# INFINITESIMAL BENDING OF CURVES ON THE RULED SURFACES

### Abstract

**In this paper we study infinitesimal bending of curves that lie on the ruled surfaces in Euclidean 3-dimensional space.We obtain an infinitesimal bending field under whose effect all bent curves remain on the same ruled surface as the initial curve. Specially, we consider infinitesimal bending of the curves which belong to the cylinder as well as to thehyperbolic paraboloid and find corresponding infinitesimal bending fields. We examine the variation of the curvature of a curve under infinitesimal bending on the hyperbolic paraboloid. Some examples are visualized using program packet**

*Mathematica*.### References

Aleksandrov, A.D. 1936. O beskonechno malyh izgibaniyah neregulyarnyh poverhnostei. Matem. sbornik, 1(43), pp. 307-321.

Alexandrov, V.A. 2010. New manifestations of the Darboux's rotation and translation fields of a surface. New Zealand Journal of Mathematics, 40, pp. 59-65.

Efimov, N. 1948. Kachestvennye voprosy teorii deformacii poverhnostei.UMN. 3. 2, pp. 47-158.

Gray, A. 1998. Modern differential geometry of curves and surfaces with Mathematica.CRC Press. 2nd ed.

Hinterleitner, I., Mikeš, J., & Stránská, J. 2008. Infinitesimal F-planar transformations. Russian Mathematics, 52(4), pp. 13-18. doi:10.3103/s1066369x08040026

Ivanova-Karatopraklieva, I., & Sabitov, I.K. 1995. Bending of surfaces. Part II. Journal of Mathematical Sciences, 74(3), pp. 997-1043. doi:10.1007/bf02362831

Kon-Fossen, S.E. 1959. Nekotorye voprosy differ. geometrii v celom.Moskva: Fizmatgiz.

Najdanović, M.S. 2014. Infinitesimal bending influence on the volume change. Applied Mathematics and Computation, 243, pp. 801-808. doi:10.1016/j.amc.2014.06.032

Najdanovic, M. 2015. Infinitesimal bending influence on the Willmore energy of curves. Filomat, 29(10), pp. 2411-2419. doi:10.2298/fil1510411n

Najdanovic, M.S., & Velimirovic, L.S. 2017. On the Willmore energy of curves under second order infinitesimal bending. Miskolc Mathematical Notes, 17(2), p. 979. doi:10.18514/mmn.2017.2133

Najdanovic, M., & Velimirovic, L. 2017. Second order infinitesimal bending of curves. Filomat, 31(13), pp. 4127-4137. doi:10.2298/fil1713127n

Rancic, S., Velimirovic, L., & Zlatanovic, M. 2009. Curvebend graphical tool for presentation of infinitesimal bending of curves. Filomat, 23(2), pp. 108-116. doi:10.2298/fil0902108r

Vekua, I. 1959. Obobschennye analiticheskie funkcii.Moskva.

Velimirović, L. 2001. Change of geometric magnitudes under infinitesimal bending. Facta universitatis - series: Mechanics, Automatic Control and Robotics, 3(11), pp. 135-148.

Velimirović, L. 2001. Infinitesimal bending of curves. Matematicki bilten Skopje, Makedonija, 25(LI), pp. 25–36.

Velimirović, L., Ćirić, M., & Zlatanović, M. 2010. Bending of spherical curves. In Proceedings of 25th International Scientific Conference MoNGeometrija, SUGIG. pp. 657-667.

Velimirović, L.S. 2009. Infinitesimal bending.University of Niš - Faculty of Sciences and Mathematics.

Yano, K., Takano, K., & Tomonaga, Y. 1946. On the infinitesimal deformations of curves in the spaces with linear connection. Proceedings of the Japan Academy, 22(10), pp. 294-309. doi:10.3792/pja/1195572193

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.