27 Feb
2007
27 Feb
'07
10:58 p.m.
Fascinating and surprisingly long-lived thread! My "natural" choice would be to make ovals by interpolating or blending two ellipses (perhaps from too much computer graphics): Standardize the graphs of the ellipses by aligning them with their major axis coincident with the X axis and scale them so their ends are at x=0 and x=1. These graphs form a family indexed by "eccentricity parameter" [e]: y = ellipse[e](x) Now define oval[e0,e1] := (1-x) ellipse[e0](x) + x ellipse[e1](x) (having "eccentricity vector" [e0,e1]) So if ellipse(x) is quadratic in x then oval(x) will be some cubic. Alternatively, we might define oval[e0,e1] := ellipse[(1-x) e0 + x e1](x) For what parameterizations of eccentricity are these the same?