Thanks for pointing out how Mathematica handles discretization—it makes sense that the rendering process involves approximations, so the surfaces aren’t infinitely differentiable in practice. But actually, my question was more about the theoretical side of things. I’m curious about how stereographic projection works with smooth parametric surfaces like cones and paraboloids from a purely mathematical perspective.

For example, I’m wondering how the smoothness (or lack of it, like at the tip of a cone) or the curvature of a paraboloid plays into applying stereographic projection. Does it still make sense to use it in those cases, or are there limitations?

Basically, I’m more interested in the math behind it—how these surfaces behave theoretically—rather than how they’re represented in software.

By smooth I meant a function or surface that is infinitely differentiable, i.e. has continuous derivatives of all orders. In the case of a cone, however, this only applies from the tip.