Complex Matroids Phirotopes and Their Realizations in Rank 2

The motivation for this article comes from the desire to link two seemingly incompatible worlds: oriented matroids and dynamic geometry. Oriented matroids have proven to be a perfect tool for dealing with sidedness information in geometric configurations (for instance for the computation of convex hulls). Dynamic geometry deals with elementary geometric constructions in which moving certain free elements controls the motion of constructively dependent elements In this field the introduction of complex coordinates has turned out to be a “key technology” for achieving a consistent continuous movement of the dependent elements. The additional freedom of an ambient complex space makes it possible to bypass disturbing singularities. Unfortunatly, complex coordinates seem to make it impossible to use oriented matroids which are heavily based on real numbers.