Cyclic polygons as critical points

We are concerned with the critical points of several natural functions on the moduli spaces of polygonal linkages. For planar linkages, it will be shown that the critical points are given by the cyclic configurations of linkage considered. It will be established that the signed area is a Morse function on the moduli space of generic planar polygonal linkage and it will be explained how one can effectively calculate its Morse indices. A few applications to the geometry of cyclic polygons and topology of moduli spaces will also be presented. In particular, we will explain how to calculate the critical values of signed area and describe relations to the classical Brahmagupta formula for cyclic polygons and conjectures of D.Robbins. Possible analogs of the aforementioned results for the signed volume and electrostatic potential will also be discussed