Duality in spherical trigonometry
I’ve written before about the law of cosines on a sphere: But that’s just one of two laws of cosines on a sphere. The equation above is the law of cosines for sides. The law of cosines for angles is typographically very similar, just inverting the case of the letters used as arguments. Let’s back up and say what the letters refer to. As is conventional in plane geometry, the sides of a triangle are a, b, and c, and the angles are A, B, and C, with a being the side opposite the angle A, and similarly for the other letters. However, unlike plane geometry, the sides are also angles, but of a different sort. The sides a, b, and c are central angles, and the “angles” A, B, and C are dihedral angles. The sides are arcs of a great circle. The side a, for example, is a portion of a great circle, running between the vertices at B and C [1]. Think of a plane that contains the center of the sphere…
