ON THE DETERMINATION OF A DEVELOPABLE SPHERICAL ORTHOTOMIC TIMELIKE RULED SURFACE

Abstract

In this paper, a method for determination of developable spherical orthotomic ruled surfaces generated by a spacelike curve on dual hyperbolic unit sphere is given by using dual vector calculus in R31 . We show that dual vectorial expression of a developable spherical orthotomic timelike ruled surface can be obtained from coordinates and the rst derivatives of the base curve. The paper concludes with an example related to this method.

Keywords

• Ruled Surface,Developable Surface,Dual Hyperbolic Unit Sphere,Minkowski Space

References

1. [1] N. Alamo, C. Criado, Generalized Antiorthotomics and their Singularities, Inverse Problems, 18(3)(2002), 881-889.
2. [2] W. Blaschke, Vorlesungen Uber Dierential Geometry I., Verlag von Julieus Springer in Berlin (1930) pp.89.
3. [3] J. W. Bruce, On Singularities, Envelopes and Elementary Dierential Geometry, Math. Proc. Cambridge Philos. Soc., 89 (1) (1981) 43-48.
4. [4] J. W. Bruce, P. J. Giblin, Curves and Singularities. A Geometrical Introduction to Singularity Theory, Second Edition, University Press, Cambridge, (1992).
5. [5] J. W. Bruce, P. J. Giblin, One-parameter Families of Caustics by Re exion in the Plane, Quart. J. Math. Oxford Ser., (2), 35(139) (1984) 243-251.
6. [6] C. Georgiou, T. Hasanis, D. Koutrouotis, On the Caustic of a Convex Mirror, Geom. Ded- icata, 28(2)(1988), 153-169.
7. [7] C. G. Gibson, Elementary Geometry of Dierentiable Curves, Cambridge University Press, May (2011).
8. [8] J. Hoschek, Smoothing of curves and surfaces. Computer Aided Geometric Design, Vol. 2, No. 1-3 (1985), special issue, 97-105.

9. [9]  O. G. Yldz, H. H. Hacsaligoglu, Study Map of Orthotomic of a Circle, International J. Math. Combin., Vol.4 (2014), 07-17.
10. [10]  O. G. Yldz, S.  O. Karakus, H. H. Hacsaligoglu,On the determination of a developable spherical orthotomic ruled surface, Bull. Math. Sci., (2014) Doi: 10.1007/s13373-014-0063-5.
11. [11]  O. Kose, A Method of the Determination of a Developable Ruled Surface, Mechanism and Machine Theory, 34 (1999), 1187-1193.
12. [12] C.Ekici, E. Ozusaglam, On the Method of Determination of a Developable Timelike Ruled Surface, Kuwait Journal of Science and Engineering, Vol:39(1A), 19-41, 2012.
13. [13] C.Y. Li, R.H. Wang, C.G. Zhu, An approach for designing a developable surface through a given line of curvature, Computer Aided Design, 45 (2013), 621-627.
14. [14] J.M. McCarthy, On The Scalar and Dual Formulations of the Curvature Theory of Line Trajectories, ASME Journal of Mechanisms, Transmissions and Automation in Design, (1987), 109/101.
15. [15] E. Study, Geometrie der Dynamen, Leibzig, (1903).
16. [16] H. H. Ugurlu, A. C aliskan, The Study mapping for directed spacelike and timelike lines in Minkowski 3-space, Mathematical and Computational Applications, Vol. 1, No:2 (1996), 142-148.
17. [17] G. R. Veldkamp, On the use of dual numbers, vectors and matrices in instantaneous, spatial kinematics, Mech. Mach. Theory, 11 (1976), no. 2, 141-156.
18. [18] J.F. Xiong, Spherical Orthotomic and Spherical Antiorthotomic, Acta Mathematica Sinica, 23 (2007), Issue 9, pp 1673-1682.
19. [19] Y.Yayl, A. C alskan, H.H. Ugurlu, The E. Study Maps of Circles on Dual Hyperbolic and Lorentzian Unit Spheres H20 and S21 ; Mathematical Proceedings of the Royal Irish Academy, 102A(2002), 1, 37-47.