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 the hyperbolic 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.

Curves on ruled surfaces under infinitesimal bending

“…The corresponding infinitesimal bending fields are obtained and some examples with graphical illustration are considered. This is a continuation of the research published in the paper (Najdanović & Velimirović, 2018).…”

“…In (Najdanović & Velimirović, 2018) the authors studied the infinitesimal bending of curves that lie on ruled surfaces in Euclidean 3-dimensional space. It was proven that it is possible infinitesimally bend such a curve so that all bent curves remain on the same surface as the initial curve.…”

Ruled surfaces obtained by bending of curves

“…The bending of curves and surfaces is related to biology and medicine [12,13]. A few suggestions for further reading about bending of curves and surfaces are [7,8,10,11]. The relation between ruled surfaces and bending of curves has motivated us to conduct this study.…”