Skip to main content
SpringerLink
Log in

Laguerre and Minkowski planes produced by dilatations

  • Published:
Journal of Geometry Aims and scope Submit manuscript

Abstract

We show that each parabolic curve f in R2 produces a Laguerre plane\(\mathbb{L}(f)\) if f and all its images under dilatations are cycles. Likewise, two hyperbolic curves f1,f2 produce a Minkowski planeM(f1,f2). We determine for which curves\(\mathbb{L}(f)\) is miquelian resp. ovoidal, and for which pairs f1,f2,M(f1,f2) is miquelian resp. satisfies the rectangle axiom, thus providing many examples of non-embeddable planes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. ARTZY, R.: A symmetry theorem for Laguerre planes. J. Geometry 5 (1974), 109–116.

    Google Scholar 

  2. BENZ, W.: Vorlesungen über Geometrie der Algebren. Springer, Berlin etc., 1973.

    Google Scholar 

  3. EWALD, G.: Aus konvexen Kurven bestehende Möbiusebenen. Abh. Math. Sem. Hamburg 30 (1967), 179–187.

    Google Scholar 

  4. GROH, H.: Ovals and non-ovoidal Laguerre planes. J. reine angew. Math. 267 (1974), 50–66.

    Google Scholar 

  5. —: R2 -planes with 2-dimensional point transitive automorphism group. Abh. Math. Sem. Hamburg 48 (1979), 171–202.

    Google Scholar 

  6. —: Pasting of R2-planes. Geometriae Dedicata 11 (1981), 69–98.

    Google Scholar 

  7. —: Isomorphism types of arc planes. Abh. Math. Sem. Hamburg 52 (1983), 133–149.

    Google Scholar 

  8. HARTMANN, E.: Eine Klasse nicht einbettbarer reeller Laguerre-Ebenen. J. Geometry 13 (1979), 49–67.

    Google Scholar 

  9. —: Beispiele nicht einbettbarer Minkowski-Ebenen. Geometriae Dedicata 10 (1981, 155–159.

    Google Scholar 

  10. -: Ovoide und Möbius-Ebenen über konvexen Funktionen. THD-Preprint Nr. 778 (September 1983). To appear in Geometriae Dedicata.

  11. -: Ebene Zykelgeometrien. Eine Einführung in die Möbius-, Laguerre- und Minkowski-Ebenen. To appear.

  12. MÄURER, H.: Laguerre-Ebenen mit Symmetrien an Punktepaaren und Zykeln. J. Geometry 8 (1976), 79–93.

    Google Scholar 

  13. PERCSY, N.: A remark on the introduction of coordinates in Minkowski planes. J. Geometry 12 (1979), 175–183.

    Google Scholar 

  14. ROBERTS, A. W., VARBERG, D. E.: Convex functions. Academic Press, New York etc. 1973.

    Google Scholar 

  15. SALZMANN, H.: Topological planes. Advances in Math. 2 (1967), 1–60.

    Google Scholar 

  16. SCHRÖDER, E. M.: Gemeinsame Eigenschaften euklidischer, galileischer und minkowskischer Ebenen. Mitt. Math. Ges. Hamburg 10 (1974), 185–217.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics, University of Haifa, 31999, Haifa, Israel

    Rafael Artzy

  2. FB 4 - AG 2 Geometrie und Algebra, Technische Hochschule Darmstadt, D-6100, Darmstadt, Germany

    Hansjoachim Groh

Authors
  1. Rafael Artzy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hansjoachim Groh
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Artzy, R., Groh, H. Laguerre and Minkowski planes produced by dilatations. J Geom 26, 1–20 (1986). https://doi.org/10.1007/BF01221003

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01221003

Keywords

Search

Navigation