A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum

Year 2021, , 7 - 18, 30.04.2021

Kahraman Esen Özen , Murat Tosun

https://doi.org/10.33187/jmsm.869698

Cited By: 5

Abstract

The theory of curves has a very long history. Moving frames defined on curves are important parts of this theory. They have never lost their importance. A point particle of constant mass moving along a trajectory in space may be seen as a point of the trajectory. Therefore, there is a very close relationship between the differential geometry of the trajectory and the kinematics of the particle moving on it. One of the most important elements of the particle kinematics is the jerk vector of the moving particle. Recently, a new resolution of the jerk vector, along the tangential direction and two special radial directions, has been presented by \"Ozen et al. (JTAM 57(2)(2019)). By means of these two special radial directions, we introduce a new moving frame for the trajectory of a moving particle with non vanishing angular momentum in this study. Then, according to this frame, some characterizations for the trajectory to be a rectifying curve, an osculating curve, a normal curve, a planar curve and a general helix are given. Also, slant helical trajectories are defined with respect to this frame. Afterwards, the necessary and sufficient conditions for the trajectory to be a slant helical trajectory (according to this frame) are obtained and some special cases of these trajectories are investigated. Moreover, we provide an illustrative numerical example to explain how this frame is constructed. This frame is a new contribution to the field and it may be useful in some specific applications of differential geometry, kinematics and robotics in the future.

Keywords

Kinematics of a particle, Plane and space curves, Moving frame, Angular momentum, Slant helices

References

• [1] K. Taskopru, M. Tosun, Smarandache curves on S2, Bol. da Soc. Parana. de Mat., 32(1) (2014), 51-59.
• [2] K. Eren, H. H. Kosal, Evolution of space curves and the special ruled surfaces with modified orthogonal frame, AIMS Math., 5(3) (2020), 2027-2039.
• [3] O. G. Yıldız, M. Akyigit, M. Tosun, On the trajectory ruled surfaces of framed base curves in the Euclidean space, Math. Methods Appl. Sci., 1-8, (2020), https://doi.org/10.1002/mma.6267
• [4] B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane, Am. Math. Mon., 110(2) (2003), 147-152.
• [5] K. Ilarslan, E. Nesovic, Some characterizations of osculating curves in the Euclidean spaces, Demonstr. Math., 41(4) (2008), 931-939.
• [6] Z. Bozkurt, ˙I. G¨ok, O. Z. Okuyucu, F. N. Ekmekci, Characterizations of rectifying, normal and osculating curves in three dimensional compact Lie groups, Life Sci. J., 10(3) (2013), 819-823.
• [7] R. L. Bishop, There is more than one way to frame a curve, Am. Math. Mon., 82 (1975), 246-251.
• [8] S. Yılmaz, M. Turgut, A new version of Bishop frame and an application to spherical images, J. Math. Anal. Appl., 371(2) (2010), 764-776.
• [9] M. A. Soliman, N. H. Abdel-All, R. A. Hussien, T. Youssef, Evolution of space curves using Type-3 Bishop frame, Caspian J. Math. Sci. 8(1) (2019), 58-73.
• [10] M. Dede, C. Ekici, H. Tozak, Directional tubular surfaces, Int. J. Algebra, 9(12) (2015), 527-535.
• [11] G. Y. Senturk, S. Yuce, Bertrand offsets of ruled surfaces with Darboux Frame, Results Math., 72(3) (2017), 1151-1159.
• [12] B. Uzunoglu, I. Gok, Y. Yaylı, A new approach on curves of constant precession, Appl. Math. Comput., 275 (2016), 317-323.
• [13] O. Keskin, Y. Yaylı, An application of N-Bishop frame to spherical images for direction curves, Int. J. Geom. Methods Mod. Phys., 14(11) (2017), 1750162.
• [14] T. Shifrin, Differential Geometry: A First Course in Curves and Surfaces, University of Georgia, Preliminary Version, 2008.
• [15] A. Menninger, Characterization of the slant helix as successor curve of the general helix, Int. Electron. J. Geom., 7(2) (2014), 84-91.
• [16] J. Casey, Siacci’s resolution of the acceleration vector for a space curve, Meccanica, 46 (2011), 471-476.
• [17] K. E. Ozen, F. S. Dundar, M. Tosun, An alternative approach to jerk in motion along a space curve with applications, J. Theor. Appl. Mech., 57(2) (2019), 435-444.
• [18] K. E. Ozen, M. Tosun, On the resolution of the acceleration vector according to Bishop frame, Univers. J. Math. Appl., 4(1) (2021), 26-32.
• [19] D. J. Struik, Lectures on Classical Differential Geometry, Dover, New-York, 1988.
• [20] S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turk. J. Math. 28 (2004), 153-163.
• [21] B. Bukcu, M. K. Karacan. The slant helices according to Bishop frame, Int. J. Comput. Math. Sci., 3 (2009), 67-70.
• [22] A. T. Ali, M. Turgut, Some characterizations of slant helices in the Euclidean space En, Hacet. J. Math. Stat. 39(3) (2010), 327-336.
• [23] O. Z. Okuyucu, I. Gok, Y. Yaylı, N. Ekmekci, Slant helices in three dimensional Lie groups, Appl. Math. Comput., 221 (2013), 672-683.
• [24] P. Lucas, J. A. Ortega-Yag¨ues, Helix surfaces and slant helices in the three-dimensional anti-De Sitter space, RACSAM 111(4) (2017), 1201-1222.
• [25] N. Macit, M. Duldul, Relatively normal-slant helices lying on a surface and their characterizations, Hacet. J. Math. Stat., 46(3) (2017), 397-408.