Partner Curves in Three Dimensional Degenerated Space Form

Year 2024, Volume: 17 Issue: 1, 213 - 220, 23.04.2024

Huili Liu Yixuan Liu

https://doi.org/10.36890/iejg.1466320

Abstract

Using elementary and effective methods we study cone curves and their associated curves or partner curves
in the three dimensional lightlike cone $\Q^3$, which is called three dimensional degenerated space form of the four dimensional Minkowski space $\E^4_1$.
We define the associated curve of the cone curve and also partner curves of
some special curves, such as a Bertrand curve and a Mannheim curve.
We consider the properties and
relations of a curve and its associated curve or partner curve.
Some geometric characterizations of these curves are also given.

Keywords

Degenerated space form, cone curve, curvature and torsion, associated curve, partner curve

References

• [1] Chen, B. Y.: When does the position vector of a space curve always lie in its rectifying plane. Amer. Math. Monthly, 110, 147-152(2003).
• [2] Chen, B. Y. and Dillen, F.:Rectifying curves as centrodes and extremal curves. Bull. Inst. Math. Acad. Sinica, 33, 77-90(2005).
• [3] Choi, J. H., Kang, T. H. and Kim,Y. H.:Bertrand curves in 3-dimensional space forms. Applied Mathematics and Computation, 219, 1040- 1046(2012).
• [4] Choi, J. H., Kang, T. H. and Kim,Y. H.: Mannheim curves in 3-dimensional space forms. Bull. Korean Math. Soc. 50 No. 4, 1099-1108(2013).
• [5] Ferrández, A.:Some variational problems on curves and applications. S. Haesen and L. Verstraelen (eds.), Topics in Modern Differential Geometry, Atlantis Transactions in Geometry 1, pp. 199-222. Springer 978-94-6239-239-7. DOI.: 10.2991/978-94-6239-240-3_8, 2017.
• [6] Ferrández, A., Gimenéz, A. and Lucas, P.: Null helices in Lorentzian space forms. Internt. J. Modern Phys. A, 16, 4845-4863(2001). [7] Koenderink, J. J.:Solid Shape. MIT Press, Cambridge 1990.
• [8] Liu,H.: Curves in the lightlike cone. Contrib. Algebr. Geom. 45, 291-303(2004).
• [9] Liu,H.: Ruled surfaces with lightlike ruling in 3-Minkowski space. J. Geom. Phys. 59, 74-78(2009).
• [10] Liu,H.: Characterizations of ruled surfaces with lightlike ruling in Minkowski 3-space. Results in Mathematics, 56, 357-368(2009).
• [11] Liu,H.: Curves in affine and Semi-Euclidean spaces. Results in Mathematics, 65, 235-249(2014).
• [12] Liu,H.: Curves in three dimensional Riemannian space forms. Results in Mathematics, 66, 469-480(2014).
• [13] Liu,H. and Jung, S. D.: Riccati equations and cone curves in Minkowski 3-space. J. Geom. 108, 623-635(2017).
• [14] Liu,H. and Jung, S. D.: Null curves and representation in three dimensional Minkowski spacetime. New Horizons in Mathematical Physics, 1 No. 1, 1-10 (2017).
• [15] Liu,H. and Jung, S. D.: Structures and properties of null scroll in Minkowski 3-space, International Journal of Geometric Methods in Modern Physics, 14 No. 5, 1750066(11pages)(2017).
• [16] Liu,H. and Liu, Y.: Curves in three dimensional Riemannian space forms. J. Geom. 112, 8(2021). https://doi.org/10.1007/s00022-021-00574-7
• [17] Liu,H., Liu, Y. and Jung, S. D.: Ruled invariants and associated ruled surfaces of a space curve. Applied Mathematics and Computation, 348, 479-486(2019).
• [18] Liu,H., Liu, Y. and Jung, S. D.: Ruled invariants and total classification of non-developable ruled surfaces. J. Geom., 113, 21(2022). https://doi.org/10.1007/s00022-022-00631-9
• [19] Liu,H. and Meng, Q.: Representation formulas of curves in a 2 and 3 dimensional lightlike cone.Results in Mathematics, 59, 437-451(2011).
• [20] Lucas, P. and Ortega-Yagües, J. A.: Bertrand curves in the three-dimensional sphere. Journal of Geometry and Physics, 62, 1903-1914(2012).
• [21] Lucas, P. and Ortega-Yagües, J. A.: Bertrand curves in non-flat three - dimensional (Riemannian or Lorentzian) space forms. Bulletin of the Korean Mathematical Society, 50, 1109-1126(2013).
• [22] Patrikalakis, N. M. and Maekawa, T.: Shape interrogation for computer aided design and manufacturing. Springer-Verlag Berlin Heidelberg 2002.
• [23] Porteous, I. R.: Geometric differentiation for the intelligence of curves and surfaces. 2nd edn. Cambridge University Press, Cambridge 2001.
• [24] Rogers, C. and Schief,W. K.: Bäcklund and Darboux transformations-Geometry and modern applications in soliton theory. Cambridge University Press 2002.