Ask A Scientist© General Science Archive

Recently, Newton has been asked twice about the formula/equation of an 
egg. It struck me as rather sad that in the spirit of the Easter season, 
we could not be more responsive to those inquiries. So I set about to see 
how one would make a model of an egg -- or at least its shape.
Here's what I came up with; you can use/not use as you see fit.

1. Start with an ellipse of revolution about the major (longer) axis as an 
approximation for an egg. The general equation for an ellipse is: (y/b)^2 
+(x/a)^2 = 1 where a>b. Without any loss of generality, set b=1. This just 
means we are measuring the ellipse in multiple units of the minor axis. 
So, y^2 = 1 - (x/a)^2. One could use other exponents besides "2", but 
higher values become too "boxy", and smaller values, say fractional 
exponents get into problem of imaginary, i.e. complex variable solutions. 
Some measurements from my refrigerator suggests that a~3/2.

2. The second property you need for an egg is a non-symmetric function, 
g(x), that makes
y(x) > 0 > y(-x), or the other way around. This non-symmetric function 
makes one end of the egg larger than the ellipse and the other end of the 
egg smaller than the ellipse. The selection of g(x) is quite broad, which 
is why I call it "non-symmetric" rather than "asymmetric" since the latter 
term is used frequently to imply that y(-x) = -y(x). This constraint, 
while useful, isn't necessary for an egg. In addition, while it makes 
computation simpler, it is not necessary that
g(0) = 0. That is, the non-symmetry can be "off center" with respect to 
the origin.

3. There needs to be a scaling parameter, (lambda) > 0, that multiplies 
g(x). This scaling parameter adjusts how much asymmetry you wish to "turn on".

4. The fourth property you need for an egg is the proper boundary 
conditions at the ends of the egg, remembering that the major axis "a" is 
chosen to lie along the x-axis. There are also several ways to do this, 
but a simple one is to select a multiplicative term:
(x-a)*(x+a) =(x^2-a^2). This guarantees that the egg function will equal 
zero at +a and -a no matter what happens to g(x), assuming it is finite at 
g(a).

Putting this all together we get: EGG(x) = (1 - (x/a)^2)^1/2 + 
(lambda)*g(x)*(x^2-a^2).

I have tested this for a number of g(x)'s, both algebraic and 
trigonometric, and with a little practice you can dial in any shape you 
want. It's especially useful to have graphing function capability of some 
sort. For those more advanced you can use the Calculus to determine the 
volume and surface area of various shaped eggs, but the latter gets messy 
because it involves
(d/dx)^2.

Vince Calder
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