Direction curves of generalized Bertrand curves and involute-evolute curves in $E^{4}$

Year 2022, Volume: 71 Issue: 2, 326 - 338, 30.06.2022

Mehmet Önder

https://doi.org/10.31801/cfsuasmas.950707

Abstract

In this study, we define (1,3)-Bertrand-direction curve and (1,3)-Bertrand-donor curve in the 4-dimensional Euclidean space $E^{4}$. We introduce necessary and sufficient conditions for a special Frenet curve to have a (1,3)-Bertrand-direction curve. We introduce the relations between Frenet vectors and curvatures of these direction curves. Furthermore, we investigate whether (1,3)-evolute-donor curves in $E^{4}$ exist and show that there is no (1,3)-evolute-donor curve in $E^{4}$ .

Keywords

Direction curve, (1,3)-Bertrand-direction curve, (0,2)-involute-direction curve

References

• Bertrand, J., Memoire sur la theorie des courbes a double courbure, Comptes Rendus 36, Journal de Mathematiques Pures et Appliquees., 15 (1850), 332-350.
• Choi, J.H., Kim, Y.H., Associated curves of a Frenet curve and their applications, Applied Mathematics and Computation, 218 (2012), 9116-9124. https://doi.org/10.1016/j.amc.2012.02.064
• Fuchs, D., Evolutes and involutes of spatial curves, American Mathematical Monthly, 120(3) (2013), 217-231. https://doi.org/10.4169/amer.math.monthly.120.03.217
• Fukunaga, T., Takahashi, M., Evolutes and involutes of frontals in the euclidean plane, Demonstratio Mathematica, 48(2) (2015), 147-166. https://doi.org/10.1515/dema-2015-0015
• Fukunaga, T., Takahashi, M., Involutes of fronts in the Euclidean plane, Beitrage zur Algebra und Geometrie/Contributions to Algebra and Geometry, 57(3) (2016), 637-653. https://doi.org/10.1007/s13366-015-0275-1
• Gere, B.H., Zupnik, D., On the construction of curves of constant width, Studies in Applied Mathematics, 22(1-4) (1943), 31-36.
• Hanif, M., Hou, Z.H., Generalized involute and evolute curve-couple in Euclidean space, Int. J. Open Problems Compt. Math., 11(2) (2018), 28-39.
• Huygens, C., Horologium oscillatorium sive de motu pendulorum ad horologia aptato, Demonstrationes Geometricae, 1673.
• Li, Y., Sun, G.Y., Evolutes of fronts in the Minkowski Plane, Mathematical Methods in the Applied Science, 42(16) 2018, 5416-5426. https://doi.org/10.1002/mma.5402
• Macit, N., Düldül, M., Some new associated curves of a Frenet curve in $E^{3}$ and $E^{4}$, Turk J Math., 38 (2014), 1023-1037. https://doi.org/10.3906/mat-1401-85
• Matsuda, H., Yorozu, S., On generalized Mannheim curves in Euclidean 4-space, Nihonkai Math. J., 20 (2009), 33-56.
• Matsuda, H., Yorozu, S., Notes on Bertrand curves, Yokohama Mathematical Journal, 50 (2003), 41-58.
• Nutbourne, A.W., Martin, R.R., Differential Geometry Applied to Design of Curves and Surfaces, Ellis Horwood, Chichester, UK, 1988.
• Önder, M., Construction of curve pairs and their applications, Natl. Acad. Sci., India, Sect. A Phys. Sci., 91(1) 2021, 21-28. https://doi.org/10.1007/s40010-019-00643-2
• Öztürk, G., Arslan, K., Bulca, B., A Characterization of involutes and evolutes of a given curve in En. Kyungpook Math. J., 58 (2018), 117-135.
• Özyılmaz, E., Yılmaz, S., Involute-evolute curve couples in the Euclidean 4-space, Int. J. Open Problems Compt. Math., 2(2) (2009), 168-174.
• Struik, D.J., Lectures on Classical Differential Geometry, 2nd ed. Addison Wesley, Dover, 1988.
• Yu, H., Pei, D., Cui, X., Evolutes of fronts on Euclidean 2-sphere, J. Nonlinear Sci. Appl., 8 (2015), 678-686. http://dx.doi.org/10.22436/jnsa.008.05.20