Generalized Helices in the Euclidean 3-space Through Flow-frame

Year 2025, Volume: 18 Issue: 2, 335 - 348

Cansu Özyurt Anar , Yusuf Yaylı , Nejat Ekmekçi

Abstract

A representation of time-dependent rotation of a usual Frenet flow is called flow-frame; the angle of rotation is exactly the current parameter. In this paper, we investigate three types of helices in the Euclidean 3-space through flow-frame and give their geometrical description with flow-frame apparatus. Then we introduce the spherical images of a curve by translating flow-frame vectors to the center of unit sphere in the Euclidean 3-space R3. Besides, we examine the relations between a generalized helix and its spherical images.

Keywords

Flow-frame , general helix , spherical indicatrix , Lancret’s theorem

Supporting Institution

TÜBİTAK (The Scientific and Technological Research Council of Turkey)

References

• Ali, A.T., López, R.: Slant helices in Minkowski space $E^3_1$ . J. Korean Math. Soc. 48(1), 159–167 (2011).
• Ali, A.T.: New special curves and their spherical indicatrices. Glob. J. Adv. Res. Class. Mod. Geom. 1(2), 28–38 (2012).
• Barros, M.: General helices and a theorem of Lancret. Proc. Am. Math. Soc. 125(5), 1503–1509 (1997).
• Barros, M., Ferrández, A., Lucas, P., Meroaño, M.A.: General helices in the 3-dimensional Lorentzian space forms. Rocky Mt. J. Math. 31(2), 373–388 (2001).
• Crasmareanu, M.: The flow-curvature of spacelike parametrized curves in the Lorentz plane. Proc. Int. Geom. Cent. 15(2), 101–109 (2022).
• Crasmareanu, M.: The flow-curvature of plane parametrized curves. Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 72(2), 1–11 (2023).
• Crasmareanu, M.: The flow-geodesic curvature and the flow-evolute of hyperbolic plane curves. Int. Electron. J. Geom. 16(2), 225–231 (2023).
• Crasmareanu, M.: The flow-curvature of curves in a geometric surface. Commun. Korean Math. Soc. 38(4), 1261–1269 (2023).
• Çiftçi, Ü.: A generalization of Lancert’s theorem. J. Geom. Phys. 59(12), 1597–1603 (2009).
• Ekmekci, N., Okuyucu, O.Z., Yaylı, Y.: Characterization of speherical helices in Euclidean 3-space. Analele Stiint. ale Univ. Ovidius Constanta 22(2), 99–108 (2014).
• Ferrández, A., Giménez, A., Lucas, P.: Null generalized helices in Lorentz-Minkowski spaces. J. Phys. A Math. Gen. 36(39), 8243–8251 (2002).
• Izumiya, S., Takeuchi, N.: New special curves and developable surfaces. Turk. J. Math. 28(2), 153–163 (2004).
• Kula, L., Yaylı, Y.: On slant helix and its spherical indicatrix. Appl. Math. Comput. 169(1), 600–607 (2005).
• Kula, L., Ekmekci, N., Yaylı, Y., ˙Ilarslan K.: Characterizations of slant helices in Euclidean 3-space. Turk. J. Math. 34, 261–273 (2010).
• Lancret, M.: Mémoire sur les courbes á double courbure. Mémoires présentés á l’Institut1. 416–454 (1806).
• Lucas, P., Ortega-Yagües, J.A.: Slant helices in the Euclidean 3-space revisited. Bull. Belg. Math. Soc. - Simon Stevin 23(1), 133–150 (2016).
• Lucas, P., Ortega-Yagües, J.A.: Slant helices in the three-dimensional sphere. J. Korean Math. Soc. 54(4), 1331–1343 (2017).
• Lucas, P., Ortega-Yagües, J.A.: Slant helices: a new approximation. Turk. J. Math. 43(1), 473–485 (2019).
• Okuyucu, O.Z, Gök, İ., Yaylı, Y., Ekmekci, N.: Slant helices in three dimensional Lie groups. Appl. Math. Comput. 221, 672–683 (2013).
• Scofield, P.D.: Curves of constant precession. Am. Math. Mon. 102(6), 531–537 (1995).
• Struik, D.J.:Lectures on classical differential geometry. Dover Publications, Inc., New York (1988).
• Yampolsky, A., Opariy, A.: Generalized helices in three-dimensional Lie groups. Turk. J. Math. 43, 1447–1455 (2019).