How to determine a point is outside or inside

Question

How could I determine a point is outside or inside of a domain with variable raduis. like this:

$$x(t)=(0.3+0.2(\sin3t))\cos t$$ $$y(t)=(0.3+0.2(\sin3t))\sin t$$

where 0$\leq t< 2\pi$.

I tried with $r=0.3+0.2(\sin3t)$ and checked $x^2+y^2-r^2$, but it does not work.

Answer 1

To test whether a point $(x_0,y_0)$ is inside the curve, test whether $x_0^2+y_0^2-r^2<0$ for $r=0.3+0.2\sin3t$ with $t=\arg(x_0,y_0)=\mbox{atan2}(y,x)$. Here atan2 is used for converting between polar and Cartesian coordinates.