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On hyperbolic virtual polytopes and hyperbolic fans

100%
Panina G.
Open Mathematics
|
2006
|
tom 4
|
nr 2
270-293
EN
Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ ℝ3 be a smooth convex body. If for a constant C, at every point of ∂K, we have R 1 ≤ C ≤ R 2 then K is a ball. (R 1 and R 2 stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic virtual polytope (and therefore, a hyperbolic hérisson) with odd an number of horns is constructed. Moreover, various properties of hyperbolic virtual polytopes and their fans are discussed.
2
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An illustrated theory of hyperbolic virtual polytopes

63%
Knyazeva M., Panina G.
Open Mathematics
|
2008
|
tom 6
|
nr 2
204-217
EN
The paper gives an illustrated introduction to the theory of hyperbolic virtual polytopes and related counterexamples to A.D. Alexandrov’s conjecture.
3
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Morse index of a cyclic polygon

63%
Panina G., Zhukova A.
Open Mathematics
|
2011
|
tom 9
|
nr 2
364-377
EN
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.
4
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Critical configurations of planar robot arms

51%
Khimshiashvili G., Panina G., Siersma D., Zhukova A.
Open Mathematics
|
2013
|
tom 11
|
nr 3
519-529
EN
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.
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