// Smallest finds the smallest circle containing three points.
//
// Arguments should represent coordinates in right ascension and declination
// or longitude and latitude. Result Δ is the diameter of the circle, typeI
// is true if solution is of type I.
//
// type I Two points on circle, one interior.
// type II All three points on circle.
func Smallest(r1, d1, r2, d2, r3, d3 unit.Angle) (Δ unit.Angle, typeI bool) {
// Using haversine formula, but reimplementing SepHav here to reuse
// the computed cosines.
cd1 := d1.Cos()
cd2 := d2.Cos()
cd3 := d3.Cos()
a := 2 * math.Asin(math.Sqrt(base.Hav(d2-d1)+cd1*cd2*base.Hav(r2-r1)))
b := 2 * math.Asin(math.Sqrt(base.Hav(d3-d2)+cd2*cd3*base.Hav(r3-r2)))
c := 2 * math.Asin(math.Sqrt(base.Hav(d1-d3)+cd3*cd1*base.Hav(r1-r3)))
if b > a {
a, b = b, a
}
if c > a {
a, c = c, a
}
if a*a >= b*b+c*c {
return unit.Angle(a), true
}
// (20.1) p. 128
return unit.Angle(2 * a * b * c /
math.Sqrt((a+b+c)*(a+b-c)*(b+c-a)*(a+c-b))), false
}
// SepHav returns the angular separation between two celestial bodies.
//
// The algorithm uses the haversine function and is superior to the naïve
// algorithm of the Sep function.
func SepHav(r1, d1, r2, d2 unit.Angle) unit.Angle {
// using (17.5) p. 115
return unit.Angle(2 * math.Asin(math.Sqrt(base.Hav(d2-d1)+
d1.Cos()*d2.Cos()*base.Hav(r2-r1))))
}