// InterpolateAtDistance returns the point X along the line segment AB whose
// distance from A is the angle ax.
func InterpolateAtDistance(ax s1.Angle, a, b Point) Point {
aRad := ax.Radians()
// Use PointCross to compute the tangent vector at A towards B. The
// result is always perpendicular to A, even if A=B or A=-B, but it is not
// necessarily unit length. (We effectively normalize it below.)
normal := a.PointCross(b)
tangent := normal.Vector.Cross(a.Vector)
// Now compute the appropriate linear combination of A and "tangent". With
// infinite precision the result would always be unit length, but we
// normalize it anyway to ensure that the error is within acceptable bounds.
// (Otherwise errors can build up when the result of one interpolation is
// fed into another interpolation.)
return Point{(a.Mul(math.Cos(aRad)).Add(tangent.Mul(math.Sin(aRad) / tangent.Norm()))).Normalize()}
}
// radiusToHeight converts an s1.Angle into the height of the cap.
func radiusToHeight(r s1.Angle) float64 {
if r.Radians() < 0 {
return emptyHeight
}
if r.Radians() >= math.Pi {
return fullHeight
}
// The height of the cap can be computed as 1 - cos(r), but this isn't very
// accurate for angles close to zero (where cos(r) is almost 1). The
// formula below has much better precision.
d := math.Sin(0.5 * r.Radians())
return 2 * d * d
}
// EarthDistance converts an angle to distance on earth in meters.
func EarthDistance(angle s1.Angle) Length {
return Length(angle.Radians() * EarthRadiusMeters)
}