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 Ω, is to define a non-negative distance function d(x,y) on
Ω×Ω 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
Ω, we show how to assign coordinates to each point of Ω 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 Ω, 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 Ω, 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