of the rectangle viewed as subset of the unit sphere, i.e. it is the
+// point in space about which this curved shape would rotate.)
+//
+// The reason for multiplying the result by the rectangle area is to make it
+// easier to compute the centroid of more complicated shapes. The centroid
+// of a union of disjoint regions can be computed simply by adding their
+// Centroid results.
+func (r Rect) Centroid() Point {
+	// When a sphere is divided into slices of constant thickness by a set
+	// of parallel planes, all slices have the same surface area. This
+	// implies that the z-component of the centroid is simply the midpoint
+	// of the z-interval spanned by the Rect.
+	//
+	// Similarly, it is easy to see that the (x,y) of the centroid lies in
+	// the plane through the midpoint of the rectangle's longitude interval.
+	// We only need to determine the distance "d" of this point from the
+	// z-axis.
+	//
+	// Let's restrict our attention to a particular z-value. In this
+	// z-plane, the Rect is a circular arc. The centroid of this arc
+	// lies on a radial line through the midpoint of the arc, and at a
+	// distance from the z-axis of
+	//
+	//     r * (sin(alpha) / alpha)
+	//
+	// where r = sqrt(1-z^2) is the radius of the arc, and "alpha" is half
+	// of the arc length (i.e., the arc covers longitudes [-alpha, alpha]).
+	//
+	// To find the centroid distance from the z-axis for the entire
+	// rectangle, we just need to integrate over the z-interval. This gives
+	//
+	//    d = Integrate[sqrt(1-z^2)*sin(alpha)/alpha, z1..z2] / (z2 - z1)
+	//
+	// where [z1, z2] is the range of z-values covered by the rectangle.
+	// This simplifies to
+	//
+	//    d = sin(alpha)/(2*alpha*(z2-z1))*(z2*r2 - z1*r1 + theta2 - theta1)
+	//
+	// where [theta1, theta2] is the latitude interval, z1=sin(theta1),
+	// z2=sin(theta2), r1=cos(theta1), and r2=cos(theta2).
+	//
+	// Finally, we want to return not the centroid itself, but the centroid
+	// scaled by the area of the rectangle. The area of the rectangle is
+	//
+	//    A = 2 * alpha * (z2 - z1)
+	//
+	// which fortunately appears in the denominator of "d".
+
+	if r.IsEmpty() {
+		return Point{}
+	}
+
+	z1 := math.Sin(r.Lat.Lo)
+	z2 := math.Sin(r.Lat.Hi)
+	r1 := math.Cos(r.Lat.Lo)
+	r2 := math.Cos(r.Lat.Hi)
+
+	alpha := 0.5 * r.Lng.Length()
+	r0 := math.Sin(alpha) * (r2*z2 - r1*z1 + r.Lat.Length())
+	lng := r.Lng.Center()
+	z := alpha * (z2 + z1) * (z2 - z1) // scaled by the area
+
+	return Point{r3.Vector{r0 * math.Cos(lng), r0 * math.Sin(lng), z}}
+}
+
+// BUG: The major differences from the C++ version are:
+//  - Get*Distance, Vertex, InteriorContains(LatLng|Rect|Point)