Inspired by an exceptionally silly article in today's newspaper I pose the following "egg parametrization problem".

Give an explicit function $ f(x,y,t) : \mathbb{R}^2\times I \to \mathbb{R}$ such that for each $t$ from interval $I$ the solution set of equation $f(x,y,t) = 0$ looks like an egg.

I'm looking for function that provides most of the various egg shapes found in nature.

Though the surface area of an egg can be difficult to solve mathematically, the enigma of how to assemble two-dimensional tiles onto a three-dimensional egg was eventually solved by Ronald Resch, a computer science professor from the University of Utah, with the assistance of computer-aided design. Resch tiled the egg uses at total of 1108 congruent equilateral triangles, 524 concave hexagons (3-pointed stars), 3,512 visible facets, 6,978 nuts and bolts and 177 internal struts.– Joel Reyes Noche Mar 29 '13 at 23:32