Generic properties of Darboux-type curves

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Abstract

In this paper, we present a method for constructing Darboux-type curves by using spherical curves and smooth functions. We show that all Darboux-type curves can be generated using this method. We discuss the relationship between Darboux-type curves and rectifying curves. Moreover, we establish a connection between Darboux-type curves and geodesics on a rectifying developable surface from the viewpoint of curves on ruled surfaces.

Keywords

Darboux-type curves / spherical curves / rectifying curves / rectifying developable surface / geodesics

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References

[1] G. Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal , Gauthier-Villars, 1915.

[2] M. Düldül, Vector fields and planes in E⁴ which play the role of Darboux vector , Turk. J. Math., 44 (2020), 274-283. https://doi.org/10.3906/mat-1908-9

[3] Z. Wang, D. Pei, Null Darboux developable and pseudo-spherical Darboux image of null Cartan curve in Minkowski 3-space, Hokkaido Math. J., 40 (2011), 219-240. https://doi.org/10.14492/hokmj/1310042829

[4] C. Ekici, M. Dede, On the Darboux vector of ruled surfaces in Pseudo-Galilean space, Math. Comput. Appl., 16 (2011), 830-838. https://doi.org/10.3390/mca16040830

[5] B. Bukcu, M. K. Karacan, Special Bishop motion and Bishop Darboux rotation axis of the space curve, J. Dyn. Syst. Geom. Theor., 6 (2008), 27-34. https://doi.org/10.1080/1726037X.2008.10698542

[6] B. Chen, F. Dillen, Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Acad. Sin., 33 (2005), 77-90.

[7] S. D. Gertzbein, J. Seligman, R. Holtby, K. H. Chan, A. Kapasouri, M. Tile, et al., Centrode patterns and segmental instability in degenerative disc disease, Spine, 10 (1985), 257-261. https://doi.org/10.1097/00007632-198504000-00014

[8] N. G. Ogston, G. J. King, S. D. Gertzbein, M. Tile, A. Kapasouri, J. D. Rubenstein, Centrode patterns in the lumbar spine, Baseline studies in normal subjects, Spine, 11 (1986), 591-595. https://doi.org/10.1097/00007632-198607000-00010

[9] P. J. Weiler, E. R. Bogoch, Kinematics of the distal radioulnar joint in rheumatoid arthritis: An In vivo study using centrode analysis, J. Hand Surg., 20 (1995), 937-943. https://doi.org/10.1016/S0363-5023(05)80140-6

[10] Y. Yu, Y. Yang, H. Liu, Darboux curves in 3-dimensional Euclidean space, J. Shenyang Norm. Univ. (Nat. Sci. Ed.), 31 (2013), 503-508. https://doi.org/10.3969/j.issn.1673-5862.2013.04.014

[11] J. Qian, M. Sun, P. Yin, Y. Kim, Null Darboux curve pairs in Minkowski 3-space, Axioms, 10 (2021), 142. https://doi.org/10.3390/axioms10030142

[12] R. Encheva, G. Georgiev, Shapes of space curves, J. Geom. Graphics, 7 (2003), 145-155.

[13] S. Izumiya, N. Takeuchi, Generic properties of helices and Bertrand curves, J. Geom., 74 (2002), 97-109. https://doi.org/10.1007/PL00012543

[14] Y. Wang, Y. Chang, Mannheim curves and spherical curves, Int. J. Geom. Methods Mod. Phys., 17 (2020), 2050101. https://doi.org/10.1142/S0219887820501017

[15] B. Chen, When does the position vector of a space curve always lie in its rectifying plane? Am. Math. Mon., 110 (2003), 147-152. https://doi.org/10.1080/00029890.2003.11919949

[16] S. Izumiya, N. Takeuchi, Special curves and ruled surfaces, Beitr. Algebra Geom., 44 (2003), 203-212.

[17] M. P. Do Carmo, Differential Geometry of Curves and Surfaces , Prentice-Hall, 1976.

[18] S. Deshmukh, B. Chen, S. H. Alshammari, On rectifying curves in Euclidean 3-space, Turk. J. Math., 42 (2018), 609-620. https://doi.org/10.3906/mat-1701-52

[19] J. J. Koenderink, Solid Shape , MIT Press, 1990.

[20] S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turk. J. Math., 28 (2004), 153-163.