Centered configurations of open linkages

Abstract

We discuss cyclic (circumscriptible) and tangential (inscriptible) configurations of a planar open polygonal linkage L. It is shown that cyclic configurations generically form a smooth onedimensional submanifold C(L) of the planar moduli space M(L) while the set of tangential configurations is two-dimensional. Location of centers of associated circumscribed and inscribed circles is described and critical points of several natural functions on M(L) and C(L) are identified. Special attention is given to the critical points of the oriented area function on moduli space M(L). In particular, we give estimates for the number of critical points of the oriented area function and formulate two related conjectures.