I just started my Calculus 3 class, which deals mainly with multi-variable Calculus. We went over the different types of 3-D surfaces, mainly Quadric Surfaces (ellipsoids, paraboloids, hyperboloids, etc.) which according to the textbook are "3D analogs of conic sections". I know for conic sections you slice cones with a plane to get ellipses, parabolas, hyperbolas, etc, so I was wondering what is the equivalent for the sliced cone when dealing with these kind of surfaces? How is such a body and the operation of "slicing" it described? And can this be generalized for higher dimensions?

I didn't quite know how to word the problem in a sentence or what field of math it's part of to do a quick google search so if anyone could point me in the right direction I would appreciate it.

Thanks