IIRC, generally speaking a beam with a transverse point load (think like a diver on the tip of a diving board) deforms into a parabolic curve, and beams with more point loads end up taking higher order polynomial curves. Engineering beam math usually much assumes that the beam is acting under small deflections though.

I don't remember as much about beziers but I think they are also often essentially a series of cubic curves. There are a couple of different types of splines- one of them is controlled locally only by it's local control points (I think that one's beziers) and one requires all control points to control the whole curve.