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

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!

• This can't be a real function as there are several $y$ for one $x$. – Yves Daoust Jun 25 '19 at 12:26
• A function? It does not pass the vertical line test so it is not a function from $x$ to $y$. Do you mean for it to be a parametric function and have it be a function from $t$ to $(x,y)$? Or how about a polar function and have it be a function from $\theta$ to $r$? – JMoravitz Jun 25 '19 at 12:26
• A polar equation that is similar, how about $f(\theta) = 1 + \frac{1}{4}\sin(10\theta)$ – JMoravitz Jun 25 '19 at 12:27

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

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

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

• This gives cusps on the inside of the curve, not smooth waving shape that the drawing seems to want. – Arthur Jun 25 '19 at 12:28

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.