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

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. 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. 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. 3.

    in the sense of Definition 4.

  4. 4.

    With respect to any reasonable metric.

References

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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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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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Keywords

  • Morse index
  • Critical point
  • Cyclic polygon
  • Flexible polygon

Mathematics Subject Classification

  • 52R70
  • 52B99