I'm trying to find out the solid angle subtended over the entirety of some closed surface S by some point P located on the surface. For a point within the surface, the answer is of course 4$\pi$, but I'm trying to get the result if that point were on the surface itself, not inside or out.
Apparently, the answer is supposed to be 2$\pi$, and this is easy to imagine if you were on a sphere or something, but I'm having trouble rigorously proving this. Also, I'm imagining a surface that's a sphere, except you have a cone-shaped indentation on it. If the vertex of that cone were in the center of the sphere, and you try to get the solid angle at the center, it seems to me that the solid angle would just be 4$\pi$ minus whatever solid angle the cone was. In other words, the answer for a point on the surface is not going to be 2$\pi$ if you have a concave/convex part on the surface. Am I missing something here?