Morse index of a cyclic polygon

Abstract

It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper, we announce an explicit formula of the Morse index for the signed area of a cyclic configuration.It depends not only on the combinatorics of a cyclic configuration, but also includes some metric characterization.

Critical configurations of planar robot arms

“…The order of the lengths The proof (which repeats the proof of the similar lemma for closed polygons from [6]) is as follows. Two consecutive edges of a configuration can be (geometrically) permuted in such a way that the oriented area remains unchanged.…”

“…These special configurations are related to critical points of functions on configuration spaces (respectively, oriented area of an arm, squared length of the closing interval, see [1], and oriented area of a polygon, see [5] Proof. The proof from [6] is applicable with some evident modifications. Namely, after introducing a local coordinate system with diagonals as coordinates, the Hessian matrix becomes tridiagonal with analytic entries.…”

“…The present paper is concerned with certain planar configurations of robot arms appearing as critical points of the oriented area considered as a function on the moduli space of the arm in question. This setting naturally arose in the framework of a general approach to extremal problems on configuration spaces of mechanical linkages developed in [2,4,5], which has led to a number of new results on the geometry of cyclic polygons [3,6] and suggested a variety of open problems. The approach and results of [2,5] provided a paradigm and basis for the developments presented in this paper.…”

On the area of a polygonal linkage

“…Theorem 2 [8]. This fact is proved by analyzing the Hessian of A calculated in a special coordinate system.…”

Morse index of a cyclic polygon. II

“…The way to count the Morse index, found in [8], is simplified. In Theorem 3.1 we present a simple formula for the Morse index of a cyclic configuration.…”