On Timelike and Spacelike Developable Ruled Surfaces

TL;DR

This paper investigates the conditions under which developable ruled surfaces generated by timelike curves are formed, focusing on the role of Frenet frames and the helix property in Lorentzian geometry.

Contribution

It provides new criteria for developability of ruled surfaces generated by timelike curves with different Frenet frames, highlighting the significance of the helix condition.

Findings

Ruled surface is developable iff the base timelike curve is a helix.

Derived the distribution parameter for ruled surfaces in Lorentzian space.

Presented theorems on similarities and differences in developability conditions.

Abstract

In this study, we have obtained the distribution parameter of a ruled surface generated by a straight line in Frenet trihedron moving along a timelike curve and also along another curve with the same parameter. At this time, the Frenet frames of these timelike curves are not the same. We have moved the director vector of the first curve along the second curve. It is shown that the ruled surface is developable if and only if the base timelike curve is helix. In addition, some smilarities and differences are presented with theorems and results.