ICN5D wrote:So, this sounds pretty straightforward. Where do I find a program that can do this?
It's only straightforward with the simplest of equations, and the simplest of slices! Once you get into more complex (i.e., interesting) toratopes, you start getting things with high-degree polynomials, and once you start doing oblique slices, the equations will turn hairy. Very hairy. For example, the equation for a plain ole innocent 3D torus requires a degree-4 polynomial. While innocent enough when you're dealing with axis-aligned slices, it quickly turns very
hairy with oblique slices... or, for that matter, when trying to solve for the intersection of a ray with a torus (e.g., done in ray-tracing or when you want to calculate projections), since you have to solve a quartic polynomial. And consider yourself lucky that it's "only" degree-4, since if you go any higher than that, then there is in general no solution to the equations in the form of +, -, *, /, and n-th root extraction. You'll need to start using heavy-duty weapons like hyperradical functions to deal with those things.
Unless, of course, you're content with axis-aligned slicing and non-analytic (i.e. numerical) solutions, then it's just a matter of choosing some value for some of the coordinates, and substituting it into the equations and seeing what comes out. You could use povray's isosurface feature to help with doing renderings of the resulting equations -- IIRC, it uses Newton's method to solve high-order polynomials so that at least something will come out when you throw something like a 5th degree polynomial at it. Good luck if you're trying to do algebraic analysis on 5th degree (or higher) polynomials, though... I don't recommend attempting that at home; your brain may catch fire.