A New Approach to Mannheim Curve in Euclidean 3-Space

Authors

  • Ali UÇUM Kırıkkale University, Faculty of Sciences and Arts, Department of Mathematics, Kırıkkale-Turkey
  • Çetin Camcı Department of Mathematics, Faculty of Sciences and Arts, Onsekiz Mart Univer- sity, Çanakkale, Turkey.
  • Kazım İlarslan Kırıkkale University, Faculty of Sciences and Arts, Department of Mathematics, Kırıkkale-Turkey.

DOI:

https://doi.org/10.5556/j.tkjm.54.2023.4085

Keywords:

Mannheim curves, general helices, anti-Salkowski curves, Euclidean 3-space

Abstract

In this article, a new approach is given for Mannheim curves in 3-dimensional Euclidean space. Thanks to this approach, the necessary and sufficient conditions including the known results have been obtained for a curve to be Mannheim curve in E³. In addition, related examples and graphs are given by showing that there can be Mannheim curves in Salkowski or anti-Salkowski curves as well as giving Mannheim mate curves, which are not in literature. Finally, the Mannheim partner curves are characterized in E³.

References

Ali A. T., Position vectors of slant helices in Euclidean 3-space, Journal of the Egyptian Mathematical Society 20 (2012), 1-6.

Camcı C., Uçum A. and İlarslan K., A new approach to Bertrand Curves in Euclidean 3-space, J. Geom. (2020) 111:49.

Izumiya S. and Takeuchi N., New special curves and developable surfaces, Turk. J. Math. 28 (2004), 531-537.

Kuhnel W., Differential geometry: curves-surfaces-manifolds. Braunschweig: Wiesbaden, 1999.

Lancret, M. A., Mémoire sur les courbes à double courbure, Mémoires présentés à l'Institut1 (1806), 416-454.

Liu H. and Wang F., Mannheim partner curves in 3-space. J. Geom. 88 (2008), 120-126.

Miller J., Note on Tortuous Curves, Proceedings of the Edinburgh Mathematical Society, 24, pp 51-55, 1905.

Monterde J., Salkowski curves revisted: A family of curves with constant curvature and non-constant torsion, Comput. Aided Geomet. Design 26 (2009) 271--278.

Salkowski E., Zur transformation von raumkurven, Mathematische Annalen 66 (1909) 517--557.

Struik D. J., Lectures on classical differential geometry, Dover publications, New York, 1961.

Tigano, O., Sulla determinazione delle curve di Mannheim, Matematiche Catania 3, 25-29, 1948.

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Published

2021-11-10

How to Cite

UÇUM, A., Camcı, Çetin, & İlarslan, K. (2021). A New Approach to Mannheim Curve in Euclidean 3-Space. Tamkang Journal of Mathematics, 54. https://doi.org/10.5556/j.tkjm.54.2023.4085

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