Find the intersection points of the polar curve function

Question

I have a lot of functions like:

r = tan(1.1θ),
r = tan(1.7θ),
r = tan(2.3θ),
r = tan(1.3θ) + 1.3,
r = tan(1.4θ) - 1

How to find for each of the functions of this type the points at which it intersects with itself?

Desmos:

r = tan(1.3θ) + 1.3

r = tan(1.1θ)

I'm not good at math, any help would be greatly appreciated.

Answer 1

First you can find the period of your functions: if they are in the form $r(\theta)=\tan(\alpha\theta)$ (I suppose $\alpha\in \mathbb{Q}^+$, so $\alpha=p/q$, with $p,q\in\mathbb{N}$ coprimes), you have to find the minimum $n\in\mathbb{N}$ that realise, for all $\theta$, \begin{align} r(\theta)&=r(\theta+2n\pi)\\ \tan\left(\frac{p}{q}\theta\right)&=\tan\left(\frac{p}{q}(\theta+2n\pi)\right)\\ \frac{p}{q}(\theta+2n\pi)-\frac{p}{q}\theta&=k\pi\qquad\text{for some }k\in\mathbb{Z}\\ \frac{2np}{q}&=k\\ n&=\frac{qk}{2p}\in\mathbb{N} \end{align} We want $n$ to be the minimum ($p$ and $q$ ar fixed, and $k$ free), so if $q$ is even, you can choose $k=p\;\Rightarrow\; n=(q/2)$ and the period will be $2n\pi=q\pi$. If $q$ is odd, you can choose $k=2p\;\Rightarrow\; n=q$ and the period will be $2n\pi=2q\pi$.

Let's take the case $q$ is even (the other case is similar, and many times I think you will have $q=10$). So we study the function $r(\theta)=\tan((p/q)\theta)$ for $\theta\in[0,q\pi)$. Now, the curve intersects itself for $\theta$ that realise

1. $r(\theta)=0$;
2. $r(\theta)=-r(\theta+(2k+1)\pi)$ for some $k\in\{0,1,\ldots,q/2-1\}$;
3. $r(\theta)=r(\theta+2n\pi)$ (but we have already found the period, so we can skip this passage).

For the first point we have $$ \tan\left(\frac{p}{q}\theta\right)=0 \;\Rightarrow\; \theta=h\frac{q}{p}\pi\quad\text{for }h=0,\ldots,p-1 $$

For the second point we have \begin{align} \tan\left(\frac{p}{q}\theta\right)&=-\tan\left(\frac{p}{q}(\theta+(2k+1)\pi)\right)\\ \frac{p}{q}\theta&=-\frac{p}{q}(\theta+(2k+1)\pi)+h\pi\quad\text{for some }h\in\mathbb{Z}\\ \theta&=\frac{h}{p}(q/2)\pi-(2k+1)\frac{\pi}{2} \end{align}
Where first you choose $k\in\{0,1,\ldots,q/2-1\}$, and then $h\in\{0,1,\ldots,2p\}$ (some points are eventually repeated).