Dual Darboux Frame of a Timelike Ruled Surface and Darboux Approach to Mannheim Offsets of Timelike Ruled Surfaces

TL;DR

This paper introduces a dual Darboux frame for timelike ruled surfaces, explores Mannheim offsets using dual geometry, and establishes conditions for developability, advancing geometric understanding of these surfaces.

Contribution

It presents a novel dual Darboux frame for timelike ruled surfaces and defines Mannheim offsets via dual geometry, providing new relationships and developability conditions.

Findings

Derived relationships between invariants of Mannheim offsets.

Established conditions for developability of surface offsets.

Connected dual Darboux frames with timelike ruled surfaces.

Abstract

In this paper, we introduce the dual geodesic trihedron (dual Darboux frame) of a timelike ruled surface. By the aid of the E. Study Mapping, we consider timelike ruled surfaces as dual hyperbolic spherical curves and define the Mannheim offsets of timelike ruled surfaces by means of dual Darboux frame. We obtain the relationships between invariants of Mannheim timelike surface offsets. Furthermore, we give the conditions for these surface offsets to be developable.