What's an equation that describes a circle, with oscillating outline?

Question

I'm looking for an equation that describes a circle with oscillating outline. See picture below. Anyone who knows a good way to do that?

Thanks a lot!

Answer 1

In polar coordinates, add some periodic perturbation to a constant radius. E.g.

$$\rho=R+r\cos(n\theta).$$

Answer 2

Take, for instance, $\displaystyle\theta\mapsto\left(\cos(\theta)+\frac{\cos(10\theta)}{10},\sin(\theta)+\frac{\sin(10\theta)}{10}\right)$.

Answer 3

The parametrisation $$ \theta\mapsto(\cos\theta, \sin\theta) $$ gives a circle. Now we want a wavey motion about along circle. Note that the parametrisation $$ \theta\mapsto\left(\frac{\cos10\theta}{10}, 0\right) $$ is a curve that goes back and forth along the $x$ axis, so if we rotate it by an angle of $\theta$: $$ \theta\mapsto \left(\frac{\cos10\theta}{10}\cdot \cos\theta, \frac{\cos10\theta}{10}\cdot\sin\theta\right) $$ this should give us the a nice radial waving motion when added to our circle.

The full expression that this gives us is $$ \theta\mapsto\left(\cos\theta + \frac{\cos10\theta}{10}\cdot \cos\theta, \sin\theta + \frac{\cos10\theta}{10}\cdot\sin\theta\right) $$ which gives the following figure:

This turns out to be exactly the same curve that some others have suggested by working in polar coordinates, and add a small sine perturbation to a constant radius. You can freely adjust the denominator $10$ to change the oscillation amplitude, or the coefficient $10$ of $\theta$ to change the oscillation frequency.