  Topological Spatial Model Checking is a recent paradigm where model checking
techniques are developed for the topological interpretation of Modal Logic. The
Spatial Logic of Closure Spaces, SLCS, extends Modal Logic with reachability
connectives that, in turn, can be used for expressing interesting spatial
properties, such as "being near to" or "being surrounded by". SLCS constitutes
the kernel of a solid logical framework for reasoning about discrete space,
such as graphs and digital images, interpreted as quasi discrete closure
spaces. Following a recently developed geometric semantics of Modal Logic, we
propose an interpretation of SLCS in continuous space, admitting a geometric
spatial model checking procedure, by resorting to models based on polyhedra.
Such representations of space are increasingly relevant in many domains of
application, due to recent developments of 3D scanning and visualisation
techniques that exploit mesh processing. We introduce PolyLogicA, a geometric
spatial model checker for SLCS formulas on polyhedra and demonstrate
feasibility of our approach on two 3D polyhedral models of realistic size.
Finally, we introduce a geometric definition of bisimilarity, proving that it
characterises logical equivalence.
