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Projective Cross-ratio on Hypercomplex Numbers

Advances in Applied Clifford Algebras volume 23, pages 1–14 (2013)Cite this article

Abstract

The paper presents a new cross-ratio of hypercomplex numbers based on projective geometry. We discuss the essential properties of the projective cross-ratio, notably its invariance under Möbius transformations. Applications to the geometry of conic sections and Möbiusinvariant metrics on the upper half-plane are also given.

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References

  1. Alan F. Beardon, The geometry of discrete groups. 1983 (English).

  2. Alan F. Beardon, Algebra and geometry. Cambridge: Cambridge University Press. xii 2005, p. 326 (English).

  3. Ewain Gwynne and Matvei Libine, On a quaternionic analogue of the cross-ratio. Advances in Applied Clifford Algebras, Online First: DOI10.1007/s00006-012-0325-9, (2012).

  4. Vladimir V. Kisil, Erlangen program at large–0: Starting with the group SL2(R). Notices Amer. Math. Soc., 54 (11) (2007), pp.1458–1465. E-print: arXiv:math/0607387, On-line. MR2361159.

  5. Vladimir V. Kisil, Erlangen program at large–1: Geometry of invariants. SIGMA, Symmetry Integrability Geom. Methods Appl. 6 No. (076) (2010), 45 pages. E-print: arXiv:math.CV/0512416.

  6. Vladimir V. Kisil, Geometry of Möbius transformations: Elliptic, parabolic and hyperbolic actions of SL2(R). Imperial College Press, (2012).

  7. I.M. Yaglom, A simple non-Euclidean geometry and its physical basis. An elementary account of Galilean geometry and the Galilean principle of relativity. Translated from the Russian by Abe Shenitzer. With the editorial assistance of Basil Gordon (1979) (English).

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Authors and Affiliations

  1. School of Mathematics, Leeds University, Leeds, LS2 9JT, England

    Sky Brewer

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  1. Sky Brewer
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Correspondence to Sky Brewer.

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Brewer, S. Projective Cross-ratio on Hypercomplex Numbers. Adv. Appl. Clifford Algebras 23, 1–14 (2013). https://doi.org/10.1007/s00006-012-0335-7

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  • DOI: https://doi.org/10.1007/s00006-012-0335-7

Keywords

  • Cross-ratio
  • Projective linear group
  • Möbius transformation
  • Cycles
  • SL(2,R)
  • Special linear group
  • Clifford algebra
  • dual numbers
  • double numbers