Title: Practical Distance Functions for Path-Planning in Planar Domains,
Abstract: Path planning is an important problem in robotics. One way to plan a path
between two points $x,y$ within a (not necessarily simply-connected) planar
domain $\Omega$, is to define a non-negative distance function $d(x,y)$ on
$\Omega\times\Omega$ such that following the (descending) gradient of this
distance function traces such a path. This presents two equally important
challenges: A mathematical challenge -- to define $d$ such that $d(x,y)$ has a
single minimum for any fixed $y$ (and this is when $x=y$), since a local
minimum is in effect a "dead end", A computational challenge -- to define $d$
such that it may be computed efficiently. In this paper, given a description of
$\Omega$, we show how to assign coordinates to each point of $\Omega$ and
define a family of distance functions between points using these coordinates,
such that both the mathematical and the computational challenges are met. This
is done using the concepts of \emph{harmonic measure} and
\emph{$f$-divergences}.
In practice, path planning is done on a discrete network defined on a finite
set of \emph{sites} sampled from $\Omega$, so any method that works well on the
continuous domain must be adapted so that it still works well on the discrete
domain. Given a set of sites sampled from $\Omega$, we show how to define a
network connecting these sites such that a \emph{greedy routing} algorithm
(which is the discrete equivalent of continuous gradient descent) based on the
distance function mentioned above is guaranteed to generate a path in the
network between any two such sites. In many cases, this network is close to a
(desirable) planar graph, especially if the set of sites is dense.