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Moduli Spaces of Bicentric Quadrilaterals

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Abstract

We deal with planar moduli spaces of polygonal linkages arising from a poristic family of bicentric polygons. For bicentric quadrilaterals, we describe the topological types of moduli spaces in poristic families and find the absolute maximum and minimum of oriented area in the union of moduli spaces. Similar results are obtained for poristic quadrilaterals associated with a pair of confocal ellipses. In conclusion we outline some research perspectives suggested by our results.

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References

  1. M. Berger, Géométrie, Vol. 2, Nathan, Paris (1990).

  2. A. Connes and D. Zagier, “A property of parallelograms inscribed in ellipses,” Am. Math. Mon., 114, No. 10, 909–914 (2007).

    MATH  MathSciNet  Google Scholar 

  3. N. Fuss, “De quadrilateris quibus circulum tam inscribere quam circumscribere licet,” Nova Acta Acad. Sci. Petrop. 10, 103–125 (1797).

  4. Z. Giorgadze, Geometric porisms and extremal problems, Diploma Thesis, Ilia State University (2013).

  5. M. Kapovich and J. Millson, “On the moduli spaces of polygons in the Euclidean plane,” J. Differ. Geom., 42, No. 1, 133–164 (1995).

    MATH  MathSciNet  Google Scholar 

  6. G. Khimshiashvili, “Cyclic polygons as critical points,” Proc. I. Vekua Inst. Appl. Math., 3, 73–78 (2008).

    MathSciNet  Google Scholar 

  7. G. Khimshiashvili, “Extremal problems on configuration spaces,” Proc. A. Razmadze Math. Inst., 155, 73–77 (2011).

    MathSciNet  Google Scholar 

  8. G. Khimshiashvili and G. Panina, “Cyclic polygons are critical points of area,” J. Math. Sci., 158, No. 6, 899–903 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Lion, “Variational aspects of Poncelet’s theorem,” Geom. Dedic., 52, 105–118 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  10. V. Radić, “Some relations cocerning triangles and bicentric quadrilaterals in connection with Poncelet’s closure theorem,” Math. Maced., 1, 35–58 (2003).

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Ilia State University, Tbilisi, Georgia

    G. Khimshiashvili

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  1. G. Khimshiashvili
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Correspondence to G. Khimshiashvili.

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. Algebra and Topology, 91, 2014.

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Khimshiashvili, G. Moduli Spaces of Bicentric Quadrilaterals. J Math Sci 211, 31–39 (2015). https://doi.org/10.1007/s10958-015-2600-5

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  • DOI: https://doi.org/10.1007/s10958-015-2600-5

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