Parameterizing Shapes: Algebraic Form Solutions

[grawil]

Does anyone have any suggestions on a functional, algebraic form to parameterize the edge of the shapes shown in the image sequence below? It begins as a circle but deforms and flattens along the edges perpendicular to the axis of symmetry. I have a crude model of it with the upper and lower edge being modeled as independent ellipses with tied end-points but it's a PIA and seems hopeless when the one edge flattens.

[PLAIN]http://img233.imageshack.us/img233/7484/shapesm.jpg

 

[Petr Mugver]

Maybe I don't understand your figure properly, but you could try with

(x + 4/5)^2 + x^2 + y^2 -1 = 0 and z >= 0

This is only the "curved part" of the picture, maybe you should add a "flat bottom" of the form

x^2 + x^2 =< 9/25 and z = 0