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


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.


Generalized complex plane Steiner formula Holditch theorem 

Mathematics Subject Classification

53A17 53B50 11E88 


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  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

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