Suppose I have a smooth object illuminated by a parallel light source (a sphere, an ellipsoid, whatever), and I wanted to be able to mathematically describe the illumination of the object (keep it simple, ignore scattering, soft shadows, all that shit, the light either touches it or not), how the hell would I begin to start to work out shit like how to describe the terminator curve, or find the centroid of the illuminated/unilluminated parts of the surface?

I've been banging my head against a wall for the last week or so trying to work this shit out (it's related to a research problem), but this kind of geometric analysis isn't my strong suite. Any chance I can get some help or even a point in the right direction?