There is a family of curves on the Cartesian plane described by this form:

If n is equal to 2, then it describes an ellipse (if a = b, then it describes a circle).

If n is greater than 2, then this is a "superellipse" (if a = b, then this is a supercircle).

As n increases, the ellipse becomes more "rectangularish", and as n approaches ∞, the limit is a rectangle (or a square if a=b).

What value must n have such that the figure has an area exactly halfway between the associated ellipse (when n=2) and rectangle (when n=∞)?

The graphs below, calculated by varying n with a = b = 1, show this property. Note that as n approaches zero, the curve degenerates into two crossed lines along the x- and y-axes.

If a=b=1, the area of the curve is A(n)=2^(2-2/n).√πΓ(1+1/n)/Γ(1/2+1/n).

For n=2, this works out to π, and for n=∞ the area is 4, so we want to solve A(n)=½(π+4).

Posted by Federico Kereki on 2004-10-28 18:56:10