  The unit disk graph (UDG) is a widely employed model for the study of
wireless networks. In this model, wireless nodes are represented by points in
the plane and there is an edge between two points if and only if their
Euclidean distance is at most one. A hop spanner for the UDG is a spanning
subgraph $H$ such that for every edge $(p,q)$ in the UDG the topological
shortest path between $p$ and $q$ in $H$ has a constant number of edges. The
hop stretch factor of $H$ is the maximum number of edges of these paths. A hop
spanner is plane (i.e. embedded planar) if its edges do not cross each other.
  The problem of constructing hop spanners for the UDG has received
considerable attention in both computational geometry and wireless ad hoc
networks. Despite this attention, there has not been significant progress on
getting hop spanners that (i) are plane, and (ii) have low hop stretch factor.
Previous constructions either do not ensure the planarity or have high hop
stretch factor. The only construction that satisfies both conditions is due to
Catusse, Chepoi, and Vax\`{e}s (2010); their plane hop spanner has hop stretch
factor at most 449.
  Our main result is a simple algorithm that constructs a plane hop spanner for
the UDG. In addition to the simplicity, the hop stretch factor of the
constructed spanner is at most 341. Even though the algorithm itself is simple,
its analysis is rather involved. Several results on the plane geometry are
established in the course of the proof. These results are of independent
interest.
