Parametric equations for regular and Reuleaux polygons

September 15, 2011

Not having any other place to note this down, I suppose this blog entry is as good as any.

From this m.SE answer, user Raskolnikov demonstrates a polar equation whose graph is a regular $n$-gon (slightly reformulated):

$r=\cos\frac{\pi}{n}\sec\left(\theta-\frac{\pi}{n}\left(2\left\lfloor\frac{n\theta}{2\pi}\right\rfloor+1\right)\right)$

Through similar considerations, one can derive parametric equations for Reuleaux polygons ($n$ of course being odd):

$x=2\cos\frac{\pi}{2n}\cos\left(\frac12\left(t+\frac{\pi}{n}\left(2\left\lfloor\frac{n t}{2\pi}\right\rfloor+1\right)\right)\right)-\cos\left(\frac{\pi}{n}\left(2\left\lfloor\frac{n t}{2\pi}\right\rfloor+1\right)\right)$
$y=2\cos\frac{\pi}{2n}\sin\left(\frac12\left(t+\frac{\pi}{n}\left(2\left\lfloor\frac{n t}{2\pi}\right\rfloor+1\right)\right)\right)-\sin\left(\frac{\pi}{n}\left(2\left\lfloor\frac{n t}{2\pi}\right\rfloor+1\right)\right)$

Here’s a side-by-side comparison of some odd-sided regular and Reuleaux polygons:

I’m not sure if there are simpler expressions for the parametric equations of the Reuleaux polygon, though, and I would be interested if you, the reader, might come up with something a bit simpler.