Thursday, April 24, 2014

Running circles around circles: hypocycloid

Running circles around circles: hypocycloid

Turning angles is the key here. The angle that the center of the small circle has turned is the sum of two angles, because it turned around its own center, and also turned around O. Total angle:

\begin{align*} \alpha &= \theta - a\theta/b \end{align*}

(Note the directions are opposite.)

The vector OP can be written as: \begin{align*} OP &= OC + CP \end{align*} while \begin{align*} OC &= \begin{pmatrix} (a-b)\cos \theta \\ (a-b)\sin \theta \\ \end{pmatrix} \\ CP &= \begin{pmatrix} b\cos \alpha \\ b\sin \alpha \\ \end{pmatrix} \end{align*}

The total turning angle of C can be deduced in a more rigorous way.

First we set up a suitable coordinate system as in the figure below:


Easily the x-axis vector is:

\begin{align*} \begin{pmatrix} \cos \theta \\ \sin \theta \\ \end{pmatrix} \end{align*} The y-axis vector can gotten by a rotation of $\pi/2$:

\begin{align*} \begin{pmatrix} \cos(\pi/2) & -\sin(\pi/2) \\ \sin(\pi/2) &\cos(\pi/2) \end{pmatrix} \begin{pmatrix} \cos \theta \\ \sin \theta \\ \end{pmatrix} &= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \cos \theta \\ \sin \theta \\ \end{pmatrix} \\ &= \begin{pmatrix} -\sin \theta \\ \cos \theta \\ \end{pmatrix} \\ \end{align*}

Thus the coord system centered at C is : \begin{align*} \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{pmatrix} \\ \end{align*} And under this, the coord of P is : \begin{align*} \begin{pmatrix} \cos(-a\theta/b) \\ \sin(-a\theta/b) \\ \end{pmatrix} \end{align*} Switching it to the normal coord: \begin{align*} I^{-1} \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{pmatrix} \begin{pmatrix} \cos(-a\theta/b) \\ \sin(-a\theta/b) \\ \end{pmatrix} &= \begin{pmatrix} \cos(\theta-a\theta/b) \\ \sin(\theta-a\theta/b) \\ \end{pmatrix} \end{align*} This is beautifully in agreement with our intuition: it's just an rotation of $\theta$.