The relation between parameter curves and lines of curvature on canal surfaces


  • Fatih Dogan Bartin University, Faculty of Science, Department of Mathematics
  • Yusuf Yayli Ankara University, Faculty of Science, Department of Mathematics


Canal surface, generalized tube, line of curvature, parameter curve, spine curve.


A canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory C(t) (spine curve) of its center and a radius function r(t). In this paper, we investigate the parameter curves which are also lines of curvature on the canal surface. Last of all, for special spine curves we obtain the radius functions of canal surfaces.


do Carmo, P.M. (1976). Differential geometry of curves and surfaces. Prentice-Hall, USA.

Dogan, F. (2012). A note on tubes. International Journal of Physical

and Mathematical Sciences, 3(1):98-105.

Dogan, F. (2015). On characteristic curves of developable surfaces

in Euclidean 3-space. arXiv:1508.05439 [math.DG].

Dogan, F. & Yayli, Y. (2011). On the curvatures of tubular surface

with Bishop frame. Communications Faculty of Science, University

of Ankara, Series A1, 60(1):59-69. tr/


Dogan, F. & Yayli, Y. (2012). Tubes with Darboux frame. InternationalJournal of Contemporary Mathematical Sciences, 7(16):751-758.

Garcia, R., Llibre, J. & Sotomayor, J. (2006). Lines of principal

curvature on canal surfaces in R3. Anais da Academia Brasileira de

Cincias, 78:405-415. Doi: 10.1590/S0001- 37652006000300002

Gray, A. 1998. Modern differential geometry of curves and surfaces

with Mathematica. second edition CrcPress, Boca Raton.

Gross, A. (1994). Analyzing generalized tubes. SPIE:422-433. Doi:


Izumiya, S., Saji, K. & Takeuchi, N. (2007). Circular surfaces.

Advances in Geometry, 7(2):295-313.

Karadag, H.B., Kilic, E. & Adag, M. (2014). On the developable

ruled surfaces kinematically generated in Minkowski 3-Space. Kuwait Journal of Science, 41(1):21-34.

Maekawa, T., Patrikalakis, N.M., Sakkalis, T. & Yu, G. (1998).

Analysis and applications of pipe surfaces. Computer Aided Geometric Design, 15:437-458.

Monterde, J. (2009). Salkowski curves revisited: A family of curves

with constant curvature and non-constant torsion. Computer Aided

Geometric Design, 26:271-278.

Salkowski, E. (1909). Zur Transformation von Raumkurven.

Mathematische Annalen, 66:517-557.

Vessiot, E. 1919. Leons de Gomtrie Suprieure. Librarie Scientifique Journal Hermann, Paris.

Xu, Z., Feng, R. & Sun, JG. (2006). Analytic and algebraic properties of canal surfaces. Journal of Computational and Applied Mathematics, 195:220-228.