Parametric equations for regular and Reuleaux polygons

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:

Regular and Reuleaux Polygons, 3-11I’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.

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5 Responses to Parametric equations for regular and Reuleaux polygons

  1. hi we talked a bit on math.stackexchange about implicit vs parametric plotting.

    Really i want you to appreciate where i’m coming from. I made this about two years ago or so http://home.arcor.de/petersheldrick/conic.html – it draws ‘the conic’ given by five points (that you can also move). It was a _huge_ pain to parametrize it (and yes this uses parametric plotting). I would be very happy if i could just plot this implicitly.

    I definitely agree with you, algorithms for implicit plotting are non-trivial. But i think the effort is worth it.

    • tpfto says:

      Oh, I can surely imagine the difficulty. I’ve had to write a conic plotter for a certain programmable calculator, and all the code for checking the discriminant, determining orientation, rotation and other sundry parameters, and then picking the proper parametrization required a fair amount of code. :) But more often than not, I’ve found it easier to derive a curve’s “properties” from parametric equations as opposed to implicit equations.

      In any event, I don’t believe I’ve seen a better method than Tupper’s…

  2. Gibran says:

    Very useful information, thanks for sharing

  3. For a regular n-gon, the equation is for r while that for the reuleaux ones, you have given both x and y values at the points. Can you explain how to get x and y for the regular n-gon also?

    • J. M. says:

      The simplest way to do it would be to use the usual polar to rectangular conversion: if r=r(\theta) is the polar equation of a given curve, then the corresponding parametric equations are x=r(\theta)\cos\theta,\;y=r(\theta)\sin\theta.

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