# Periodicity of Polar Curves

I am a math educator preparing a unit on the calculus of polar curves. This is my first time teaching this particular unit, so it was also the first time I noticed that the "periods" of different polar curves were not what I expected. By "period" of a polar curve I mean the minimum angle that must be spanned by $$\theta$$ in order to trace the polar curve such that it may intersect itself but not overlap itself.

Here are some examples:

The period of $$r(\theta)=cos(2\theta)$$ is $$2\pi$$, even though the period of $$y=cos(2x)$$ is $$\pi$$.

The period of $$r(\theta)=cos(\theta/2)$$ is $$4\pi$$, and so also is the period of $$y=cos(x/2)$$.

The period of $$r(\theta)=tan(\theta/3)$$ is $$6\pi$$, even though the period of $$y=tan(x/3)$$ is $$3\pi$$.

Can anyone explain a general method for determining the "period" of a polar curve and perhaps even its relation to the period of the related trigonometric function?

So as you have probably seen geometrically, the periodicity of the function $$r(\theta)$$ tells you that you're going to get the same "shape" every so often, but it might be rotated based on what the period actually is; for example, the shape of $$r = \cos(2\theta)$$ for $$\theta\in[\pi,2\pi]$$ is the same shape as $$r=\cos(2\theta)$$ for $$\theta\in[0,\pi]$$, but rotated to be upside down (i.e. rotated by $$\pi$$) and thus doesn't overlap.
If you want the points $$(x,y)$$ to overlap, you should look at the points $$(x,y)$$, that is $$(r\cos\theta,r\sin\theta)$$, and see what the period of those are. So in your first example, the $$(x,y)$$ coordinates are given by $$(\cos(2\theta)\cos\theta, \cos(2\theta)\sin\theta)$$ which have period $$2\pi$$. (Of course if your curve isn't given as $$r=f(\theta)$$ you maybe have a bit more work to do.)