Non-euclidean geometry and degenerate conic sections

[Bohandas]

Two questions, regarding the relationship between conic sections and non-euclidean geometry:

1.) The the assignment of the names of conic sections to different geometries (ie. hyperbolic geometry*, parabolic geometry**, elliptic geometry***) meaningful, or is it just a fanciful naming convention like the color charges in quark physics?

2.) If its meaningful, are there geometries corresponding to the degenerate conic sections?

The "X" shaped one is of particular interest to me. To a lesser extent so is the one where the cone itself is degenerate, yielding parallel lines, as this doesn't seem to correspond to any of the normal sections the way the other three do.


*Lobachevskian geometrt
**Euclidean geometry
***Riemannian geometry

[GeoffWatson]

They're named after the mathematicians who first studied them.

[DavidSh]

The conic section names are probably related to surfaces of 3-d objects. Ellipsoids have positive intrinsic curvature, while hyperboloids of one sheet have negative intrinsic curvature. The 3-d forms corresponding to the degenerate conic sections are cones and cylinders, the surfaces of which don't have intrinsic curvature.

This falls apart a bit when you look at hyperboloids of two sheets.