Suppose I have a spherical triangle (no special properties; in particular, not equilateral) with a known solid angle. Now, I divide it into four new spherical triangles by bisecting each edge:
Appallingly, the solid angle of the new triangles is NOT simply 1/4 that of the original triangle. My question: is there a way to determine how the solid angle is allocated?
Specifically, I could recompute the solid angle (compute edge angles, then vertex angles, then spherical excess, then surface area, then steradians) at a terrible cost in trigonometry, but I'm wondering if knowing the original solid angle and the geometry of bisection makes the problem more tractable.