x2 + y2 = C
An obvious generalization of this is to allow higher values for the exponent:
|x|n + |y|n = C : n >= 2
This produces a family of nicely rounded closed curves which are asymptotic to squares. For n = 4 the curve looks like this:
It turns out this geometric shape has been given a cute name: squircles (Who said geometry has to be dull?) The entire family is known as supercircles. And of course, there is a analogous generalization for ellipses known as superellipses. (I don't think there's a "sqellipse", however - and maybe that's just as well).
This page allows you to generate supercircles for different values of n. (Incidentally it uses Walter Zorn's very slick jsGraphics library for generating pixel & vector graphics on web pages).
Useful? Sort of - Wikipedia has a few examples, mostly in the decorative arts. And I'm betting that a concept this elegant is bound to turn up in lots more places. So maybe I should add this to JTS, just to - ahem - get ahead of the curve.