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One might ask, whether it is possible to provide a regular polygon as a continuous curve, possibly given in polar coordinates?

Indeed, this is an easy exercise. We just need some values from the page on the general regular polygon:

Dynkin diagram
| x-n/d-o |

Schläfli symbol
| {n/d} |

Circumradius
| 1/(2 sin(π d/n)) |

Inradius
| 1/(2 tan(π d/n)) |

First consider a vertical side for some positive `x`.

From `x = r cos(θ)` we obviously get `r(θ) = x / cos(θ)`.

As that side will be centered at its midpoint at the distance of the inradius, one further has `x = 1/(2 tan(π d/n))`.

Thus this correctly displaced vertical line would be given within polar coordinates as

r(θ) = 1/[2 tan(π d/n) cos(θ)]

Next one has to determine the 2 vertices of this side.

These will be given at `θ = ±π d/n`, up and down from the `x`-axis.

The remainder then just is to apply that very angular region over and over again periodically.

r(θ) = 1/[2 tan(π d/n) cos(θ - 2π d/n floor[(θ n + π d)/(2π d)])]

(where `floor(x) = x - (x mod 1)`, for sure).

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