"Verify angular separation formulas" x1 = r1 sin(theta1) cos(phi1) y1 = r1 sin(theta1) sin(phi1) z1 = r1 cos(theta1) x2 = r2 sin(theta2) cos(phi2) y2 = r2 sin(theta2) sin(phi2) z2 = r2 cos(theta2) check(r1^2 == x1^2 + y1^2 + z1^2) check(r2^2 == x2^2 + y2^2 + z2^2) r12 = sqrt((x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2) d = sin(theta1) sin(theta2) cos(phi1 - phi2) + cos(theta1) cos(theta2) check(d == (r1^2 + r2^2 - r12^2) / (2 r1 r2)) check(d == (x1 x2 + y1 y2 + z1 z2) / (r1 r2)) "ok" Run