rect.CapBound: Fix for tall and wide rects (#233)

For tall (>90 degrees) and wide (>180 degrees) lat/lng rects,
rect.CapBound returned an incorrect cap that didn't contain the full
rect. The function intends to return a polar cap in this case, but due
to an apparent typo (2 * Pi instead of Pi), returned a cap centered at
the center of the rect instead.

This fix has recently been applied to C++ and Java in google3, but has
not been exported to those github repos yet.

Author: jmr

s2/rect.go

@@ -247,7 +247,7 @@ func (r Rect) CapBound() Cap {
 	// 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 {
+	if math.Remainder(r.Lng.Hi-r.Lng.Lo, 2*math.Pi) >= 0 && r.Lng.Hi-r.Lng.Lo <= math.Pi {
 		midCap := CapFromPoint(PointFromLatLng(r.Center())).AddPoint(PointFromLatLng(r.Lo())).AddPoint(PointFromLatLng(r.Hi()))
 		if midCap.Height() < poleCap.Height() {
 			return midCap

s2/rect_test.go

@@ -395,6 +395,16 @@ func TestRectCapBound(t *testing.T) {
 			rectFromDegrees(-30, -150, -10, 50),
 			CapFromCenterAngle(Point{r3.Vector{X: 0, Y: 0, Z: -1}}, s1.Angle(80)*s1.Degree),
 		},
+		{
+			// Longitude span > 180 degrees and latitude span > 90 degrees. This results
+			// in a polar cap that is larger than the "midpoint cap" (centered at the
+			// center of the rect and containing the vertices), but is nonetheless the
+			// correct result. The "midpoint cap" must not be returned since it doesn't
+			// contain the entire rect due the rect being wider than 180 degrees.
+			// In this example, (-34, 49) is in the rect but not the midpoint cap.
+			rectFromDegrees(-60, -150, 70, 50),
+			CapFromCenterAngle(Point{r3.Vector{X: 0, Y: 0, Z: 1}}, s1.Angle(150)*s1.Degree),
+		},
 	}
 	for _, test := range tests {
 		if got := test.r.CapBound(); !test.want.ApproxEqual(got) {