We investigate critical points of several natural functions on the configuration space of polygonal linkage. The main attention is given to the signed area A and electrostatic potential E considered as functions on the planar configuration space C(L) of a generic polygonal linkage L. The following three settings will be discussed in some detail. Firstly, we consider A as a function on C(L) and show that its critical points are given by the cyclic configurations of L (as usual a configuration is called cyclic if it can be inscribed in a circle, i.e., there exists a point in the reference plane equidistant from all vertices of L). We also show that, generically, A is a Morse function and find the Morse indices of cyclic configurations. This, in particular, enables us to obtain useful information about C(L) by merely examining the cyclic configurations of L, which can be effectively done in many cases. Next, we consider A as a function on the configuration space of an n-arm (open polygonal n-chain) and show that its critical points are again given by the cyclic configurations. In this case the structure of critical points can be more complicated and we'll illustrate certain typical phenomena by concrete examples. Similar results hold for electrostatic potential E on the planar configuration space of polygonal linkages satisfying some additional conditions which guarantee boundedness of E. The same problems become drastically more complicated in the case of spatial linkage and we'll end by presenting a few conjectures and open problems arising in this context.
Go to day