Advances in Applied Clifford Algebras

, Volume 26, Issue 1, pp 31–38 | Cite as

Canal Surfaces with Quaternions

  • Selahattin Aslan
  • Yusuf Yaylı


Quaternions are more usable than three Euler angles in the three dimensional Euclidean space. Thus, many laws in different fields can be given by the quaternions. In this study, we show that canal surfaces and tube surfaces can be obtained by the quaternion product and by the matrix representation. Also, we show that the equation of canal surface given by the different frames of its spine curve can be obtained by the same unit quaternion. In addition, these surfaces are obtained by the homothetic motion. Then, we give some results.


Canal surfaces Tubular surfaces Generalized tubes Bishop frame Darboux frame Quaternions Rotation matrices 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Babaarslan, M., Yaylı, Y.: A new approach to constant slope surfaces with quaternion. ISRN Geometry (ISRN GEOM.), Volume 2012, Article ID 126358, pp. 8 (2012). doi: 10.5402/2012/126358
  2. 2.
    Doğan, F.: Generalized Canal Surfaces. Ankara University Graduate School, Ph.D. Thesis (2012)Google Scholar
  3. 3.
    Doğan, F., Yaylı, Y.: On the curvatures of tubular surface with Bishop frame. Commun. Fac. Sci. Univ. Ank. Ser. A1 60(1), 59–69 (2011)Google Scholar
  4. 4.
    Doğan F., Yaylı Y.: Tubes with Darboux frame. Int. J. Contemp. Math. Sci. 7(16), 751–758 (2012)MathSciNetMATHGoogle Scholar
  5. 5.
    Doğan F., Yaylı Y.: Parametrization of perspective silhouettes on canal surfaces in Minkowski 3-space. J. Math. Comput. Sci. 2(2), 374–385 (2012)MathSciNetGoogle Scholar
  6. 6.
    Düldül M.: Two and three dimensional regions from homothetic motions. Appl. Math. ENotes 10, 86–93 (2010)MATHGoogle Scholar
  7. 7.
    Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd edn. CRC Press, Boca Raton (1999)Google Scholar
  8. 8.
    Gross, A.: Analyzing generalized tubes, SPIE, pp. 422–433 (1994)Google Scholar
  9. 9.
    Hacisalihoglu, H.H.: On the rolling of one curve or surface upon another. In: Proceeding of the Royal Irish Academy, vol. 71, Section A, Number 2, pp. 13–17 (1971)Google Scholar
  10. 10.
    Hamilton W.R.: On quaternions; or on a new system of imagniaries in algebra. Lond. Edinb. Dublin Philos. Mag. J. Sci. 25(3), 489–495 (1844)Google Scholar
  11. 11.
    Karacan, M.K., Yaylı, Y.: On the geodesics of tubular surfaces in Minkowskı 3-space. Bull. Malays. Math. Sci. Soc. (2) (SCI) 31(1), 1–10 (2008)Google Scholar
  12. 12.
    Shoemake, K.: Animating rotation with quaternion curves. In: Proceedings of the 12th Annual Conference on Computer Graphics and Interactive Techniques (SIG-GRAPH ’85), vol. 19, pp. 245–254, ACM, New York, NY, USA, 1985Google Scholar
  13. 13.
    Yayli Y.: Homothetic motions at \({{\mathbb{R}}^{4}}\). Mech. Mach. Theory 27(3), 303–305 (1992)CrossRefGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsAnkara UniversityAnkaraTurkey

Personalised recommendations