Mannheim B-Curves in the Euclidean 3-Space E^3
Keywords:
Bishop frame, Euclidean space, Frenet frame, Mannheim B-pair, Mannheim B-curvesAbstract
In this study we have defined Mannheim curves according to a Bishop frame and called Mannheim B-curves in the Euclidean 3-space E^3. Then the relations have been found between the Bishop and Frenet vectors of these curves and some conditions have been obtained which are Mannheim B-curves. Also the relation k_2^*=k_1/1+lambdak_1 has been obtained between the Bishop curvatures of the Mannheim B-curves.References
Akyigit M., Ersoy S., Uzgur, I. & Tosun M. 2011. Generalized timelike Mannheim curves in Minkowski Space-Time . Mathematical Problems in Engineering. Article ID 539378: 19 pages, doi 10.1155/2011/539378.
Bishop R. L. 1975. There is more than one way to Frame a curve. American Mathematical Monthly 82(3): 246-251, 1975.
Blum R. 1996. A remarkable class of Mannheim-curves. Canadian Mathematical Bulletin 9, 223-228.
Gungor M. A. & Tosun M. 2010. A study on dual Mannheim partner curves. International Mathematical Forum 5( 47): 2319-2330.
Karacan M. K. 2011. Weakened Mannheim curves. International Journal of the Physical Sciences 6(20): 4700-4705.
K?z?ltug, S. & Yayl?, Y. (2015). On the quaternionic Mannheim curves of Aw (k) type in Euclidean space E3. Kuwait Journal of Science, 42(2):128140.
Lee J. W. 2011. No null-Helix Mannheim curves in the Minkowski Space .International Journal of Mathematics and Mathematical Sciences, Article ID 580537: 7 pages, doi.10.1155/2011/580537
Liu H. & Wang F. 2008. Mannheim partner curves in 3-space. Journal of Geometry 88: 120-126.
Matsuda H. & Yorozu S. 2009. On generalized Mannheim curves in Euclidean 4-space, Nihonkai Math Journal 20: 33-56.
Okuyucu O. Z. 2013. Characterizations of the Quaternionic Mannheim curves in Euclidean space . International Journal of Mathematical Combinatorics 2: 44-53.
Orbay K. & Kasap E. 2009. On Mannheim partner curves in , International Journal of Physical Sciences 4(5): 261-264.
Orbay K., Kasap E. & Aydemir I. 2009. Mannheim Offsets of ruled surfaces, Mathematical Problems in Engineering, Article ID 160917: 9 Pages, doi: 10.1155/2009/160917.
Onder, M. & K?z?ltug S. 2012. Bertrand and Mannheim partner D-curves on parallel surfaces in Minkowski 3-Space. International Journal of Geometry 1(2): 34-45.
Ozkald?, S., larslan K. & Yayh Y. 2009. On Mannheim partner curve in Dual space, Ananele Stiintifice Ale Universitatii Ovidius Constanta 17(2): 131-142.
Oztekin, H. B. & Ergut, M. 2011. Null Mannheim curves in the Minkowski 3-space .Turkish Journal of Mathematics 35(1): 107-114.
Serret, J.A. (1851). Sur quelques formules relatives a la theorie
des courbes a double courbure. Journal de mathematiques pures et
appliquees. 16:193-207.