magazinelogo

Journal of Applied Mathematics and Computation

ISSN Online: 2576-0653 Downloads: 177303 Total View: 1998880
Frequency: quarterly ISSN Print: 2576-0645 CODEN: JAMCEZ
Email: jamc@hillpublisher.com
Article Open Access http://dx.doi.org/10.26855/jamc.2018.04.005

Surfaces family with a common Mannheim geodesic curve

Gülnur ŞAFFAK ATALAY*

Education Faculty, Department of Mathematics and Science Education, Ondokuz Mayis University, Samsun, Turkey.

*Corresponding author: Gülnur ŞAFFAK ATALAY

Published: April 15,2018

Abstract

In this paper, we analyzed surfaces family possessing a Mannheim partner curve of a given curve as a geodesic. Using the Frenet frame of the curve in Euclidean 3-space, we express the family of surfaces as a linear combination of the components of this frame and derive the necessary and sufficient conditions for coefficients to satisfy both the geodesic and isoparametric requirements. The extension to ruled surfaces is also outlined. Finally, examples are given to show the family of surfaces with common Mannheim geodesic curve.

References

Akyiğit, M., Ersoy, S., Özgür, İ. & 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.

Atalay G.Ş, E. Kasap. (2016). Surfaces Family with Common Smarandache Geodesic Curve According to Bishop Frame in Euclidean Space, Mathematical Sciences and Applications E-Notes 4 (1) 164-174. 

Atalay G.Ş, E. Kasap. (2017). Surfaces Family with Common Smarandache Geodesic Curve According to Frenet Frame in Euclidean Space, (in review: Journal of Science and Arts).

Bishop, R.L. (1975). There is more than one way to Frame a curve. American Mathematical Monthly, 82(3):246-251.

Blum, R. (1966). A remarkable class of Mannheim-curves. Canadian Mathematical Bulletin, 9:223-228.

Deng, B. (2011). Special Curve Patterns for Freeform Architecture Ph.D. thesis, submitted to the Vienna University of Technology, Faculty of Mathematics and Geoinformation of.

Do Carmo , M.P . (1976). Differential Geometry of Curves and Surfaces, Prentice Hall, Inc., Englewood Cliffs, New Jersey.

Güngör, 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.

Kızıltuğ, S. & Yaylı, Y. (2015). On the quaternionic Mannheim curves of Aw (k) – type in Euclidean space E3. Kuwait Journal of Science, 42(2):128–140.

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 Mathematical Journal, 20:33-56.

Okuyucu, O.Z. (2013). Characterizations of the quaternionic Mannheim curves in Euclidean space E4. International Journal of Mathematical Combinatorics, 2:44-53.

Orbay, K. & Kasap, E. (2009). On Mannheim partner curves in E3. International Journal of Physical Sciences, 4(5):261-264.

Orbay, K., Kasap, E. & Aydemir, İ. (2009). Mannheim offsets of ruled surfaces. Mathematical Problems in Engineering, Article ID 160917: 9 Pages, doi:10.1155/2009/160917.

O’Neill, B. (1966). Elementary Differential Geometry, Academic Press Inc., New York.

Önder, M. & Kızıltuğ, S. (2012). Bertrand and Mannheim partner D-curves on parallel surfaces in Minkowski 3-Space. International Journal of Geometry, 1(2):34-45.

Özkaldı, S., İlarslan, K. & Yaylı, Y. (2009). On Mannheim partner curve in Dual space. Ananele Stiintifice Ale Universitatii Ovidius Constanta, 17(2):131-142.

Öztekin, H.B. & Ergüt, 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 à la théorie des courbes à double courbure. Journal de mathématiques pures et appliquées. 16:193-207.

Wang, F., Liu, H. (2007). “Mannheim partner curves in 3-Euclidean space”, Mathematics in Practice and Theory, vol. 37, no. 1, pp. 141-143.

Wang, G. J., Tang, K. (2004). C. L. Tai, Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Des. 36 (5) 447-459.

Kasap, E., Akyildiz, F.T., Orbay, K. (2008). A generalization of surfaces family with common spatial geodesic, Applied Mathematics and Computation, 201781-789. 

Guan Z, Ling J, Ping X, Rongxi T (1997). Study and Application of Physics-Based Deformable Curves and Surfaces, Computers and Graphics 21: 305-313.

How to cite this paper

Surfaces family with a common Mannheim geodesic curve

How to cite this paper: Gülnur ŞAFFAK ATALAY. (2018). Surfaces family with a common Mannheim geodesic curve. Journal of Applied Mathematics and Computation, 2(4), 155-165.

DOI: http://dx.doi.org/10.26855/ jamc.2018.04.005