On the Mannheim Surface Offsets

TL;DR

This paper investigates Mannheim surface offsets in dual space using dual spherical curves and the dual Darboux frame, establishing relationships between invariants and conditions for developability.

Contribution

It introduces a novel approach to analyze Mannheim surface offsets via dual space and dual spherical curves, providing new relationships and developability conditions.

Findings

Dual spherical radius of curvature equals dual offset angle.

Derived relationships between invariants of Mannheim ruled surfaces.

Established conditions for the offsets to be developable.

Abstract

In this paper, we study Mannheim surface offsets in dual space. By the aid of the E. Study Mapping, we consider ruled surfaces as dual unit spherical curves and define the Mannheim offsets of the ruled surfaces by means of dual geodesic trihedron (dual Darboux frame). We obtain the relationships between the invariants of Mannheim ruled surfaces. Furthermore, we give the conditions for these surface offset to be developable. Furthermore, we obtained that the dual spherical radius of curvature of offset surface is equal to dual offset angle.