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

TL;DR

This paper introduces a dual Darboux frame for spacelike ruled surfaces and explores Mannheim offsets in dual Lorentzian space, establishing relationships between invariants and conditions for developability.

Contribution

It defines a dual Darboux frame for spacelike ruled surfaces and analyzes Mannheim offsets using dual Lorentzian geometry, providing new geometric relationships and developability conditions.

Findings

Derived relationships between invariants of Mannheim offsets.

Established conditions for the developability of offset surfaces.

Represented spacelike ruled surfaces via dual Lorentzian unit spherical curves.

Abstract

In this paper, we define dual geodesic trihedron(dual Darboux frame) of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim offsets of the spacelike ruled surfaces by means of dual Darboux frame. We obtain relationships between the invariants of Mannheim spacelike offset surfaces and offset angle, offset distance. Furthermore, we give conditions for these surface offsets to be developable.