Points on a lemniscate

Question

I have this lemniscate I plotted $r^2 = a^2 \cos 2\theta$ and that $a = 1$.

And I have these questions:

1) Are there values of $\theta$ that do not give any points on the curve? - My guess is a no since I know that the lemniscate rotates at $2\pi$.

2) Are there any values of θ that gives more than 2 points? - My guess is also no since every value $\theta$ gives a unique value.

Please correct me if I'm wrong. Thanks in advance!

Answer 1

You can only plot the lemniscate in the range

$$\theta \in \left[-\frac{\pi}{4},\frac{\pi}{4}\right]\cup \left[\frac{3\pi}{4},\frac{5\pi}{4}\right]$$.

For other values of $\theta$, you have that $\cos(2\theta)<0$, so you cannot compute $r^2$ in polar coordinates.

So the first question is true.

I don't understand the second question.