  We study self-approaching paths that are contained in a simple polygon. A
self-approaching path is a directed curve connecting two points such that the
Euclidean distance between a point moving along the path and any future
position does not increase, that is, for all points $a$, $b$, and $c$ that
appear in that order along the curve, $|ac| \ge |bc|$. We analyze the
properties, and present a characterization of shortest self-approaching paths.
In particular, we show that a shortest self-approaching path connecting two
points inside a polygon can be forced to use a general class of non-algebraic
curves. While this makes it difficult to design an exact algorithm, we show how
to find a self-approaching path inside a polygon connecting two points under a
model of computation which assumes that we can calculate involute curves of
high order. Lastly, we provide an algorithm to test if a given simple polygon
is self-approaching, that is, if there exists a self-approaching path for any
two points inside the polygon.
