Free Access

On the Curvature Functions of Ruled Surface

Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia

https://doi.org/10.1142/S021819592450002XCited by:0 (Source: Crossref)

Abstract

We considered differential geometries of helical curves for ruled and developable surfaces in Euclidean 3-space E 3. We clarify the conditions for ruled surface to become stationary-axis ruled surface. Then, we concentrate on ruled surface generated by a line undergoing a one-parameter screw movement. Finally, we provide some modules to illustrate the main results.

Communicated by Young J. Kim

Keywords:

AMSC: 53A04, 53A05, 53A17

Remember to check out the Most Cited Articles!
Check out these titles in image analysis!

References

1. M. P. Do Carmo , Differential Geometry of Curves and Surfaces (Pearson Education, Upper Saddle River, 1976). Google Scholar
2. O. Bottema and B. Roth , Theoretical Kinematics (North-Holland Press, New York, 1979). Google Scholar
3. A. Karger and J. Novak , Space Kinematics and Lie Groups (Gordon and Breach Science Publishers, New York, 1985). Google Scholar
4. H. Pottman and J. Wallner , Computational Line Geometry (Springer-Verlag, Berlin, Heidelberg, 2001). Crossref, Google Scholar
5. J. A. Schaaf , Structure Curvature Function of Line Trajectories in Spatial Kinematics (Doctoral dissertation, University of California, Davis, CA, 1988). Google Scholar
6. W. Kühnel and M. Steller , On closed Weingarten surfaces, Monatsh. Math. 146 (2005) 113–126. Crossref, Web of Science, Google Scholar
7. Y. Yuan , The Binormal Surface of the Space Curves (Master thesis of Northeastern University, 2006). Google Scholar
8. K. S. Sprott and B. Ravani , Cylindrical milling of ruled surfaces, Int. J. Adv. Manuf. Technol. 38 (2008) 649–656. Crossref, Web of Science, Google Scholar
9. R. A. Abdel-Baky and F. R. Al-Solamy , A new geometrical approach to one-parameter spatial motion, J. Eng. Maths. 60 (2008) 149–172. Crossref, Web of Science, Google Scholar
10. R. A. Abdel-Baky and R. A. Al-Ghefari , On the one-parameter dual spherical motions, Computer Aided Geometric Design 28 (2011) 23–37. Crossref, Web of Science, Google Scholar
11. Y. Yu, H. Liu and S. D. Jung , Structure and characterization of ruled surfaces in Euclidean 3-space, AMC. 233 (2014) 252–259. Google Scholar
12. Y. Yu, H. Liu and S. D. Jung , Invariants of non-developable ruled surfaces in Euclidean 3-space, Beitr. Algebra Geom. 55 (2014) 189–199. Crossref, Google Scholar
13. R. A. Al-Ghefari and R. A. Abdel-Baky , Kinematic geometry of a line trajectory in spatial motion, J. Mechanical Science and Technology 29(9) (2015) 3597–3608. Crossref, Web of Science, Google Scholar
14. R. A. Abdel-Baky , Developable surfaces through sweeping surfaces, Bulletin of the Iranian Mathematical Society 45 (2018) 951–963. Crossref, Google Scholar
15. R. A. Abdel-Baky , On the curvature theory of a line trajectory in spatial kinematics, Commun. Korean Math. Soc. 34(1) (2019) 333–349. Web of Science, Google Scholar
16. M. C. Aslan and G. A. Sekerci , Dual curves associated with the Bonnet ruled surfaces, Int. J. Geom. Methods Mod. Phys. 17(13) (2020) 2050204. Link, Web of Science, Google Scholar
17. N. Alluhaibi , Ruled surfaces with constant Disteli-axis, AIMS Mathematics 5(6) (2020) 7678–7694. Crossref, Web of Science, Google Scholar
18. D. L. Wang, J. Liu and D. Z. Xiao , Geometrical characteristics of some typical spatial constraints, Mechanism and Machine Theory 35 (2000) 1413–1430. Crossref, Web of Science, Google Scholar
19. H. Liu, Y. Liu and S. D. Jung , Ruled invariants and total classiffication of non-developable ruled surfaces, J. Geom. 113(1) (2022) 21, p. 13. Crossref, Google Scholar
20. Y. L. Li, Y. S. Zhu and Q. Y. Sun , Singularities and dualities of pedal curves in pseudo-hyperbolic and de Sitter space, Int. J. Geom. Methods Mod. Phys. 18 (2021) 1–31. Link, Web of Science, Google Scholar
21. Y. L. Li, S. Nazra and R. A. Abdel-Baky , Singularities Properties of timelike sweeping surface in Minkowski 3-Space, Symmetry 2022 (1996). https://doi.org/10.3390/sym14101996. Google Scholar
22. Y. L. Li, Z. Chen, S. Nazra and R. A. Abdel-Baky , Singularities for timelike developable surfaces in Minkowski 3-Space, Symmetry 15 (2023) 277. Crossref, Web of Science, Google Scholar