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.

  • $\begingroup$ A swallow's egg? and if so, African or European? I see no reason why ths question belongs here instead of, say, math.stackexchange.com $\endgroup$ – Yemon Choi Mar 29 '13 at 22:42
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    $\begingroup$ They had a hard time creating a model for the Vegreville egg. (en.wikipedia.org/wiki/Vegreville_egg) 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. $\endgroup$ – Joel Reyes Noche Mar 29 '13 at 23:32
  • $\begingroup$ I made some effort ~two Easters ago to make the image in this question look like an Easter egg: "Which convex bodies roll along closed geodesics?" mathoverflow.net/questions/61386. Even though it was only an ellipsoid, I was inspired by Easter-egg rolling contests. $\endgroup$ – Joseph O'Rourke Mar 30 '13 at 0:07

Here is a nice Mathematica Demonstration (incidentally colored according to the Riemann zeta function), with the shape based on the mathematics of egg shapes described at an egg curves website www.mathematische-basteleien.de:
           Riemannian Egg
One of the neat constructions explained there is the Gardener's Egg:



You might e.g. look at Cartesian ovals


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