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.

(In reply to Quad I by nikki)

For a=b=1, in the first quadrant, which has 1/4 of the area (and the area would be proportional for other a's and b's), the formula would be

y = (1-x^n)^(1/n)

When the Wolfram Integrator is asked to integrate this, it comes out with

x Hypergeometric2F1(1/n, -1/n, 1+1/n, x^n)

That's where I'm lost.

Posted by Charlie on 2004-10-28 18:24:25