My instructor asked us to find the polar form of the elliptic curve defined by the equation $$y^2=x^3+ax+b$$
What I did:
Using $x=r\cos\theta$ and $y=r\sin\theta$, I got
$$r^2\sin^2\theta=r^3\cos^3\theta+ar\cos\theta+b$$
That's all I got so far. I want to derive an equation of $r$ in terms of $\theta$, so I'm not sure how to advance from this.
Any help will be greatly appreciated.
You can solve the cubic equation in $r$ by means of the Cardano formula, after depression. Nothing really nice. In fact, plain awful.
https://www.wolframalpha.com/input/?i=r%5E2+(sin+t)%5E2%3Dr%5E3+(cos+t)%5E3%2Ba+r+cos+t+%2Bb,+solve+for+r