Abstract
All cycles (points, oriented circles, and oriented lines of a Euclidean plane) are represented by points of a three dimensional quadric in four dimensional real projective space. The intersection of this quadric with primes and planes are, respectively, two- and one-dimensional systems of cycles. This paper is a careful examination of the interpretation, in terms of systems of cycles in the Euclidean plane, of fundamental incidence configurations involving this quadric in projective space. These interpretations yield new and striking theorems of Euclidean geometry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blaschke, W.:Vorlesungen über Differentialgeometrie, III.Differentialgeometrie der Kreise und Kugeln, Springer-Verlag, Berlin, 1929.
Cecil, T. E.:Lie Sphere Geometry, Springer-Verlag, New York, 1992.
Klein, F.:Vorlesungenüber höhere Geometrie, 3. Aufl., Springer-Verlag, Berlin, 1926, and Chelsea Publ. Co., New York, 1957.
Lie, S.:Geometrie der Berührungstransformationen, Leipzig, 1896, and Chelsea, New York, 1977.
Rigby, J. F.: The geometry of cycles, and generalized Laguerre inversion, in Davis, Grünbaum and Sherk (eds),The Geometric Vein, The Coxeter Festschrift, Springer-Verlag, New York, 1981, pp. 355–378.
Yaglom, I. M.: On the circular transformations of Möbius, Laguerre, and Lie, in Davis, Grünbaum, and Sherk (eds),The Geometric Vein, The Coxeter Festschrift, Springer-Verlag, New York, 1981, pp. 345–353.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fillmore, J.P., Springer, A. Planar sections of the quadric of Lie cycles and their Euclidean interpretations. Geom Dedicata 55, 175–193 (1995). https://doi.org/10.1007/BF01264928
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01264928