// intersectsLatEdge reports if 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 := a.PointCross(b)
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 := z.PointCross(PointFromCoords(0, 0, 1))
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.IntervalFromEndpoints(
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
}
