# Is a shape 'polarizable'?

Given a point $p$ inside a shape $S$ described as an $n$-vertex polygon, let us say that $S$ is polar with respect to $p$ if S can be described by a polar equation $r(\theta)$ with $p$ as the origin. This is equivalent to saying that every ray cast from $p$ outwards intersects $S$ exactly once.

Furthermore, let us say that a shape $S$ is polarizable if there exists at least one $p$ inside $S$ such that $S$ is polar with respect to $p$. Otherwise, we say that $S$ is non-polarizable.

Intuitively it seems to me that any convex polygon is polar with respect to any point inside it (figure A). Some concave shapes are polar with respect to some points inside them (figure B1) but not with respect to others (figure B2). Yet other concave shapes are non-polarizable (figure C).

I am making all the above remarks intuitively. My question is: are there algorithms for solving these problems formally? In particular, given $S$ can we find which points inside $S$ make $S$ polar with respect to them, if any?

It is easy to find out if a given polygon $S$ is polarizable, also called star-shaped:
Take an edge of $S$ and extend it into a line. This edge separates the plane into two half-planes, an inner and an outer one. Mark all points in the outer half-plane. Note that if $S$ is not convex, this may mark some points in the interior of $S$.
Repeat this for every edge of $S$. Let $P$ be the set of all points that were never marked. Now $P$ is the set of all possible poles for $S$, and $S$ is star-shaped iff $P$ is nonempty.