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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.

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A swallow's egg? and if so, African or European? I see no reason why ths question belongs here instead of, say, – Yemon Choi Mar 29 '13 at 22:42
They had a hard time creating a model for the 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. – Joel Reyes Noche Mar 29 '13 at 23:32
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?" Even though it was only an ellipsoid, I was inspired by Easter-egg rolling contests. – Joseph O'Rourke Mar 30 '13 at 0:07
up vote 10 down vote accepted

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
           Riemannian Egg
One of the neat constructions explained there is the Gardener's Egg:


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You might e.g. look at Cartesian ovals

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