Cyclides are really cool. It's basically like a quadric except that one of the terms is the square of the radius, so it's actually a quartic.

So a cyclide in 3D would be described by P(x, y, z, r^2) = 0, where P is a second order polynomial and r^2 = x^2 + y^2 + z^2. As far as I can see there's no standard definition for a four dimensional cyclide.

Here's one example that's equivalent to a torus:

(x^2 + y^2 + z^2 - 1)^2 - 2 ( x^2 + y^2 - z^2 ) + 1 = 0

There's a subset of cyclides called Dupin cyclides, which seem to look very similar to toruses, but wider in some parts than others.

There's a few articles about these things, but most of them are very old. I think it would be interesting to classify cyclides the same way we classify quadrics. We could extend the definition to 4D and see how they relate to toratopes. Note that since cylides are quartic surfaces, but ((II)(II)) and (((II)I)I) are octic surfaces, those toratopes will not be cyclides.