Advertisement

Plane-Geometric Investigation of a Proof of the Pohlke’s Fundamental Theorem of Axonometry

  • Thomas L. Toulias
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 131)

Abstract

Consider a bundle of three given coplanar line segments (radii) where only two of them are permitted to coincide. Each pair of these radii can be considered as a pair of two conjugate semidiameters of an ellipse. Thus, three concentric ellipses Ei, i = 1, 2, 3, are then formed. In a proof by G.A. Peschka of Karl Pohlke’s fundamental theorem of axonometry, a parallel projection of a sphere onto a plane, say, \(\mathbb E\), is adopted to show that a new concentric (to Ei) ellipse E exists, “circumscribing” all Ei, i.e., E is simultaneously tangent to all \(E_i\subset \mathbb E\), i = 1, 2, 3. Motivated by the above statement, this paper investigates the problem of determining the form and properties of the circumscribing ellipse E of Ei, i = 1, 2, 3, exclusively from the analytic plane geometry’s point of view (unlike the sphere’s parallel projection that requires the adoption of a three-dimensional space). All the results are demonstrated by the actual corresponding figures as well as with the calculations given in various examples.

Notes

Acknowledgements

The author would like to thank Prof. G.E. Lefkaditis for posing the problem and providing the historical background and Prof. C.P. Kitsos for the useful discussions during the preparation of this article.

References

  1. 1.
    Emch, A.: Proof of Pohlke’s theorem and its generalizations by affinity. Am. J. Math. 40(2), 366–374 (1918)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Fishburn, P.C., Trotter, W.T.: Containment orders for similar ellipses with a common center. Discret. Math. 256, 129–136 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Lefkaditis, G.E., Toulias, T.L., Markatis, S.: The four ellipses problem. Int. J. Geom. 5(2), 77–92 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Müller, E., Kruppa, E.: Lehrbuch der Darstellenden Geometrie. Springer, Wien (1961)zbMATHGoogle Scholar
  5. 5.
    Peschka, G.A.: Elementarer beweis des Pohlke’schen fundamentalsatzes der Axonometrie. Stzgsb. Math. Nat., Akad. Wien LXXVIII, II Abth, 1043–1054 (1878)Google Scholar
  6. 6.
    Sklenáriková, Z., Pémová, M.: The Pohlke-Schwarz theorem and its relevancy in the didactics of mathematics. Quaderni di Ricerca in Didattica 17 (2007). http://math.unipa.it/~grim/quad17_sklenarikova-pemova_07.pdf

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Thomas L. Toulias
    • 1
  1. 1.Technological Educational Institute of AthensEgaleo, AthensGreece

Personalised recommendations