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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? sketch of the intended result

Thanks a lot!

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    $\begingroup$ This can't be a real function as there are several $y$ for one $x$. $\endgroup$ – Yves Daoust Jun 25 '19 at 12:26
  • $\begingroup$ 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 $r$? $\endgroup$ – JMoravitz Jun 25 '19 at 12:26
  • $\begingroup$ A polar equation that is similar, how about $f(\theta) = 1 + \frac{1}{4}\sin(10\theta)$ $\endgroup$ – JMoravitz Jun 25 '19 at 12:27
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In polar coordinates, add some periodic perturbation to a constant radius. E.g.

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

enter image description here

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

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

enter image description here

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.

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