Abstract:In this paper, Frenet vector fields, curvature and torsion of the natural lift curve of a given curve is calculated by using the angle between Darboux vector field and the binormal vector field of the given curve in . Also, a similar calculation is made in considering timelike or spacelike Darboux vector field.

In this paper, we study the spherical indicatrices of involutes of a spacelike curve with spacelike binormal. Then we give some important relationships between arc lengths and geodesic curvatures of the spherical indicatrices of involute-evolute curve couple in Minkowski 3-space. Also, we give some important results about curve couple.

We construct a surface family possessing an involute of a given curve as an asymptotic curve. We express necessary and sufficient conditions for that curve with the above property. We also present natural results for such ruled surfaces. Finally, we illustrate the method with some examples, e.g. circles and helices as given curves.

Abstract. The purpose of this paper is to describe ruled surfaces generated by a Frenet trihedron of closed dual involute for a given dual curve. We identify relations between the pitch, the angle of the pitch, and the drall of these surfaces. Some new results related to the developability of these surfaces are also obtained. Finally, we illustrate these surfaces by presenting one example.

This paper concentrates on the requirements of being an integral curve for the geodesic spray of the natural lift curves of spherical indicatrices of the timelike-spacelike involute-evolute curve pair in Lorentz 3-space. In addition, the obtained results were supported by one example.