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Universal hyperbolic geometry I: trigonometry

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Abstract

Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective point of view, with trigonometric laws that extend to ‘points at infinity’, here called ‘null points’, and beyond to ‘ideal points’ associated to a hyperboloid of one sheet. The theory works over a general field not of characteristic two, and the main laws can be viewed as deformations of those from planar rational trigonometry. There are many new features; this paper gives 92 foundational theorems.

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References

  1. Artzy R.: Linear Geometry. Addison-Wesley, Reading, MA (1974)

    Google Scholar 

  2. Angel J.: Finite upper half planes over finite fields. Finite Fields Appl. 2, 62–86 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beardon A.F.: The Geometry of Discrete Groups, GTM 91. Springer, New York (1983)

    Book  Google Scholar 

  4. Behnke, H., Bachmann, F., Fladt, K., Kunle, H. (eds): Fundamentals of Mathematics vol. 2 Geometry. MIT Press, Cambridge (1974)

    Google Scholar 

  5. Coxeter H.S.M.: Introduction to Geometry. Wiley, New York (1961)

    MATH  Google Scholar 

  6. Fenchel W.: Elementary Geometry in Hyperbolic Space, Studies in Mathematics 11. Walter de Gruyter, Berlin (1989)

    Book  Google Scholar 

  7. Goh, S.: Chebyshev polynomials and spread polynomials, Honours Thesis. School of Mathematics, UNSW (2005)

  8. Goh, S., Wildberger, N.J.: Spread polynomials, rotations and the butterfly effect. arXiv:0911.1025vl (2009)

  9. Greenberg M.J.: Euclidean and Non-Euclidean Geometries: Development and History. W. H. Freeman, San Francisco (1972)

    Google Scholar 

  10. Gunn, C.: Geometry, Kinematics and Rigid Body Mechanics in Cayley–Klein Geometries. PhD thesis, Berlin (2011)

  11. Hestenes D., Li H., Rockwood A.: A unified algebraic approach for classical geometries. In: Sommer, G. (eds) Geometric Computing with Clifford Algebra, pp. 3–27. Springer, Berlin (2001)

    Google Scholar 

  12. Katok S.: Fuchsian Groups, Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1992)

    Google Scholar 

  13. Klein, F.: Vorlesungen über nicht-euklidische Geometrie. Springer, Berlin, reprinted 1968 (orig. 1928)

  14. Klingenberg W.: Eine Begründung der hyperbolischen Geometrie. Math. Ann. 127, 340–356 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ramsay A., Richtmyer R.D.: Introduction to Hyperbolic Geometry. Springer, New York (1995)

    Book  MATH  Google Scholar 

  16. Richter-Gebert J.: Perspectives on Projective Geometry: A Guided Tour through Real and Complex Geometry. Springer, Heidelberg (2010)

    Google Scholar 

  17. Szmielew, W.: Some Metamathematical Problems Concerning Elementary Hyperbolic Geometry. In: Studies in Logic and the Foundations of Mathematics, vol. 27, pp. 30–52 (1959)

  18. Soto-Andrade, J.: Geometrical Gel’fand models, tensor quotients, and Weil representations. In: Proc. Symp. Pure Math. vol. 47, Am. Math. Soc., Providence, pp. 305–316 (1987)

  19. Sommerville D.M.Y.: The Elements of Non-Euclidean Geometry. G. Bell and Sons, London (1914)

    MATH  Google Scholar 

  20. Terras A.: Fourier Analysis on Finite Groups and Applications, London Math. Soc. Student Texts 43. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  21. Thurston W.P.: Threedimensional geometry and topology. In: Levy, S. (eds) Princeton mathematical series 35, vol. 1., Princeton University Press, Princeton NJ (1997)

    Google Scholar 

  22. Ungar, A.A.: Hyperbolic trigonometry in the Einstein relativistic velocity model of hyperbolic geometry. In: Computers and Mathematics with Applications vol. 40, Issues 2–3, July–August 2000, pp. 313–332

  23. Wildberger N.J.: Divine Proportions: Rational Trigonometry to Universal Geometry. Wild Egg Books, Sydney (2005)

    MATH  Google Scholar 

  24. Wildberger N.J.: One dimensional metrical geometry. Geometriae Dedicata 128(1), 145–166 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wildberger N.J.: Chromogeometry and Relativistic Conics. KoG 13, 43–50 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Wildberger, N.J.: Universal affine and projective geometry. J. Geom (to appear)

  27. Wildberger N.J.: A Rational Approach to Trigonometry. Math Horizons, 16–20 (Nov. 2007)

  28. Wildberger N.J.: Universal hyperbolic geometry II: a pictorial overview. KoG 14, 3–24 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Wildberger N.J.: Universal hyperbolic geometry III: first steps in projective triangle geometry. KoG 15, 25–49 (2011)

    MathSciNet  MATH  Google Scholar 

  30. Wildberger N.J.: Chromogeometry. Math. Intel. 32(1), 26–32 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. School of Mathematics and Statistics, UNSW, Sydney, 2052, Australia

    N. J. Wildberger

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  1. N. J. Wildberger
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Correspondence to N. J. Wildberger.

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Wildberger, N.J. Universal hyperbolic geometry I: trigonometry. Geom Dedicata 163, 215–274 (2013). https://doi.org/10.1007/s10711-012-9746-9

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  • DOI: https://doi.org/10.1007/s10711-012-9746-9

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