Why does Desmos outlines $r=\cos(2\theta)$ differently from $r\le\cos(2\theta)$?

Question

Why does Desmos draw different perimeters for the graphs of $$r=\cos(2\theta)$$ and $$r\le\cos(2\theta)$$ (2 leaves versus 4 leaves)? Why do the numbers of leaves differ?

$r=\cos(2\theta)$

$r\le\cos(2\theta)$

Answer 1

In the second case, you are throwing out the cases where $\cos(2\theta)$ goes negative. In particular, the vertical petals are drawn when $r$ goes negative. For example, the lower petal corresponds to $\frac{\pi}{4}<\theta<\frac{3\pi}{4}$.

Answer 2

When drawing polar curves, Desmos uses $$(r,\theta)\overset{\text{def}}\equiv(-r,\theta+\pi),$$ but when graphing polar inequalities, it ignores negative values of $r$.

Its official website suggests inputting $$r\le|\cos(2\theta)|$$ instead of $$r\le\cos(2\theta)$$ if you want to see four shaded petals instead of two.