Circles, parabolas, ellipses, hyperbolas are all quadratic. However an ellipse is never a parabola and vice versa. But they can all be pretty close. Perhaps even insignificantly different. The term "quadratic" when used in skiing or snowboarding has come to mean parabolic.
They're all conic sections:
from here.
I've ridden radial (Burton, Prior), parabolic (Burton, Donek), elliptical (Madd), tri-radial (Volkl) and I believe I enjoy the parabolic and elliptical best. However flex pattern and construction have a lot to do with it too. Very hard to credit just the sidecut geometry with the difference. However I will say stepping off a Prior wcr175 (11.5m) and onto a Donek 171fcII (11.2m) was dramatic.
I've never ridden a progressive board - one that has one radius for the front half of the board, and a different one for the back. In theory it makes very little sense to me. I don't think you should be pressuring one foot or the other so much that half the board isn't doing anything. Nitro and I believe Oxygen were like this. Probably others. Longer sidecut towards the nose, tighter towards the tail. I hear they are now building pro GS race boards that are the opposite - tight nose, long tail. *shrug*.
Whatever your sidecut is, I think it should be symmetrical front to back, at least in theory. Suppose I'll have to demo one of them thar new fangled reverse progressive thangs.
Here is an article on some of the physics of sidecut:
http://www.bomberonline.com//articles/physics.cfm