Mannheim Partner D-Curves in the Euclidean 3-space

Abstract

In this paper, we consider the idea of Mannheim partner curves for curves lying on surfaces. By considering the Darbouxframes of surface curves, we define Mannheim partner D-curves and give the characterizations for these curves. We also find therelations between geodesic curvatures, normal curvatures and geodesic torsions of these associated curves. Furthermore, we show thatdefinition and characterizations of Mannheim partner D-curves include those of Mannheim partner curves in some special cases

Keywords

• Mannheim partner D-curves,Darboux frame; geodesic curvature; normal curvature; geodesic torsion

Mannheim partner D-curves in the Euclidean 3-space E3

Öz

Kaynakça

1. Burke J. F., Bertrand Curves Associated with a Pair of Curves, Mathematics Magazine, 34(1) (1960) 60-62.
2. G¨org¨ul¨u, E., ¨Ozdamar, E., A generalizations of the Bertrand curves as general inclined curves in En, Communications de la Fac. Sci. Uni. Ankara, Series A1, 35 (1986) 53-60.
3. Hacisaliho˘glu, H. H., Diferansiyel Geometri, ˙In¨on¨u ¨Universitesi Fen-Edebiyat Fak¨ultesi Yayınları No:2, 1983.
4. Izumiya, S., Takeuchi, N., Special Curves and Ruled surfaces, Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry, 44(1) (2003) 203-212.
5. Izumiya, S., Takeuchi, N., Generic properties of helices and Bertrand curves, Journal of Geometry 74 (2002) 97–109.
6. ¨Onder, M., U˘gurlu, H.H., On the Developable Mannheim Offsets of Timelike Ruled Surfaces, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 84(4) (2014) 541-548
7. ¨Onder, M., U˘gurlu, H.H., On the Developable Mannheim offsets of spacelike ruled surfaces, arXiv:0906.4660v25 [math.DG].
8. Kahraman, T., ¨Onder, M., Kazaz, M., U˘gurlu, H.H., Some Characterizations of Mannheim Partner Curves in Minkowski 3-space E3, Proceedings of the Estonian Academy of Sciences, 60(4) (2011) 210–220. 1, Proceedings of the Estonian Academy of Sciences, 60(4) (2011) 210–220.

9. Liu, H., Wang, F., Mannheim partner curves in 3-space, Journal of Geometry, 88(1-2) (2008) 120-126.
10. O’Neill, B., Elemantery Differential Geometry, Academic Press Inc. New York, 1966.
11. Orbay, K., Kasap, E., Aydemir, I, Mannheim Offsets of Ruled Surfaces, Mathematical Problems in Engineering, Volume 2009, Article ID 160917.
12. Ravani, B., Ku, T.S., Bertrand Offsets of ruled and developable surfaces, Comp. Aided Geom. Design, 23(2) (1991) 145-152.
13. Struik, D.J., Lectures on Classical Differential Geometry, 2nded. Addison Wesley, Dover, 1988.
14. Wang, F., Liu, H., Mannheim partner curves in 3-Euclidean space, Mathematics in Practice and Theory, 37(1) (2007) 141-143.
15. Whittemore, J.K., Bertrand curves and helices,Duke Math. J.6(1) (1940) 235-245.