I'm curious about what I did in Geogebra while I was experimenting with animation. Here it is:
- Construct a circle with radius $r$
- Define two points $A(r\cos a\theta , r \sin a\theta)$ and $B(r\cos b\theta, r \sin b\theta)$. Let $(r,0)$ be the initial position of both $A$ and $B$.
- Get the midpoint of $AB$. Name this point $M$.
- Change $\theta$ continuously starting from $0$ until $A$ and $B$ are both at $(r,0)$ again.
The midpoint, by doing the fourth step, seem to make some form of curve. I noticed that the curved traced by the midpoint does not change as long as $a/b$ does not change, even if $a$ and $b$ does so. Also, it doesn't matter if the values of $a$ and $b$ are swapped, but to avoid such ambiguities, we will avoid swapping of values and $a$ is always greater than $b$.
For the file, see this graph in Desmos.
For the angle, let $p$ be the numerator of $a/b$ in lowest terms. Then, we have $0 \leq \theta \leq 2p\pi$. (I don't know how to this in Desmos, by the way)
Examples
And for $a = 1.1$ and $b = 0.007$ $(0 \leq \theta \leq 98\pi)$,
As for the midpoint, the coordinates is $$\left(\frac{r}{2}\left(\cos(a\theta) + \cos(b\theta)\right), \frac{r}{2}\left(\sin(a\theta) + \sin(b\theta)\right)\right)$$
This means that the parametric equation is \begin{align*} x(\theta) &= \frac{r}{2}\left(\cos(a\theta) + \cos(b\theta)\right) \\[10pt] y(\theta) &= \frac{r}{2}\left(\sin(a\theta) + \sin(b\theta)\right) \end{align*} Is there a name for these curves?
Update 1: It seems like this is somehow related to the mathematical basis of a spirograph.
... and therefore the trajectory equations take the form \begin{align*} x(t) &= R\left[(1-k)\cos t+lk\cos {\frac{1-k}{k}}t\right],\\y(t)&=R\left[(1-k)\sin t-lk\sin {\frac {1-k}{k}}t\right]\end{align*}