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Polygon Curve

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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