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Morse index of a cyclic polygon

100%

Panina G. , Zhukova A.

Open Mathematics

2011

tom 9

nr 2

364-377

It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper we give an explicit formula of the Morse index for the signed area of a cyclic configuration. We show that it depends not only on the combinatorics of a cyclic configuration, but also on its metric properties.

Critical configurations of planar robot arms

80%

Khimshiashvili G. , Panina G. , Siersma D. , Zhukova A.

Open Mathematics

2013

tom 11

nr 3

519-529

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.