In this paper, we analyze a certain class of timelike ruled surface in one-parameter hyperbolic dual spherical motions by means of the E. Study map. Then, some new formulae of surfaces theory into Minkowski line space and their geometrical explanations are derived. In addition to that, timelike Pl¨ucker conoid associated with the motion has been obtained and investigated in detail. Finally, we have obtained the necessary and sufficient condition for a timelike ruled surface to be a constant timelike Disteli-axis.

The developable surface is a surface that can be unfolded on a plane without tearing or stretching, which is widely used in many fields of engineering and manufacturing. This work presents a new version of developable ruled surfaces in Euclidean 3-space. First, we establish an adapted frame along a spatial curve, denoted by the quasi-frame. We then introduce a parametric representation of a developable ruled surface and call it a directional developable ruled surface. At the core of this paper, we investigate the existence and uniqueness of such developable surfaces, then study their classification by singularity theory and unfolding method. Some examples are given in the final.

The main aim of this paper is the use of the rotation minimizing frame for the singularity type classification of timelike sweeping surface in Minkowski 3–Space $\mathbb{E}_{1}^{3}$. Then, we give conditions for a timelike sweeping surface to be developable ruled surfaces. To apply and illustrate the main results, some examples are given.%

In this study, we construct ruled surfaces whose Disteli-axis is constant in dual 3-space D^3_1 . Then we attain a general system characterizing these surfaces, and also give characterizations for a ruled surface to get a constant Disteli-axis. As a result, the ruled surface formed by a line undergoing a screw motion is investigated.