  While structural width parameters (of the input) belong to the standard
toolbox of graph algorithms, it is not the usual case in computational
geometry. As a case study we propose a natural extension of the structural
graph parameter of clique-width to geometric point configurations represented
by their order type. We study basic properties of this clique-width notion, and
relate it to the monadic second-order logic of point configurations. As an
application, we provide several linear FPT time algorithms for geometric point
problems which are NP-hard in general, in the special case that the input point
set is of bounded clique-width and the clique-width expression is also given.
