Sunday, September 23, 2007

Geometry Safari - Squircles, Supercircles, and Superellipses

Everyone knows that the equation of a circle is the beautifully simple:

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:

x4 + y4 = C

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.

1 comment:

Antonio said...

Thank you, I enjoy your Geometry Safari article. I would like to invite you to visit Geometry from the Land of the Incas at:
www.agutie.com

Antonio