Journal of Mathematical Sciences

, Volume 158, Issue 6, pp 899–903

Cyclic polygons are critical points of area

Article

It is shown that typical critical points of the signed area function on the moduli space of a generic planar polygon are given by cyclic configurations, i.e., configurations that can be inscribed in a circle. Several related problems are briefly discussed in conclusion. Bibliography: 14 titles.

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References

  1. 1.
    V. Arnold, A. Varchenko, and S. Gusein-Zade, Singularities of Differentiable Mappings [in Russian], Nauka, Moscow (2005).Google Scholar
  2. 2.
    J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry, Springer, Berlin–Heidelberg–New York (1998).MATHGoogle Scholar
  3. 3.
    R. Connelly and E. Demaine, “Geometry and topology of polygonal linkages,” in: Handbook of Discrete and Computational Geometry, 2nd edition, CRC Press, Boca Raton (2004), pp. 197–218.Google Scholar
  4. 4.
    H. Coxeter and S. Greitzer, Geometry Revisited, Amer. Math. Soc. (1967).Google Scholar
  5. 5.
    E. Elerdashvili, M. Jibladze, and G. Khimshiashvili, “Cyclic configurations of pentagon linkages,” Bull. Georgian Nat. Acad. Sci., 2, No. 4, 13–16 (2008).Google Scholar
  6. 6.
    M. Fedorchuk and I. Pak, “Rigidity and polynomial invariants of convex polytopes,” Duke Math. J., 129, No. 2, 371–404 (2005).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    C. Gibson and P. Newstead, “On the geometry of the planar 4-bar mechanism,” Acta Appl. Math., 7, 113–135 (1986).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Kapovich and J. Millson, “Universality theorems for configuration spaces of planar linkages,” Topology, 41, 1051–1107 (2002).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Kempe, “A method of describing curves of the nth degree by linkwork,” Proc. London Math. Soc., 7, 213–216 (1876).CrossRefGoogle Scholar
  10. 10.
    G. Khimshiashvili, “On configuration spaces of planar pentagons,” Zap. Nauchn. Semin. POMI, 292, 120–129 (2002).Google Scholar
  11. 11.
    G. Khimshiashvili, “Signature formulae and configuration spaces,” J. Math. Sci. 59 (2009), in press.Google Scholar
  12. 12.
    D. Robbins, “Areas of polygons inscribed in a circle,” Discrete Comput. Geom., 12, 223–236 (1994).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    W. Thurston, “Shapes of polyhedra and triangulations of the sphere,” Geom. Topol. Monogr., 1, 511–549 (1998).CrossRefMathSciNetGoogle Scholar
  14. 14.
    V. Varfolomeev, “Inscribed polygons and Heron polynornials,” Sb. Math., 194, 311–321 (2003).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Institute for Informatics and AutomationSt. PetersburgRussia
  2. 2.Ilia Chavchavadze State UniversityTbilisiGeorgia

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