How do you find out if a polar curve retraces itself?

Question

I was solving a question where you were supposed to find the area of petal formed by a polar curve. But the answer I got was greater than the actual answer by a factor of 2. Apparenty few polar curves retrace themselves, how do you find out wherther a polar curve retraces itself?

Answer 1

If your function is $r(\theta)$, with the polar radius expressed as a function of the polar angle, then it retraces itself if $r(\theta) = r(\theta + 2\pi)$ (or possibly if $r(\theta) = -r(\theta + \pi)$).