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Path planning is an important problem in robotics. One way to plan a pathbetween two points $x,y$ within a (not necessarily simply-connected) planardomain $\Omega$, is to define a non-negative distance function $d(x,y)$ on$\Omega\times\Omega$ such that following the (descending) gradient of thisdistance function traces such a path. This presents two equally importantchallenges: A mathematical challenge -- to define $d$ such that $d(x,y)$ has asingle minimum for any fixed $y$ (and this is when $x=y$), since a localminimum 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 $\Omega$ anddefine a family of distance functions between points using these coordinates,such that both the mathematical and the computational challenges are met. Thisis 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 finiteset of \emph{sites} sampled from $\Omega$, so any method that works well on thecontinuous domain must be adapted so that it still works well on the discretedomain. Given a set of sites sampled from $\Omega$, we show how to define anetwork connecting these sites such that a \emph{greedy routing} algorithm(which is the discrete equivalent of continuous gradient descent) based on thedistance function mentioned above is guaranteed to generate a path in thenetwork 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.

Practical Distance Functions for Path-Planning in Planar Domains