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I'm curious about what I did in Geogebra while I was experimenting with animation. Here it is:

  1. Construct a circle with radius $r$
  2. 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 $B$.
  3. Get the midpoint of $AB$. Name this point $M$.
  4. 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

For $a = 2.1$ and $b = 1.4$, enter image description here

For $a = 3$ and $b = 1$, enter image description here

And for $a = 1.1$ and $b = 0.007$ $(0 \leq \theta \leq 98\pi)$, enter image description here

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*}

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  • $\begingroup$ It looks like what they call the evolute of a cardioid in this page (second figure) mathcurve.com/courbes2d.gb/largeur%20constante/… $\endgroup$
    – Evariste
    Jul 7 at 13:31
  • $\begingroup$ Depending on a and b the curves can take many forms with multiple loops (b=1, a = 3) or spirals (a=2.3, b=2) so not cardioid, though one may be a special case. $\endgroup$
    – Paul
    Jul 7 at 14:31
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    $\begingroup$ This has the properties of several of the so-called transcendental curves, such as the sinusoidal spirals, Archimedean spiral, involute of a circle, and cochleoid. Curves are defined by equations, Cartesian and polar. You need to express this curve as an equation in order to define it, rather than a set of instructions. My go-to reference for plane curves is A Catalog of Special Plane Curves, by J. Dennis Lawrence, Dover Publications, 1972. $\endgroup$ Jul 9 at 14:59
  • $\begingroup$ @CyeWaldman The parametric equation can be found before Update 1. $\endgroup$
    – soupless
    Jul 9 at 16:24
  • $\begingroup$ @soupless Thank you. From those equations, I would conclude that this is an unnamed curve. On a quick search the closest curve I can find to this is the hypotrochoid, described by $x=n\cos t+h\cos(n/b\cdot t)$ and $y=n\sin t-h\sin(n/b\cdot t)$. $\endgroup$ Jul 9 at 20:25
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I am posting this since my updates that can answer my question should be posted as an answer, not as a part of the question.


We restrict the values of $a$ and $b$ to be integers where $a$ and $b$ are both nonzero. Then, the curve seems to represent an epitrochoid given by the parametric equation \begin{align*} x(t) &= m \cos t - h \cos\left(\frac{mt}{c}\right) \\ y(t) &= m \sin t - h \sin\left(\frac{mt}{c}\right). \end{align*}

where $m = 1$, $h = 1$, and $c = \frac{a}{b}$ for $0 \leq t \leq 2a\pi$.. For the parametric equation in the question, we let $r = 2$.

We can say that the two curves match if they are in the same magnitude and angle, and that they don't match if they don't have the same angle and magnitude. However, we'll say that they are a pseudomatch if they have the same magnitude but not having the same angle.

Now, for odd $a$, the two curve are a pseudomatch regardless of the value of $b$. For even $a$, there are two cases:

  1. If $a$ is a power of $2$, then the curves are a pseudomatch if $b$ is a multiple of $a$. Otherwise, the curves are a match.
  2. If $a$ is not a power of $2$, then the curves are a pseudomatch if $b$ is a multiple of the greatest power of $2$ that divides $a$. An example would be $a = 28$, where the curves are a pseudomatch if $b$ is a multiple of $4$.

As of the moment, I don't know how to prove that this works. I am just relying on graphs from the file.

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