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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.

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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

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  • $\begingroup$ I tried to graph the result and it looks nothing like an elliptic curve. $\endgroup$ – Lhoel Busano Jul 12 at 4:49
  • $\begingroup$ @LhoelBusano: do you know how an elliptic curve looks like ? $\endgroup$ – Yves Daoust Jul 12 at 8:02

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