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A Spatial Version of the Theorem of the Angle of Circumference

  • Georg GlaeserEmail author
  • Boris Odehnal
  • Hellmuth Stachel
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 809)

Abstract

We try a generalization of the theorem of the angle of circumference to a version in three-dimensional Euclidean space and ask for pairs \(({\varepsilon },{\varphi })\) of planes passing through two (different skew) straight lines \(e\ni {\varepsilon }\) and \(f\ni {\varphi }\) such that the angle \(\alpha \) enclosed by \({\varepsilon }\) and \({\varphi }\) is constant. It turns out that the set of all such intersection lines is a quartic ruled surface \(\varPhi \) with \(e\cup f\) being its double curve. We shall study the surface \(\varPhi \) and its properties together with certain special appearances showing up for special values of some shape parameters such as the slope of e and f (with respect to a fixed plane) or the angle \(\alpha \).

Keywords

Ruled surface Angle of circumference Quartic ruled surface Thaloid Isoptic surface 

References

  1. 1.
    Glaeser, G., Stachel, H., Odehnal, B.: The Universe of Conics. From the ancient Greeks to 21st century developments. Springer-Spektrum, Springer-Verlag, Heidelberg (2016)Google Scholar
  2. 2.
    Müller, E.: Vorlesungen über Darstellende Geometrie. III. Band: Konstruktive Behandlung der Regelflächen. Deuticke, Leipzig und Wien (1931)Google Scholar
  3. 3.
    Wieleitner, H.: Spezielle Ebene Kurve. G.J. Göschen’sche Verlagshandlung, Leipzig, Sammlung Schubert LVI (1908)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Georg Glaeser
    • 1
    Email author
  • Boris Odehnal
    • 1
  • Hellmuth Stachel
    • 2
  1. 1.University of Applied Arts ViennaViennaAustria
  2. 2.TU WienViennaAustria

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