May 22, 2017
Consider the set \(U\) of all points on the unit circle \(r(\theta) = 1\) in \(\mathbb{R}^2\).
Consider a rotation about the origin by an angle \(\beta\) given by
$$R[\beta](r,\theta) = (r,\theta+\beta)$$in polar coordinates.
Notice that the unit circle remains invariant under the rotation \(R[\beta]\), regardless of the choice of \(\beta\), since for every \((r(\theta),\theta) \in U\),
$$ R[\beta](r(\theta),\theta) = R[\beta](1,\theta) = (1,\theta+\beta) = (r(\theta+\beta),\theta+\beta)$$Let \(kU = \{(kr,\theta) : (r,\theta) \in U\}\), where \(k \geq 0\).
\(kU\) is the set of all points located on the circle of radius \(k\), centered at the origin. It is also invariant under the rotation \(R[\beta]\) for arbitrary \(\beta\), since for every \((kr(\theta),\theta) \in kU\),
$$ R[\beta](kr(\theta),\theta) = R[\beta](k,\theta) = (k,\theta+\beta) = (kr(\theta+\beta),\theta+\beta)$$Now consider the set \(T\) of all points on an equilateral triangle centered at the origin.
Notice that \(T\) is only invariant under \(R[\beta]\) for the three values \(\beta = 0,\frac{2\pi}{3},\frac{4\pi}{3}\), since equilateral triangles have only threefold rotational symmetry.
We would like to define some operation \(R_{T}[\beta]\), similar to a rotation, so that \(T\) is invariant under \(R_{T}[\beta]\) for any choice of \(\beta\), and we would also like any multiple \(kT\) to be similarly invariant.
Let \(\rho(\theta)\) be the radial function that describes \(T\).
Then we want the operator \(R_{T}[\beta]\) to have the property
$$ R_{T}[\beta](k\rho(\theta),\theta) = (k\rho(\theta+\beta),\theta+\beta).$$Note that for any \((r,\theta) \in \mathbb{R}^2\) there exists some \(k\) such that \(k\rho(\theta) = r\).
Specifically \(k = \frac{r}{\rho(\theta)}\).
Therefore,
$$ R_{T}[\beta](r,\theta) = R_{T}[\beta](k\rho(\theta),\theta) = (k\rho(\theta+\beta),\theta+\beta) = \left(\frac{r}{\rho(\theta)}\rho(\theta+\beta),\theta+\beta\right).$$So the desired operation is \(R_{T}[\beta](r,\theta) = \left(r\frac{\rho(\theta+\beta)}{\rho(\theta)},\theta+\beta\right)\)
Notice that nothing we have done so far has actually been specific to an equilateral triangle, but could also be applied to just about any curve, (presuming it is well-behaved, whatever that entails).
The radial component for a regular n-gon with outer radius 1 and vertex at \(\theta = 0\) is given by
$$ \rho(\theta) = \frac{\cos{\frac{\pi}{n}}}{\cos{((\theta \bmod \frac{2\pi}{n})-\frac{\pi}{n}})}.$$Therefore,
$$R_{T}[\beta](r,\theta) = \left(\frac{r\cos{((\theta \bmod \frac{2\pi}{3})-\frac{\pi}{3}})}{\cos{((\theta +\beta \bmod \frac{2\pi}{3})-\frac{\pi}{3}})},\theta\right)$$The following animation shows the image of an equilateral triangle, an inscribed circle, and a circumscribing circle under \(R_{T}[\beta]\) for \(\beta\) from 0 to 360 degrees, incrementing by 1 degree for frame.