Advertisement

Advances in Applied Clifford Algebras

, Volume 26, Issue 1, pp 97–113 | Cite as

A New Generalization of the Steiner Formula and the Holditch Theorem

  • Tülay Erişir
  • Mehmet Ali Güngör
  • Murat Tosun
Article

Abstract

In this study, we first obtained the Steiner area formula in the generalized complex plane. Then, with the aid of this formula, we determined a new approach for the Holditch theorem giving the relationship between the areas formed by points in the generalized complex plane (or p-complex plane). Finally, according to the special values of p = −1, 0, 1 we examined the cases of the Steiner Formula and Holditch Theorem. In this way, for \({p \in \mathbb{R}}\) we generalized the Steiner Formula and Holditch theorem consisting the Euclidean \({\left({p = -1}\right)}\), Galilean \({\left({p = 0}\right)}\) and Lorentzian \({\left({p = 1}\right)}\) cases.

Keywords

Generalized complex plane Steiner formula Holditch theorem 

Mathematics Subject Classification

53A17 53B50 11E88 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blaschke W.H., Müller R.: Ebene Kinematik. Verlag Oldenbourg, München (1956)MATHGoogle Scholar
  2. 2.
    Broman A.: A fresh look at a long-forgotten theorem. Math. Mag. 54(3), 99–108 (1981)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Gürses, N.B., Yüce, S.: One-parameter planar motions in generalized complex number plane \({\mathbb{C}_j}\). Adv. Appl. Clifford Algebra (2015). doi: 10.1007/s00006-015-0530-4
  4. 4.
    Hacisalihoglu, H.H.: On the geometry of motion of Lorentzian plane. In: Proc. of Assiut First International Conference of Mathematics and Statistics, Part I, pp. 87–107. University of Assiut, Assiut, Egypt (1990)Google Scholar
  5. 5.
    Harkin A.A., Harkin J.B.: Geometry of generalized complex numbers. Math. Mag. 77(2), 118–129 (2004)MathSciNetMATHGoogle Scholar
  6. 6.
    Hering L.: Sätze vom Holditch-typ für ebene kurven. Elem. Math. 38, 39–49 (1983)MathSciNetMATHGoogle Scholar
  7. 7.
    Holditch H.: Geometrical theorem. QJ Pure Appl. Math. 2, 858 (1858)Google Scholar
  8. 8.
    Koru, G.: Manifolds and the Holditch theorem. Phd thesis, Ankara University, Ankara, Turkey (2000)Google Scholar
  9. 9.
    Kuruoǧlu N., Yüce S.: The generalized Holditch theorem for the homothetic motions on the planar kinematics. Czechoslov. Math. J. 54(129), 337–340 (2004)CrossRefGoogle Scholar
  10. 10.
    Müller H.R.: Verallgemeinerung einer Formel von Steiner. Abh. d. Brschw. Wiss. Ges. Bd. 29, 107–113 (1978)MATHGoogle Scholar
  11. 11.
    Parapatits L., Schuster F.E.: The Steiner formula for Minkowski valuations. Adv. Math. 230, 978–994 (2012)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Potmann H.: Holditch–Sicheln. Arc. Math. 44, 373–378 (1985)CrossRefGoogle Scholar
  13. 13.
    Potmann H.: Zum Satz von Holditch in der euklidischen Ebene. Elem. Math. 41, 1–6 (1986)MathSciNetGoogle Scholar
  14. 14.
    Sachs, H.: Ebene Isotrope Geometrie. Fiedr. Vieweg-Sohn (1987)Google Scholar
  15. 15.
    Spivak M.: Calculus on Manifolds: a Modern Approach to Classical Theorems of Advanced Calculus. WA Benjamin, New York (1965)MATHGoogle Scholar
  16. 16.
    Steiner, J.: Über parallele Flächen, Monatsber. Preuss. Akad. Wiss. pp. 114–118 (1840), [Ges. Werke, Vol II (Georg Reimer, Berlin, 1882) 245–308]Google Scholar
  17. 17.
    Steiner, J.: Gesammelte Werke I. Georg Reimer, Berlin (1881)Google Scholar
  18. 18.
    Tutar A., Kuruoǧlu N.: The Steiner formula and the Holditch theorem for the homothetic motions on planar kinematics. Mech. Mach. Theory 34, 1–6 (1999)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Yaglom I.M.: Complex Numbers in Geometry. Academic Press, New York (1968)Google Scholar
  20. 20.
    Yaglom I.M.: A Simple non-Euclidean Geometry and its Physical Basis. Springer, New-York (1979)MATHGoogle Scholar
  21. 21.
    Yüce S., Kuruoǧlu N.: Holditch-type theorems under the closed planar homothetic motions. Ital. J. Pure Appl. Math. 21, 105–108 (2007)Google Scholar
  22. 22.
    Yüce S., Kuruoǧlu N.: Steiner formula and Holditch-type theorems for homothetic Lorentzian Motions. Iran. J. Sci. Technol. Trans. A Sci. 31(A2), 207–212 (2007)MathSciNetGoogle Scholar
  23. 23.
    Yüce S., Kuruoǧlu N.: Holditch theorem and Steiner formula for the planar hyperbolic motions. Adv. Appl. Clifford Algebra. 20, 195–200 (2010)CrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Tülay Erişir
    • 1
  • Mehmet Ali Güngör
    • 1
  • Murat Tosun
    • 1
  1. 1.Department of MathematicsSakarya UniversitySakaryaTurkey

Personalised recommendations