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SOME CHARACTERIZATIONS OF CANAL SURFACES

  • Kim, Young Ho (Department of Mathematics, Kyungpook National University) ;
  • Liu, Huili (Department of Mathematics, Northeastern University) ;
  • Qian, Jinhua (Department of Mathematics, Northeastern University)
  • Received : 2015.01.08
  • Published : 2016.03.31

Abstract

This work considers a particular type of swept surface named canal surfaces in Euclidean 3-space. For such a kind of surfaces, some interesting and important relations about the Gaussian curvature, the mean curvature and the second Gaussian curvature are found. Based on these relations, some canal surfaces are characterized.

Keywords

canal surface;Gaussian curvature;mean curvature;second Gaussian curvature;Weingarten surface;linear Weingarten surface

Acknowledgement

Supported by : National Research Foundation of Korea (NRF), NSFC

References

  1. J. A. Galvez, A. Martinez, and F. Milan, Linear Weingarten Surfaces in $R^3$, Monatsh. Math. 138 (2003), no. 2, 133-144. https://doi.org/10.1007/s00605-002-0510-3
  2. S. Haesen, S. Verpoort, and L. Verstraelen, The mean curvature of the second funda-mental form, Houston J. Math. 34 (2008), no. 3, 703-719.
  3. Y. H. Kim and D. W. Yoon, On non-developable ruled surface in Lorentz-Minkowski 3-spaces, Taiwanese J. Math. 11 (2007), no. 1, 197-214. https://doi.org/10.11650/twjm/1500404646
  4. S. N. Krivoshapko and C. A. Bock Hyeng, Classification of cyclic surfaces and geomet-rical research of canal surfaces, Int. J. Res. Rev. Appl. Sci. 12 (2012), no. 3, 360-374.
  5. R. Lopez, Linear Weingarten surfaces in Euclidean and hyperbolic space, Mat. Contemp. 35 (2008), 95-113.
  6. R. Lopez, Rotational linear Weingarten surfaces of hyperbolic type, Israel J. Math. 167 (2008), 283-302. https://doi.org/10.1007/s11856-008-1049-3
  7. T. Maekawa, M. N. Patrikalakis, T. Sakkalis, and G. Yu, Analysis and applications of pipe surfaces, Comput. Aided Geom. Design 15 (1998), no. 5, 437-458. https://doi.org/10.1016/S0167-8396(97)00042-3
  8. A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Second edition, Chapman and Hall/CRC, 2003.
  9. J. S. Ro and D. W. Yoon, Tubes of Weingarten types in a Euclidean 3-space, J. Chungcheong Math. Soc. 22 (2009), 360-366.
  10. S. Verpoort, The Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects, PhD. Thesis, Katholieke Universiteit Leuven, Belgium, 2008.
  11. Z. Q. Xu, R. Z. Feng, and J. G. Sun, Analytic and algebraic properties of canal surfaces, J. Comput. Appl. Math. 195 (2006), no. 1-2, 220-228. https://doi.org/10.1016/j.cam.2005.08.002