[Math] Formula to create a Reuleaux polygon

algorithmsgeometryplane-curvespolygons

The Wikipedia articles for Reuleaux triangle and curve of constant width do a good job of describing the properties of a Reuleaux polygon, but they don't give a straightforward formula for computing or drawing such a figure, except in terms of the manual compass-and-straightedge construction.

Is there a formula or algorithm that, given the number of sides and the width/diameter, would give some data representation of a Reuleaux polygon that could be used to recreate it programmatically?

In particular, I'm looking for the coordinates of the vertices (or the angle/direction from one vertex to another) and the details of the arc connecting them.

Best Answer

I derived a parametric formula for Reuleaux polygons some time ago in this blog entry.

To make this post self-contained, here are the equations:

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

Here is a Mathematica demonstration:

Table[ParametricPlot[2 Cos[π/(2 n)] Exp[I (t + π (2 Floor[n t/(2 π)] + 1)/n)/2] -
                     Exp[I π (2 Floor[n t/(2 π)] + 1)/n] // ReIm, {t, 0, 2 π}],
      {n, 3, 7, 2}] // GraphicsRow

Reuleaux polygons

(Note the use of the complex form of the parametric equations.)

For some applications, a polar equation (like the one in this answer) might be more convenient. One can use the usual distance formula in polar coordinates to derive the polar equation of an $n$-sided Reuleaux polygon:

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

In Mathematica, one can do this:

Table[PolarPlot[With[{c = Cos[θ - 2 π Floor[n (θ - π)/(2 π) + 1/2]/n]}, 
                     c + Sqrt[1 + 2 Cos[π/n] + c^2]], {θ, 0, 2 π}],
      {n, 3, 7, 2}] // GraphicsRow

to get a picture identical to the one above.

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