func (l *Loop) initBound() {
// The bounding rectangle of a loop is not necessarily the same as the
// bounding rectangle of its vertices. First, the loop may wrap entirely
// around the sphere (e.g. a loop that defines two revolutions of a
// candy-cane stripe). Second, the loop may include one or both poles.
// Note that a small clockwise loop near the equator contains both poles.
bounder := NewRectBounder()
for i := 0; i <= l.NumVertices(); i++ {
bounder.AddPoint(l.Vertex(i))
}
b := bounder.GetBound()
// Note that we need to initialize bound with a temporary value since
// contains() does a bounding rectangle check before doing anything else.
l.bound = FullRect()
if l.ContainsPoint(PointFromCoordsRaw(0, 0, 1)) {
b = Rect{r1.IntervalFromPointPair(b.Lat.Lo, math.Pi/2), s1.FullInterval()}
}
// If a loop contains the south pole, then either it wraps entirely
// around the sphere (full longitude range), or it also contains the
// north pole in which case b.lng().isFull() due to the test above.
if b.Lng.IsFull() && l.ContainsPoint(PointFromCoordsRaw(0, 0, -1)) {
b = Rect{r1.IntervalFromPointPair(-math.Pi/2, b.Lat.Hi), b.Lng}
}
l.bound = b
}
// RectBound returns a bounding latitude-longitude rectangle that contains
// the region. The bounds are not guaranteed to be tight.
func (c Cell) RectBound() Rect {
if c.level > 0 {
// Except for cells at level 0, the latitude and longitude extremes are
// attained at the vertices. Furthermore, the latitude range is
// determined by one pair of diagonally opposite vertices and the
// longitude range is determined by the other pair.
//
// We first determine which corner (i,j) of the cell has the largest
// absolute latitude. To maximize latitude, we want to find the point in
// the cell that has the largest absolute z-coordinate and the smallest
// absolute x- and y-coordinates. To do this we look at each coordinate
// (u and v), and determine whether we want to minimize or maximize that
// coordinate based on the axis direction and the cell's (u,v) quadrant.
u := c.uv.X.Lo + c.uv.X.Hi
v := c.uv.Y.Lo + c.uv.Y.Hi
var i, j int
if uAxis(int(c.face)).Z == 0 {
if u < 0 {
i = 1
}
} else {
if u > 0 {
i = 1
}
}
if vAxis(int(c.face)).Z == 0 {
if v < 0 {
j = 1
}
} else {
if v > 0 {
j = 1
}
}
lat := r1.IntervalFromPointPair(c.latitude(i, j), c.latitude(1-i, 1-j))
lat = lat.Expanded(MAX_ERROR).Intersection(validRectLatRange)
if lat.Lo == validRectLatRange.Lo || lat.Hi == validRectLatRange.Hi {
return Rect{lat, s1.FullInterval()}
}
lng := s1.IntervalFromPointPair(c.longitude(i, 1-j), c.longitude(1-i, j))
return Rect{lat, lng.Expanded(MAX_ERROR)}
}
switch c.face {
case 0:
return Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{-math.Pi / 4, math.Pi / 4}}
case 1:
return Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{math.Pi / 4, 3 * math.Pi / 4}}
case 2:
return Rect{r1.Interval{POLE_MIN_LAT, math.Pi / 2}, s1.Interval{-math.Pi, math.Pi}}
case 3:
return Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{3 * math.Pi / 4, -3 * math.Pi / 4}}
case 4:
return Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{-3 * math.Pi / 4, -math.Pi / 4}}
default:
return Rect{r1.Interval{-math.Pi / 2, -POLE_MIN_LAT}, s1.Interval{-math.Pi, math.Pi}}
}
}
// RectBound returns a bounding latitude-longitude rectangle.
// The bounds are not guaranteed to be tight.
func (c Cap) RectBound() Rect {
if c.IsEmpty() {
return EmptyRect()
}
capAngle := c.Radius().Radians()
allLongitudes := false
lat := r1.Interval{
Lo: latitude(c.center).Radians() - capAngle,
Hi: latitude(c.center).Radians() + capAngle,
}
lng := s1.FullInterval()
// Check whether cap includes the south pole.
if lat.Lo <= -math.Pi/2 {
lat.Lo = -math.Pi / 2
allLongitudes = true
}
// Check whether cap includes the north pole.
if lat.Hi >= math.Pi/2 {
lat.Hi = math.Pi / 2
allLongitudes = true
}
if !allLongitudes {
// Compute the range of longitudes covered by the cap. We use the law
// of sines for spherical triangles. Consider the triangle ABC where
// A is the north pole, B is the center of the cap, and C is the point
// of tangency between the cap boundary and a line of longitude. Then
// C is a right angle, and letting a,b,c denote the sides opposite A,B,C,
// we have sin(a)/sin(A) = sin(c)/sin(C), or sin(A) = sin(a)/sin(c).
// Here "a" is the cap angle, and "c" is the colatitude (90 degrees
// minus the latitude). This formula also works for negative latitudes.
//
// The formula for sin(a) follows from the relationship h = 1 - cos(a).
sinA := math.Sqrt(c.height * (2 - c.height))
sinC := math.Cos(latitude(c.center).Radians())
if sinA <= sinC {
angleA := math.Asin(sinA / sinC)
lng.Lo = math.Remainder(longitude(c.center).Radians()-angleA, math.Pi*2)
lng.Hi = math.Remainder(longitude(c.center).Radians()+angleA, math.Pi*2)
}
}
return Rect{lat, lng}
}
