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Polygons with prescribed edge slopes: configuration space and extremal points of perimeter

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Abstract

We describe the configuration space \(\mathbf {S}\) of polygons with prescribed edge slopes, and study the perimeter \({\mathcal {P}}\) as a Morse function on \(\mathbf {S}\). We characterize critical points of \({\mathcal {P}}\) (these are tangential polygons) and compute their Morse indices. This setup is motivated by a number of results about critical points and Morse indices of the oriented area function defined on the configuration space of polygons with prescribed edge lengths (flexible polygons). As a by-product, we present an independent computation of the Morse index of the area function (obtained earlier by Panina and Zhukova).

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Notes

  1. The space \(\mathbf {L}\) appears in the literature as “configuration space of a flexible polygon”, or “configuration space of a polygonal linkage”, or just as “space of polygons”.

  2. In more detail, \(\mathbf {L}\) is a smooth manifold if and only if none of configurations fits in a straight line. The oriented area is a Morse function for all \((l_1,\ldots ,l_n)\) except for some measure zero closed subspace of the parameter space.

  3. in the sense of Definition 4.

  4. With respect to any reasonable metric.

References

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  • Khimshiashvili, G., Panina, G.: On the area of a polygonal linkage. Dokl. Akad. Nauk. Math. 85(1), 120–121 (2012)

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Acknowledgements

Sections 3, 4, and 5 are supported by the Russian Science Foundation Grant N 14-21-00035. Gaiane Panina is supported by the RFBR Grant 17-01-00128 and the Program of the Presidium of the Russian Academy of Sciences N 01 ’Fundamental Mathematics and its Applications’ under Grant PRAS-18-01. We are indebted to Alexander Gaifullin who was the first to point out the vanishing perimeter of a bifurcating polygon. We also thank Mikhail Khristoforov for useful discussions. Support to Joseph Gordon from the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged.

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Authors and Affiliations

  1. Mathematics and Mechanics Department, St. Petersburg State University, St. Petersburg, Russia

    Gaiane Panina

  2. St. Petersburg Department of Steklov Mathematical Institute, St. Petersburg, Russia

    Gaiane Panina

  3. Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29B, St. Petersburg, 199178, Russia

    Joseph Gordon & Yana Teplitskaya

  4. International Laboratory of Game Theory and Decision Making, National Research University Higher School of Economics, Soyuza Pechatnikov str., 16, St. Petersburg, Russia

    Joseph Gordon

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  1. Joseph Gordon
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  2. Gaiane Panina
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  3. Yana Teplitskaya
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Correspondence to Gaiane Panina.

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Gordon, J., Panina, G. & Teplitskaya, Y. Polygons with prescribed edge slopes: configuration space and extremal points of perimeter. Beitr Algebra Geom 60, 1–15 (2019). https://doi.org/10.1007/s13366-018-0409-3

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  • DOI: https://doi.org/10.1007/s13366-018-0409-3

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