Through testing, I have convinced myself that these two shapes are identical:
$$(x,y) = \frac{1}{\sqrt{a\cos^2\theta + b\sin^2\theta}}( a\cos\theta, b\sin\theta)$$
$$r(\phi) = \sqrt{\frac{ab}{a\sin^2\phi+b\cos^2\phi}}$$
However, I can't figure out how to prove this. I've just managed to prove that the former is an ellipse. I also tried writing the former on polar form, but it was so messy that there must be another way that is escaping me. How can I approach the problem?