I have a polar curve $r = \frac{2}{\theta}$ (which is a hyperbolic spiral) and I need to find out where it self-intersects. When $\theta$ is restricted to positive values it never intersects, but when it is allowed to be negative it intersects itself infinitely many times. My question is how can we prove that the curve does in face self-intersect, and, if possible, find the coordinates of a particular intersection point (a general formula would be lovely, but it seems like a far shot.). I have a (rather strong) feeling that this is really easy but I'm not getting how for some reason.
[Math] Finding self-intersections on a polar curve
polar coordinates
Best Answer
Polar coordinates $(r_1,\theta_1)$ and $(r_2,\theta_2)$ represent the same point if $$\theta_2=\theta_1+2k\pi\,,\ r_2=r_1\quad\hbox{or}\quad \theta_2=\theta_1+(2k+1)\pi\,,\ r_2=-r_1\,.$$ For your curve the first case gives $$\frac{2}{\theta_1}=\frac{2}{\theta_1+2k\pi}$$ which has no solution except for the trivial $k=0$. The second case gives $$\frac{2}{\theta_1}=-\frac{2}{\theta_1+(2k+1)\pi}\ ,$$ which gives $$\theta_1=-\Bigl(k+\frac{1}{2}\Bigr)\pi\ .$$ That is, for any integer $k$ $$\Bigr(\frac{-4}{(2k+1)\pi},\,-\frac{(2k+1)\pi}{2}\Bigr)\quad\hbox{and}\quad \Bigr(\frac{4}{(2k+1)\pi},\,\frac{(2k+1)\pi}{2}\Bigr)$$ are the same point.