Abstract
This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown analytically, through a framework comparing the measurement of distances and angles in Cayley–Klein geometries, including Lorentzian geometries, as done by F. Bachmann and later R. Struve. On the other hand, such a relationship may also be expressed in a purely linear algebraic manner, as explained by D. Hestens, H. Li and A. Rockwood. The model described in this article unifies these approaches via a generalization of Lie sphere geometry. Like the work of N. Wildberger, it is a purely algebraic construction, and as such it works over any field of odd characteristic. It is shown that measurement of distances and angles is an inherent property of the model that is easy to identify, and the possible models are classified over the real, complex and finite fields, and partially in characteristic 2, revealing a striking analogy between the real and finite geometries. This is an abbreviated version of a previous manuscript, with certain expository parts removed for the sake brevity. The original manuscript is available on the website arXiv, with more motivation, examples, properties. Several definitions and theorems can be extended to include the characteristic 2 case, which are also omitted here.
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References
Bachmann, F.: Aufbau der Geometrie aus dem Spiegelungsbegriff. Springer, Heidelberg (1973)
Benz, W.: Vorlesungen über Geometrie der Algebren. Springer, Heidelberg (1973)
Chen, Y.: Der Satz von Miquel in der Möbius-Ebene. Math. Ann. 186(2), 81–100 (1970)
Fillmore, J.P., Springer, A.: New Euclidean theorems by the use of Laguerre transformations—some geometry of Minkowski (2 + 1)-space. J. Geom. 52(1–2), 74–90 (1995)
Fillmore, J.P., Springer, A.: Planar sections of the quadric of Lie cycles and their Euclidean interpretations. Geom. Dedicata 55(2), 175–193 (1995)
Fillmore, J.P., Springer, A.: Determining circles and spheres satisfying conditions which generalizes tangency. Preprint (2000)
Hestens, D., Li, H., Rockwood, A.: A Unified Algebraic Framework for Classical Geometry. In: Sommer, G. (ed.) Geometric Computing with Clifford Algebra. Springer, Heidelberg (2010)
Horváth, Á.G.: Constructive curves in non-Euclidean planes. Stud. Univ. Žilina 28, 13–42 (2016)
Kitaoka, Y.: Arithmetic of Quadratic Forms. Cambridge University Press, Cambridge (1993)
Klein, F.: Vorlesungen über höhere Geometrie. Springer, Berlin (1926)
Lie, S., Scheffers, G.: Geometrie der Berührungstransformationen. Taubner, Leipzig (1896)
Macdonald, A.: A survey of geometric algebra and geometric calculus. Adv. Appl. Clifford Algebras 27(1), 853–891 (2017)
Molnár, E.: A tükrözésfogalom abszolút geometriai alkalmazásai. CSc dissertation (1975)
Molnár, E.: Kreisgeometrie und Konforme Interpretation des mehrdimensionalen metrischen Raumes. Period. Math. Hung. 10(4), 237–259 (1979)
Molnár, E.: Inversion auf der Idealebene der bachmannschen metrische Ebene. Acta Math. Acad. Sci. Hung. Tomus 37(4), 451–470 (1981)
Molnár, E.: Constructions in the absolute plane—to the memory of Gyula Strommer. J. Geom. Graphics 20(1), 63–73 (2016)
Onishchik, A.L., Sulanke, R.: Projective and Cayley–Klein Geometries. Springer, Berlin (2006)
Rigby, J.F.: The geometry of cycles, and generalized Laguerre inversions. In: The Coxeter Festschrift, The Geometric Vein, pp. 355–378 Davis, C., Grünbaum, B., Sherk, F. A. (eds.) Springer, New York (1981)
Struve, R.: An axiomatic foundation of Cayley–Klein geometries. J. Geom. 107(2), 225–248 (2016)
Wildberger, N.J.: Affine and Projective Universal Geometry (2006). arXiv:math/0612499
Wildberger, N.J.: Universal Hyperbolic Geometry I: Trigonometry Geom Dedicata 163(1), 215–274 (2013)
Yaglom, I.M.: Galilei’s Relativity Principle and Non-Euclidean Geometry. Nauka, Moscow (1969)
Yaglom, I.M.: A Simple Non-Euclidean Geometry and Its Physical Basis. Springer, Heidelberg (1979)
Yaglom, I.M.: On the circular transformation of Möbius, Laguerre, and Lie. In: The Geometric Vein, The Coxeter Festschrift, pp. 345–353 Davis, C., Grünbaum, B., Sherk, F. A. (eds.) Springer, New York (1981)
Acknowledgements
I would like to thank Ákos G. Horváth for encouraging me to write this article. He has given me thorough guidance and I am thankful for his time. Acknowledgements are due to András Hraskó who introduced me to Lie sphere geometry, and provided me with important references. Finally, I would like to thank Professor Emil Molnár for his comments and for providing some essential references, and the Budapest University of Technology and Economics for their support.
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Juhász, M.L. A universal linear algebraic model for conformal geometries. J. Geom. 109, 48 (2018). https://doi.org/10.1007/s00022-018-0451-1
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DOI: https://doi.org/10.1007/s00022-018-0451-1