CURVES ON RULED SURFACES UNDER INFINITESIMAL BEND- ING

Infinitesimal bending of curves lying with a given precision on ruled surfaces in 3-dimensional Euclidean space is studied. In particular, the bending of curves on the cylinder, the hyperbolic paraboloid and the helicoid are considered and appropriate bending fields are found. Some examples are graphically presented.


INTRODUCTION
Infinitesimal bending is a kind of deformations of geometric objects under which the arc length is stationary with appropriate precision which is described by the following equation It means that the difference of the squares of the line elements of deformed and initial object is an infinitesimal of the order higher then the first with respect to the infinitesimal parameter . Many other geometric magnitudes stay invariant in the sense that they don't get the variations of the first order (for example, the coefficients of the first fundamental form, Cristoffel's symbols, Gaussian curvature etc.). Many papers are devoted to the infinitesimal bending of curves, surfaces and manifolds (see (Aleksandrov, 1936;Efimov, 1948;Kon-Fossen, 1959;Vekua, 1959;Ivanova-Karatopraklieva & Sabitov, 1995;Velimirović, 2001a,b;Hinterleitner et al., 2008;Rančić et al., 2009;Alexandrov, 2010;Najdanović, 2015;Najdanović & Velimirović, 2017;Kauffman et al., 2019;Najdanović et al., 2019;Rančić et al., 2019;Rýparová & Mikeš, 2019;Belova et al., 2021;Maksimović et al., 2021).
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. Corresponding infinitesimal bending field under whose effect all bent curves remain on the same ruled surface was obtained.
The connection between ruled surfaces and infinitesimal bending of curves is also considered in (Gözütok et al., 2020). Some interesting papers on ruled surfaces are (Li & Pei, 2016;Li et al., 2021).
In this paper we observe a curve on a ruled surface and set the condition that all bent curves remain on the initial surface with a given precision. More precisely, let be the curve on the surface S given by the implicit equation is an infinitesimal bending of the curve C and we get C for = 0. The problem we pose is to determinate an infinitesimal bending of C so that all bent curves C are on the surface S with a given precision, ie. so that the following condition is valid: (1) Below we are going to consider infinitesimal bending of curves on a cylinder, on a hyperbolic paraboloid and on a helicoid. Some examples are graphically presented using program packet Mathematica.

INFINITESIMAL BENDING OF CURVES
At the beginning we are giving basic definitions and theorems regarding infinitesimal bending of curves according to (Efimov, 1948;Vekua, 1959;Velimirović, 2001a).
Let a regular curve C be given in the vector form included in the family of the curves Corresponding author: marija.najdanovic@pr.ac.rs

MATHEMATICS, COMPUTER SCIENCE AND MECHANICS
where > 0, → 0 is an infinitesimal parameter and we get C for = 0 (C = C 0 ).
Definition 1. A family of curves C given by (3) is called infinitesimal deformation of the curve C given by (2). The field z = z(t) , z ∈ C 1 , is infinitesimal deformation field of C.
Definition 2. An infinitesimal deformation C given by (3) is called infinitesimal bending of the curve C given by (2) if where the field z = z(t) , z ∈ C 1 , is infinitesimal bending field of C.
Theorem 3. (Efimov, 1948) Necessary and sufficient condition for z(t) to be an infinitesimal bending field of the curve C is to be where · stands for the scalar product in R 3 .
Definition 4. An infinitesimal bending field is trivial if it can be given in the form where a and b are constant vectors.
According to Vekua (1959) we have the next theorem.
Theorem 5. Under infinitesimal bending of curves each line element undertakes a nonnegative addition, which is the infinitesimal value of at least the second order with respect to , i. e.
Infinitesimal bending field of a curve C is determined in the following theorem.
Theorem 6. (Velimirović, 2001a) Infinitesimal bending field for the curve C given by (2) is where p(t) and q(t) are arbitrary integrable functions and vectors n 1 (t) and n 2 (t) are respectively unit principal normal and binormal vector fields of the curve C.

INFINITESIMAL BENDING OF CURVES ON CYLINDER
Let be given a cylinder by the implicit equation a > 0, or by the vector parametric equation be the curve on the cylinder S . Suppose that (9) is an infinitesimal bending of C, where z 1 (t), z 2 (t), z 3 (t) are real continuous differentiable functions. In order to stay on the cylinder S with a given precision, it is necessary to apply the condition (1), so we have From the last equation we obtain the condition which allows the bent curves to stay on the cylinder S with a given precision. For cos u(t) 0 we can express z 1 as a function of z 2 : Therefore, we are looking for the infinitesimal bending field in the following form In order for the field (12) to be an infinitesimal bending field, it is necessary that the conditionṙ ·ż = 0 is valid. Sinceṙ(t) = (−a sin u(t)u(t), a cos u(t)u(t),v(t)) andż(t) = (−˙u (t) cos 2 u(t) z 2 (t) − tan u(t)ż 2 (t),ż 2 (t),ż 3 (t)), we obtain This is the relationship between z 2 and z 3 . Let us choose arbitrarily z 3 and solve the linear differential equation by z 2 . The solution is u(t) const, cos u(t) 0 and c is a constant. Ifv(t) = 0 ⇒ v(t) = const, the curve C is a circle on the cylinder. In that case we choose z 3 arbitrarily and determine z 2 from z 2 (t) = ce − u(t) tan u(t) dt .
Based on the previous considerations, the following theorems hold.
Theorem 7. The field z(t) = (z 1 (t), z 2 (t), z 3 (t)) whose components z 1 and z 2 satisfy the condition (10) includes the curve (8) under infinitesimal deformation into the family of deformed curves on the cylinder (6) with a given precision.
Theorem 8. The field z(t) given by (12) where z 3 (t) is arbitrary real continuous differentiable function, and z 2 (t) is given in (14), is infinitesimal bending field of the curve (8) so that all bent curves are on the cylinder (6) with a given precision.
Example 9. Let be u(t) = t, v(t) = 0. Then the curve C is a circle r(t) = (a cos t, a sin t, 0). We have z 2 (t) = ce − tan t dt =c cos t, z 1 (t) = −c tan t cos t = −c sin t, z 3 -arbitrarily, c,c are constants. So, the infinitesimal bending field is z(t) = (−c sin t,c cos t, z 3 (t)). By a simple check, we conclude that the conditionsṙ ·ż = 0 and (a cos t − c sin t) 2 + (a sin t + c cos t) 2 − a 2 = 2c2 = o( ) are satisfied. An illustration of the infinitesimal bending is shown in Figures 1 and 2.

INFINITESIMAL BENDING OF CURVES ON HYPER-BOLIC PARABOLOID
Let the curve C : r = r(t) lie on the hyperbolic paraboloid with the vector parametric equation u, v, uv).
If we arbitrarily choose z 1 , we obtain the function z 2 by solving the linear differential equation (20), and z 3 from Eq. (18). The equation (20) reduces tȯ ,v + (uv) . u 0 whose solution is c is a constant.
In this way we have proved the following theorems.
Theorem 10. The field z(t) = (z 1 (t), z 2 (t), z 3 (t)) whose components z 1 , z 2 and z 3 satisfy the condition (17) includes the curve (16) under infinitesimal deformation into the family of deformed curves on the hyperbolic paraboloid (15) with a given precision.
Theorem 11. The field z(t) given by (19) where z 1 (t) is arbitrary real continuous differentiable function, and z 2 (t) is given in (21), is infinitesimal bending field of the curve (16) so that all bent curves are on the hyperbolic paraboloid (15) with a given precision.
For simplicity, let us consider a helix which is obtained for u = 1 and v = t. Therefore, the vector parametric equation of a helix is r(t) = (cos t, sin t, ct).