Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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ON SOME GEOMETRIC PROPERTIES OF QUADRIC SURFACES IN EUCLIDEAN SPACE

  • Ali, Ahmad T. (King Abdul Aziz University, Faculty of Science, Department of Mathematics, Mathematics Department, Faculty of Science, Al-Azhar University) ;
  • Aziz, H.S. Abdel (Department of Mathematics, Faculty of Science, Sohag University) ;
  • Sorour, Adel H. (Department of Mathematics, Faculty of Science, Sohag University)
  • Received : 2015.03.23
  • Accepted : 2015.08.20
  • Published : 2016.09.25
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Abstract

This paper is concerned with the classifications of quadric surfaces of first and second kinds in Euclidean 3-space satisfying the Jacobi condition with respect to their curvatures, the Gaussian curvature K, the mean curvature H, second mean curvature $H_{II}$ and second Gaussian curvature $K_{II}$. Also, we study the zero and non-zero constant curvatures of these surfaces. Furthermore, we investigated the (A, B)-Weingarten, (A, B)-linear Weingarten as well as some special ($C^2$, K) and $(C^2,\;K{\sqrt{K}})$-nonlinear Weingarten quadric surfaces in $E^3$, where $A{\neq}B$, A, $B{\in}{K,H,H_{II},K_{II}}$ and $C{\in}{H,H_{II},K_{II}}$. Finally, some important new lemmas are presented.

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References

  1. J. Weingarten, Ueber eine Klasse auf einander abwickelbarer Flaachen, J. Reine Angew. Math. 59 (1861) 382-393.
  2. J.Weingarten, Ueber die Flachen derer Normalen eine gegebene Flache beruhren, J. Reine Angew. Math. 62 (1863) 61-63.
  3. Van-Brunt and K. Grant, Potential applications of Weingarten surfaces in CAGD, Part I: Weingarten surfaces and surface shape investigation, Comput. Aided Geom. Des. 13 (1996) 569-582. https://doi.org/10.1016/0167-8396(95)00046-1
  4. W. Kuhnel, Ruled W-surfaces, Arch. Math. 62 (1994) 475-480. https://doi.org/10.1007/BF01196440
  5. D. W. Yoon, Some properties of the helicoid as ruled surfaces, JP J. Geom. Topology 2(2) (2002) 141-147.
  6. F. Dillen and W. Kuhnel, Ruled Weingarten surfaces in a Minkowski 3-space, Manuscripta Math. 98 (1999) 307-320. https://doi.org/10.1007/s002290050142
  7. H. Y. Kim and D. W. Yoon, Classification of ruled surfaces in a Minkowski 3-space J. Geom. Phys. 49 (2004) 89-100. https://doi.org/10.1016/S0393-0440(03)00084-6
  8. W. Sodsiri, Ruled surfaces of Weingarten type in Minkowski 3-space, PhD. thesis, Katholieke Universiteit Leuven, Belgium, (2005).
  9. M. I. Munteanu and A. I. Nistor, Polynomial translation Weingarten surfaces in 3-dimensional Euclidean space, Differential geometry, 316-320, Worled Sci. Publ., Hackensack, NJ, (2009).
  10. R. Lopez, On linear Weingarten surfaces, International J. Math. 19 (2008) 439-448. https://doi.org/10.1142/S0129167X08004728
  11. R. Lopez, Special Weingarten surfaces foliated by circles, Monatsh. Math. 154 (2008) 289-302. https://doi.org/10.1007/s00605-008-0557-x
  12. R. Lopez, Parabolic Weingarten surfaces in hyperbolic space, Publ. Math. Debrecen 74 (2009) 59-80.
  13. J. Ro and D. W.Yoon, Tubes of Weingarten types in a Euclidean 3-space, J. Chungcheong Math. Soc. 22 (3) (2009) 359-366.
  14. Y. H. Kim and D. W. Yoon, Weingarten quadric surfaces in a Euclidean 3-space, Turk. J. Math. 35 (2011) 479-485.
  15. D. W. Yoon and J. S. Jun, Non-degenerate quadric surfaces in Euclidean 3-space, Int. J. Math. Anal. 6 (52) (2012) 2555-2562.
  16. H. S. Abdel-Aziz and M. Khalifa Saad, Weingarten timelike tube surfaces around a spacelike curve, Int. J. of Math. Analysis, 25 (5) (2011) 1225-1236.
  17. H. S. Abdel-Aziz and M. Khalifa Saad and Sezai Kiziltug, Parallel Surfaces of Weingarten Type in Minkowski 3-Space, Int. Math. Forum, 4 (7) (2012) 2293-2302.
  18. D. A. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Now York, (1990).
  19. M. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Englewood Cliffs, (1976).
  20. S. Verpoort, The Geometry of the second fundamental form: Curvature properties and variational aspects, PhD. Thesis, Katholieke Universiteit Leuven, Belgium, (2008).
  21. C. Baikoussis and T. Koufogiorgos, On the inner curvature of the second fundamental form of helicoidal surfaces, Arch. Math. 68 (2) (1997) 169-176. https://doi.org/10.1007/s000130050046
  22. T. Yilmaz and K. K. Murat, Differential Geometry of three dimensions, Syndic of Cambridge University press, (1981).
  23. Y. B. Chen and F. Dillen, Quadrics of finite type, J. Geom. 38 (1990) 16-22. https://doi.org/10.1007/BF01222891