  The game of cops and robbers is a pursuit game on graphs where a set of
agents, called the cops try to get to the same position of another agent,
called the robber. Cops and robbers has been studies on several classes of
graphs including geometrically represented graphs. For example, it has been
shown that string graphs, including geometric graphs, have cop number at most
15. On the other hand, little is known about geometric graphs of any cop number
less than 15 and there is only one example of a geometric graph of cop number
three that has as many as 1440 vertices. In this paper we present a
construction for subdividing planar graphs of maximum degree $\le 5$ into
geometric planar graphs of at least the same cop number. Indeed, our
construction shows that there are infinitely many planar geometric graphs of
cop number three. We also present another construction that consists in clique
substitutions alongside subdividing the edges in a planar graph of maximum
degree $\le 9$, resulting in geometric, but not necessarily planar, graphs of
at least the same cop number as the starting graphs.
