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Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations

Year 2017, Volume: 10 Issue: 2, 32 - 36, 29.10.2017

Oğuzhan Demirel

https://doi.org/10.36890/iejg.545044
Cited By: 1

Abstract

In this paper, we define a new geometric concept that we will call “degenerate Saccheri
quadrilateral” and use it to give a new characterization of Möbius transformations. Our proofs
are based on a geometric approach.

Keywords

Saccheri quadrilateral Möbius transformation

References

  • [1] Aczél, J. and McKiernan, M.A., On the characterization of plane projective and complex Möbius transformations. Math. Nachr. 33, (1967), 315–337.
  • [2] Beardon, A.F. and Minda, D., Sphere-preserving maps in inversive geometry. Proc. Amer. Math. Soc. 130 (2002), no. 4, 987–998.
  • [3] Beardon, A.F., The geometry of discrete groups, Springer-Verlag, New York, 1983.
  • [4] Carathéodory, C., The most general transformations of plane regions which transform circles into circles. Bull. Am. Math. Soc. 43, (1937), 573–579.
  • [5] Demirel, O. and Seyrantepe, E.S., A characterization of Möbius transformations by use of hyperbolic regular polygons. J. Math. Anal. Appl. 374 (2011), no. 2, 566–572.
  • [6] Demirel, O., A characterization of Möbius transformations by use of hyperbolic regular star polygons. Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 14 (2013), no. 1, 13–19.
  • [7] Demirel, O., Degenerate Lambert quadrilaterals and Möbius transformations. Bull. Math. Soc. Sci. Math. Roumanie, (accepted for publication).
  • [8] Haruki, H and Rassias, T.M., A new characteristic of Möbius transformations by use of Apollonius quadrilaterals. Proc. Amer. Math. Soc. 126 (1998), no. 10, 2857–2861.
  • [9] Höfer, R., A characterization of Möbius transformations. Proc. Amer. Math. Soc. 128 (2000), no. 4, 1197–1201.
  • [10] Jing, L., A new characteristic of Möbius transformations by use of polygons having type A. J. Math. Anal. Appl. 324 (2006), no. 1, 281–284.
  • [11] Jones, G.A and Singerman, D., Complex functions. An algebraic and geometric viewpoint. Cambridge University Press, Cambridge, 1987.
  • [12] Ungar, A.A., Analytic hyperbolic geometry. Mathematical foundations and applications. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005
  • [13] Ungar, A.A., The hyperbolic square and Möbius transformations, Banach J. Math. Anal. 1 (2007), no. 1, 101–116.
  • [14] Yang, S. and Fang, A., A new characteristic of Möbius transformations in hyperbolic geometry. J. Math. Anal. Appl. 319 (2006), no. 2, 660–664.
  • [15] Yang, S. and Fang, A., Corrigendum to "A new characteristic of Möbius transformations in hyperbolic geometry, J. Math. Anal. Appl. 319 (2) (2006) 660-664" J. Math. Anal. Appl. 376 (2011), no. 1, 383–384.

Year 2017, Volume: 10 Issue: 2, 32 - 36, 29.10.2017

Oğuzhan Demirel

https://doi.org/10.36890/iejg.545044
Cited By: 1

Abstract

References

  • [1] Aczél, J. and McKiernan, M.A., On the characterization of plane projective and complex Möbius transformations. Math. Nachr. 33, (1967), 315–337.
  • [2] Beardon, A.F. and Minda, D., Sphere-preserving maps in inversive geometry. Proc. Amer. Math. Soc. 130 (2002), no. 4, 987–998.
  • [3] Beardon, A.F., The geometry of discrete groups, Springer-Verlag, New York, 1983.
  • [4] Carathéodory, C., The most general transformations of plane regions which transform circles into circles. Bull. Am. Math. Soc. 43, (1937), 573–579.
  • [5] Demirel, O. and Seyrantepe, E.S., A characterization of Möbius transformations by use of hyperbolic regular polygons. J. Math. Anal. Appl. 374 (2011), no. 2, 566–572.
  • [6] Demirel, O., A characterization of Möbius transformations by use of hyperbolic regular star polygons. Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 14 (2013), no. 1, 13–19.
  • [7] Demirel, O., Degenerate Lambert quadrilaterals and Möbius transformations. Bull. Math. Soc. Sci. Math. Roumanie, (accepted for publication).
  • [8] Haruki, H and Rassias, T.M., A new characteristic of Möbius transformations by use of Apollonius quadrilaterals. Proc. Amer. Math. Soc. 126 (1998), no. 10, 2857–2861.
  • [9] Höfer, R., A characterization of Möbius transformations. Proc. Amer. Math. Soc. 128 (2000), no. 4, 1197–1201.
  • [10] Jing, L., A new characteristic of Möbius transformations by use of polygons having type A. J. Math. Anal. Appl. 324 (2006), no. 1, 281–284.
  • [11] Jones, G.A and Singerman, D., Complex functions. An algebraic and geometric viewpoint. Cambridge University Press, Cambridge, 1987.
  • [12] Ungar, A.A., Analytic hyperbolic geometry. Mathematical foundations and applications. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005
  • [13] Ungar, A.A., The hyperbolic square and Möbius transformations, Banach J. Math. Anal. 1 (2007), no. 1, 101–116.
  • [14] Yang, S. and Fang, A., A new characteristic of Möbius transformations in hyperbolic geometry. J. Math. Anal. Appl. 319 (2006), no. 2, 660–664.
  • [15] Yang, S. and Fang, A., Corrigendum to "A new characteristic of Möbius transformations in hyperbolic geometry, J. Math. Anal. Appl. 319 (2) (2006) 660-664" J. Math. Anal. Appl. 376 (2011), no. 1, 383–384.
There are 15 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Oğuzhan Demirel

Publication Date October 29, 2017
Published in Issue Year 2017 Volume: 10 Issue: 2

Cite

APA Demirel, O. (2017). Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations. International Electronic Journal of Geometry, 10(2), 32-36. https://doi.org/10.36890/iejg.545044
AMA Demirel O. Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations. Int. Electron. J. Geom. October 2017;10(2):32-36. doi:10.36890/iejg.545044
Chicago Demirel, Oğuzhan. “Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations”. International Electronic Journal of Geometry 10, no. 2 (October 2017): 32-36. https://doi.org/10.36890/iejg.545044.
EndNote Demirel O (October 1, 2017) Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations. International Electronic Journal of Geometry 10 2 32–36.
IEEE O. Demirel, “Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations”, Int. Electron. J. Geom., vol. 10, no. 2, pp. 32–36, 2017, doi: 10.36890/iejg.545044.
ISNAD Demirel, Oğuzhan. “Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations”. International Electronic Journal of Geometry 10/2 (October 2017), 32-36. https://doi.org/10.36890/iejg.545044.
JAMA Demirel O. Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations. Int. Electron. J. Geom. 2017;10:32–36.
MLA Demirel, Oğuzhan. “Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations”. International Electronic Journal of Geometry, vol. 10, no. 2, 2017, pp. 32-36, doi:10.36890/iejg.545044.
Vancouver Demirel O. Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations. Int. Electron. J. Geom. 2017;10(2):32-6.

Cited By

The hyperbolic polygons of type (ϵ, n) and Möbius transformations
Open Mathematics
https://doi.org/10.1515/math-2020-0015
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