On the Determination of Ruled and Developable Surfaces in Euclidean 3-space E³

Abstract

Depending on E. Study’s map, the method of determination of ruled and developable surfaces is introduced. We show that dual vectorial expression of ruled and developable surfaces will be gained from coordinates and the first derivatives of the base curve. We illustrated this method by giving some representative examples.

Tangent bundle of pseudo-sphere and Non-null slant ruled surfaces

“…)), (6.5) According to [4], slant ruled surfaces are unique ruled surfaces where the Frenet vectors form a constant angle with a few fixed directions in the space. We can sum up some of the findings in this work by considering the definition of slant ruled surfaces in the literature: 1.…”

“…The helices on a Lorentzian manifold were denoted in [6]. In [14], the authors dealt with timelike curves of a constant slope in E 4 1 . In [10], the authors constructed an entirely novel curve known as the slant helix by using the general helix notion.…”

“…The geometric interpretation of the ruled surfaces is a topic that many authors are interested in. Some of them are as follows: [4] describes how the author used dual vector analysis to investigate the idea of "slant" for ruled surfaces. The author defined the slant ruled surface in E 3 in [18].…”

On the Timelike Sweeping Surfaces and Singularities in Minkowski 3-Space E 1 3

“…One of the noteworthy facts linked with the sweeping surface is that the sweeping surface can be developable surface, that is, can be developed onto a plane without tearing and stretching. Therefore, sweeping surfaces have great usefulness in considerable product design which uses leather, paper, and sheet metal as materials (see, e.g., [5][6][7][8]). The developable surface can be represented using the Serret-Frenet frame of space curves from the viewpoint of singularity theory.…”