Mapping 3D data on a unit sphere

Question

I have a three D data set and I am trying to map it on the surface of a unit sphere. I don't know how to start but I found some papers in web but it is not clear to me how to use them. Can you share you thoughts on this algorithm.

In my data I have three value at each point. For example a coffee mug. I have the exact coordinates of each point on the mug surface. Now I want to map mug on a unit sphere. Any help in this regard.

I also found a similar post here in stackExchange at Messed up Sphere but they are using opengl. I want to use this in simple plane c or fortran code.

Thank you.