Oh, yes, I see! The dihedral angles are just the angles of the triangle on the surface of the sphere. It hadn't occurred to me to treat the center of the sphere as a polyhedron vertex. Thanks! Tom Allan Wechsler writes:
See https://en.wikipedia.org/wiki/Law_of_sines#Spherical_case, The vertex you have in mind is the center of the sphere.
On Sat, Oct 6, 2018 at 4:18 AM Tom Karzes <karzes@sonic.net> wrote:
Is anyone familiar with the following?
Consider a polyhedron vertex with three faces (and three edges). Let the polygon angles at the vertex be A, B, and C. Let the dihedral angles at the vertex be a, b, and c, where a is opposite A, b is opposite B, and c is opposite C. Then:
sin(A)/sin(a) = sin(B)/sin(b) = sin(C)/sin(c)
This is just another Law of Sines, and seems pretty fundamental, but I can't find any reference to it (or proof of it).
I can only find the version taught in high school trig (which uses the ratios of edge lengths to the sines of the opposite angles), and the spherical case (which uses the ratios of the sines of the surface triangle angles to the sines of the angles of the arcs from the center of the sphere).
Is this well known? Does anyone have a reference for it? Or a simple proof?
Tom