Conic sections are to cones as quadric surfaces are to what? [closed]

Question

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

Answer 1

The thing that makes quadric surfaces "3D analogs of conic sections" is just that they are defined by a single equation of degree 2. It's not a particularly helpful characterization though, I would say. It strikes me more as something a pedagogue would say in a (poor) attempt to relate a new concept to one already known.

[One could see quadric surfaces as "slices" of a certain geometric object, analogously to conic sections, but only if you are only interested in them up to isomorphism (as algebraic varieties). Then a quadric surface can be regarded as a hyperplane section of the image $V \subset \mathbb{P}^9$ of the Veronese embedding $\mathbb{P}^3 \to \mathbb{P}^9$. Note that this is only partially analogous to the representation of a conic section (considered as a plane curve) as the intersection of cone and plane, since the conic section is related to the aforementioned intersection by a projective transformation, which is a lot stronger than saying they are isomorphic as algebraic varieties.]