Analogues to the Conic Sections?

I was a bit frustrated when Beyond The Third Dimension failed to go beyond the third dimension in its discussion of conic sections. It brings them up, presumably as it is pertinent to the subject of slicings, but fails to extend it at all! C'mon; I wanted to see nifty 3-d computer rendered images of these surfaces, but they weren't even mentioned.

What happens when we slice a 4-d analogue of the double cone with a hyperplane? Presumably we end up with curved surfaces. What sorts of surfaces, though? It's not even clear to me exactly what 4-d figure we should be slicing; presumably a coned cone. But what geometric properties can we expect from these surfaces? What sorts of equations have these surfaces for graphs? I see two possibilities for that; either it continues to be two degree equations, and the conic section of any space produces graphs of two degree equations in the dimension below it, or the graphs or of three degree equations, in which case each dimension's cone produces graphs of n-1 degree equations in n-1 dimensions. Which is it? Or is this one of those nefarious things that just doesn't extend particularly well beyond a handful (two or three) dimensions?

If anbody can point me to a cogent treatment of the subject, as I'm sure someone must have written this up by now, I would appreciate it.

In a somewhat non-sequitorial note, I am looking for some graphics programs to help me with this stuff. If anybody knows of a freely distributed program for Mac or PC which is capable of making relatively simple work of things like doing timesliced animations of four dimensional figures, slicing and sectioning figures, rotating figures in four-space, and the like, please drop me a line. Sorry, UNIXheads; I'm not running Linux, so it's got to be mac or windows.