Zehra İşbilir, Kahraman Esen Özen, Murat Tosun

DOI Number
First page
Last page


In this study, we consider the concept of Mannheim partner trajectories related to the Positional Adapted Frame on Regular Surfaces (PAFORS) for the particles moving on the different regular surfaces in Euclidean 3-space. We give the relations between the PAFORS elements of these aforementioned trajectories. Also, we obtain the relations between Darboux basis vectors of these trajectories. Furthermore, some special cases of these trajectories are written.


Mannheim partner trajectories, Positional Adapted Frame on Regular Surfaces, Darboux basis vectors.

Full Text:



R. L. Bishop: There is more than one way to frame a curve. Amer. Math. Monthly 82 (1975), 246-251.

R. Blum: A remarkable class of Mannheim-curves. Can. Math. Bull. 9 (1966), 223-228.

G. Darboux: Lecons Sur La Thorie Gnrale Des Surfaces I-II-III-IV. Gauthier-Villars, Paris, 1896.

M. Dede: A new representation of tubular surfaces. Houston J. Math. 45 (2019), 707-720.

M. Dede, C. Ekici and H. Tozak: Directional tubular surfaces. Int. J. Algebra 9 (2015), 527-535.

F. Dogan ˘ and Y. Yaylı: Tubes with Darboux frame. Int. J. Contemp. Math. Sci. 7 (2012), 751-758.

M. A. Gung ¨ or ¨ and M. Tosun: A study on dual Mannheim partner curves. Int. Math. Forum. 5 (2010), 2319-2330.

N. E. Gurb ¨ uz ¨ : The evolution of an electric field with respect to the type-1 PAF and the PAFORS frames in R3 1. Optik 250 (2022), 168285.

M. Kazaz, H. H. Ugurlu, M. ˘ Onder ¨ and T. Kahraman: Mannheim partner Dcurves in the Euclidean 3-Space E3. New Trend. Math. Sci. 3 (2015), 24-35.

O. Keskin and Y. Yaylı: An application of N-Bishop frame to spherical images for direction curves. Int. J. Geom. Methods Mod. Phys. 14 (2017), 1750162.

T. Korpnar ¨ and Y. Unl ¨ ut ¨ urk ¨ : An approach to energy and elastic for curves with extended Darboux frame in Minkowski space. AIMS Mathematics 5 (2020), 1025-1034.

H. Liu and F. Wang: Mannheim partner curves in 3-space. Journal of Geometry 88 (2008), 120-126.

A. Mannheim: Paris C.R. 86 (1878), 1254-1256.

M. Masal and A. Z. Azak: Mannheim B-curves in the Euclidean 3-space. Kuwait J. Sci. 44 (2017), 36-41.

B. O'Neil: Elemantary Differential Geometry. Academic Press, New York, 1966.

K. Orbay, E. Kasap and I. Aydemir: Mannheim offsets of ruled surfaces. Mathematical Problems in Engineering 2009 (2009), 160917.

K. E. Ozen ¨ and M. Tosun: A new moving frame for trajectories on regular surfaces. Ikonion Journal of Mathematics 3 (2021), 20-34.

K. E. Ozen ¨ and M. Tosun: A new moving frame for trajectories with non-vanishing angular momentum. J. Math. Sci. Model. 4 (2021), 7-18.

K. E. Ozen ¨ and M. Tosun: Some characterizations on geodesic, asymptotic and slant helical trajectories according to PAFORS. Maltepe Journal of Mathematics 3 (2021), 74-90.

K. E. Ozen, M. Tosun ¨ and M. Akyigit ˘ : Siaccis theorem according to Darboux frame. An. S¸t. Univ. Ovidius Constant¸a 25 (2017), 155-165.

S. Ozkaldı, K. ¨ Ilarslan _ and Y. Yaylı: On Mannheim partner curve in dual space. An. S¸t. Univ. Ovidius Constant¸a 17 (2009), 131-142.

S. P. Radzevich: Geometry of Surfaces: A Practical Guide for Mechanical Engineers. Wiley, 2013.

P. D. Scofield: Curves of constant precession. Amer. Math. Monthly 102 (1995), 531-537.

T. Shifrin: Differential Geometry: A First Course in Curves and Surfaces. University of Georgia, Preliminary Version, 2008.

M. A. Soliman, N. H. Abdel-All, R. A. Hussien and T. Youssef: Evolution of space curves using type-3 Bishop frame. Caspian J. Math. Sci. 8 (2019), 58-73.

G. Y. S¸enturk ¨ and S. Yuce ¨ : Bertrand offsets of ruled surfaces with Darboux frame. Results in Mathematics 72 (2017), 1151-1159.

Y. Unl ¨ ut ¨ urk, M. C¸ imdiker ¨ and C. Ekici: Characteristic properties of the parallel ruled surfaces with Darboux frame in Euclidean 3-space. Communication in Mathematical Modeling and Applications 1 (2016), 26-43.

F. Wang and H. Liu: Mannheim partner curves in 3-Euclidean space. Mathematics in Practice and Theory 37 (2007), 141-143.

S. Yılmaz and M. Turgut: A new version of Bishop frame and an application to spherical images. J. Math. Anal. Appl. 371 (2010), 764-776.



  • There are currently no refbacks.

© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)