On 31/12/2015 04:56, James Propp wrote:
Is there a version of the central angle theorem that applies to spheres and solid angles?
I don't think so. Draw a picture like this. You have the unit sphere, centre O. You have an area element dA somewhere on it, say at P. You have a point Q somewhere else on the sphere. Let alpha be angle OPQ (= angle OQP). The solid angle subtended at O is (by definition) dA. The solid angle subtended at Q is smaller because of two factors: - The distance is 2 cos alpha instead of 1. - The area element's normal isn't along PQ but at an angle alpha to it. This means that the ratio of subtended angles is 1/(2 cos alpha)^2 from the first factor, times cos alpha from the second factor. The cosine factors don't cancel the way they do in two dimensions; you get 1 / 4 cos alpha. -- g