On the differential geometric elements of bertrandian darboux ruled surface in E

Year 2017, , 572 - 576, 01.06.2017

Süleyman Senyurt , Şeyda Kılıçoğlu

https://doi.org/10.16984/saufenbilder.306867

Abstract

In this paper, we consider two special ruled surfaces associated to a Bertrand curve  and Bertrand mate   . First,
Bertrandian Darboux Ruled surface with the base curve  has been defined and examined in terms of the FrenetSerret apparatus of the curve  , in E3 . Later, the differential geometric elements such as, Weingarten map S,
Gaussian curvature K and mean curvature H, of Bertrandian Darboux Ruled the surface and Darboux ruled surface has
been examined relative to each other. Further, first, second and third fundamental forms of Bertrandian Darboux Ruled
surface have been investigated in terms of the Frenet apparatus of Bertrand curve  , too.  

Keywords

ruled surface, Darboux vector, Bertrand curves

Öklid uzayında bertrandian darboux regle yüzeyin diferensiyel geometrik elemanlar

Abstract

Bu çalışmada Bertrand eğrisi ve Bertrand eşi olan eğriler üzerinde Darboux vektörleri ile üretilen iki özel regle yüzeyi
gözönüne alındı. İlk olarak,  eğrisinin Bertrand Darboux regle yüzeyi, Bertrand eğrisinin Frenet-Serret aparatlar
cinsinden tanımlandı ve araştırıldı. Daha sonra, Bertrand Darboux regle yüzeyi ile Darboux regle yüzeyinin
Weingarten dönüşümü, Gauss eğriliği ve ortalama eğriliği gibi diferensiyel geometrik değişmezleri birbirleri ile ilişkili
olarak incelendi. Son olarak, Bertrand Darboux regle yüzeyinin birinci, ikinci ve üçüncü temel formlar  Bertrand
eğrisinin Frenet-Serret aparatlar cinsinden ifadeleri verildi.  

Keywords

Regle yüzey, Darboux vektörü, Bertrand eğrileri

References

• [1] do Carmo, M. P., Differential Geometry of Curves and Surfaces. Prentice-Hall, ISBN 0-13-212589-7, 1976.
• [2] Eisenhart, Luther P., A Treatise on the Differential Geometry of Curves and Surfaces. Dover, ISBN 0-486-43820-1, 2004.
• [3] Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 205, 1997.
• [4] Hacısalihoğlu, H.H., Differential Geometry(in Turkish), vol.1, Inönü University Publications, 1994.
• [5] Hoschek, J., Liniengeometrie. BI-Hochschulskripte, Zurich 1971.
• [6] Izumiya, S. and Takeuchi, N., Special curves and ruled surfaces . Beitr age zur Algebra und Geometrie Contributions to Algebra and Geometry, vol.44, no. 1, 203-212, 2003.
• [7] Kılıçoğlu Ş., Şenyurt S., and Hacısalihoğlu H. H., An Examination on the Positions of Frenet ruled Surface Along Bertrand Pairs and according to Their Normal Vector Fields in E^3 Applied Mathematical Sciences, Vol. 9, no. 142, 7095 - 7103 2015.
• [8] Şenyurt, S., and Kılıçoğlu S. On the differential geometric elements of the involute ˜D scroll, Adv. Appl. Clifford Algebras (2015) Springer Basel, doi:10.1007/s00006-015-0535-z. 25(4), pp. 977-988, 2015.
• [9] Kılıçoğlu Ş. and Şenyurt S., On the Differential Geometric Elements of Mannheim Darboux Ruled surface in E3 (accepted)
• [10] Springerlink, Encyclopaedia of Mathematics. Springer-Verlag, Berlin Heidelberg New York 2002.
• [11] Strubecker, K., Differential geometrie II. Sammlung G oschen, 2. Aufl., Berlin 1969.