Year 2024,
Volume: 17 Issue: 1, 252 - 258, 23.04.2024
Alexander Navarro
,
Yun Myung Oh
References
- [1] Alghanemi, A., Khan, M.A.: Position Vectors of the Natural Mate and Conjugate of a Space Curve. Adv. Math. Phys., 1-5 (2023).
https://doi.org/10.1155/2023/7565988
- [2] Bertrand, J.M.: M’emoire sur la th’eorie des courbes a’ double courbure. Comptes Rendus. 15 (1), 332-350 (1850).
- [3] Camci, Ç., Chen, B.-Y., ˙Ilarslan, K. et al.: Sequential natural mates of Frenet curves in Euclidean 3-space. J. Geom. 112, 46 (2021).
https://doi.org/10.1007/s00022-021-00610-6
- [4] 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).
- [5] Choi, J. H., Kim, Y. H.: Associated curves of a Frenet curve and their applications. Appl. Math. Comput. 218 (18), 9116-9124 (2012).
https://doi.org/10.1016/j.amc.2012.02.064
- [6] Deshmukh S., Chen B.-Y., Alghanemi A.: Natural mates of Frenet curves in Euclidean 3-space. Turk. J. Math. 42, 2826–2840 (2018).
- [7] Deshmukh, S., Chen B.-Y., Turki N.B.: A differential equations for Frenet curves in Euclidean 3-space and its applications. Rom. J. Math. Comput.
Sci. 8, 1-6 (2018).
- [8] Menninger, T.: Characterization of the slant helix as successor curve of the general helix. Int. Electron. J. Geom. 7, 84-91 (2014).
Extending Natural Mates in Euclidean 3-space and Applications to Bertrand Pairs
Year 2024,
Volume: 17 Issue: 1, 252 - 258, 23.04.2024
Alexander Navarro
,
Yun Myung Oh
Abstract
In Euclidean 3-space, a family of curves, the co-successor, is motivated and then introduced in relation to the natural mate. A complete characterization of co-successors is proved, followed by an application of the co-successor towards describing Bertrand curves and their mates.
Ethical Statement
This is joint work with Alex and has not been submitted anywhere else.
Supporting Institution
Andrews University
Thanks
Thank you for this exceptional opportunity!
References
- [1] Alghanemi, A., Khan, M.A.: Position Vectors of the Natural Mate and Conjugate of a Space Curve. Adv. Math. Phys., 1-5 (2023).
https://doi.org/10.1155/2023/7565988
- [2] Bertrand, J.M.: M’emoire sur la th’eorie des courbes a’ double courbure. Comptes Rendus. 15 (1), 332-350 (1850).
- [3] Camci, Ç., Chen, B.-Y., ˙Ilarslan, K. et al.: Sequential natural mates of Frenet curves in Euclidean 3-space. J. Geom. 112, 46 (2021).
https://doi.org/10.1007/s00022-021-00610-6
- [4] 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).
- [5] Choi, J. H., Kim, Y. H.: Associated curves of a Frenet curve and their applications. Appl. Math. Comput. 218 (18), 9116-9124 (2012).
https://doi.org/10.1016/j.amc.2012.02.064
- [6] Deshmukh S., Chen B.-Y., Alghanemi A.: Natural mates of Frenet curves in Euclidean 3-space. Turk. J. Math. 42, 2826–2840 (2018).
- [7] Deshmukh, S., Chen B.-Y., Turki N.B.: A differential equations for Frenet curves in Euclidean 3-space and its applications. Rom. J. Math. Comput.
Sci. 8, 1-6 (2018).
- [8] Menninger, T.: Characterization of the slant helix as successor curve of the general helix. Int. Electron. J. Geom. 7, 84-91 (2014).