On Dual Curves of DAW(k)-Type and Their Evolutes

Article Sidebar

PDF
Cite as:
H. S. Abdel-Aziz, M. Khalifa Saad, S. A. Mohamed, On Dual Curves of DAW(k)-Type and Their Evolutes, Int. J. Anal. Appl., 16 (5) (2018), 614-627.

Main Article Content

H. S. Abdel-Aziz, M. Khalifa Saad, S. A. Mohamed

Abstract

In this paper, we study to express the theory of curves including a wide section of Euclidean geometry in terms of dual vector calculus which has an important place in the three -dimensional dual space $\mathbb{D}^{3}$. In other words, we study $DAW(k)$-type curves $\left( 1\leq k\leq 3\right)$ by using Bishop frame defined as alternatively of these curves and give some of their properties in $\mathbb{D}^{3}$. \ Moreover, we define the notion of evolutes of dual spherical curves for ruled surfaces. Finally, we give some examples to illustrate our findings.

Article Details

References

  1. E. Study, Geometry der Dynamen. Lepzig, 1901.Clifford, W. K., Preliminary Sketch of Biquaternions, Proc. Lond. Soc., 4 (64) (1873), 381-395.
  2. H. W. Guggenheimer, Differential geometry. McGraw-Hill Book Co., New York, 1963.
  3. C.E. Weatherburn, Differential Geometry of Three Dimensions. Syndic of Cambridge University press, 1981.
  4. K. Arslan, A. West, Product submanifolds with pointwise 3-planar normal sections. Glasgow Math. J., 37(1) (1995), 73-81.
  5. K. Arslan, C.Özgür, Curves and surfaces of AW(k) Type in Geometry and Topology of Submanifolds. IX (Valenci- ennes/Lyon/Leuven, 1997), World Sci. Publ., 1999, 21-26.
  6. C.Özgür, F. Gezgin, On some curves of AW(k)-type. Differ. Geom. Dyn. Syst., 7 (2005), 74-80.
  7. Ë™Ilim Ki ¸si, GünayÖztürk, AW (k)-Type Curves According to the Bishop Frame. arXiv preprint arXiv:1305.3381, 2013.
  8. G.Öztürk, A.Kü ¸cük, K. Arslan, Some Characteristic Properties of AW(k)-type Curves on Dual Unit Sphere. Extracta Mathematicae, 29(1-2) (2014), 167-175.
  9. W. K. Clifford, Preliminary sketch of bi-quaternions. Proc. London Math. Soc., s1-4(1) (1871), 381 - 395.
  10. M. P. Do Carmo, Differential geometry of curves and surfaces. Prentice Hall, Englewood Cliffs, N. J., 1976.
  11. T. Shifrin , Differential Geometry, A first Course in Curves and Surfaces. (Preliminary Version), University of Georgia, 2010.
  12. S. Suleyman, M. Bilici and M. Caliskan, Some characterizations for the involute curves in dual space. Math. Combin. Book Ser., Vol. 1, 2015, 113-125.
  13. N. H. Abdel-All, R. A. Huesien and A. Abdela Ali, Dual construction of Developable Ruled Surface. J. Amer. Sci., 7(4) (2011), 789-793.
  14. M. ¯Onder, H. U˘ gurlu, Dual Darboux frame of a timelike ruled surface and Darboux approach to Mannheim offsets of timelike ruled surfaces. Proc. Nat. Acad. Sci., India Sect. A: Phys. Sci., 83 (2013), 163-169.
  15. R. Baky, R. Ghefari, On the one-parameter dual spherical motions. Computer Aided Geometric Design, 28 (2011), 23-37.
  16. Li. Yanlin, Pei. Donghe, Evolutes of dual spherical curves for ruled surfaces. Math. Meth. Appl. Sci., 39 (2016), 3005-3015.
  17. F. M. Dimentberg, The Screw Calculus and its Applications in Mechanics. (Izdat. Nauka, Moscow, USSR) English trans- lation: AD680993, Clearinghouse for Federal and Scientific Technical Information, 1965.
  18. M. T. Aldossary, R. Baky, On the Bertrand offsets for ruled and developable surfaces. Boll. Unione Mat. Ital., 8 (2015), 53-64.
  19. J.A. Schaaf, Curvature theory of line trajectories in spatial kinematics. Doctoral dissertation, University of California, Davis, CA, 1988.
  20. G.R. Veldkamp, On the use of dual numbers, vectors, and matrices in instantaneous spatial kinematics. Mech. Mach. Theory, 11 (1976), 141-156.