Mannheim Offsets of Ruled Surfaces

Abstract

In a recent works Liu and Wang (2008; 2007) study the Mannheim partner curves in the three dimensional space. In this paper, we extend the theory of the Mannheim curves to ruled surfaces and define two ruled surfaces which are offset in the sense of Mannheim. It is shown that, every developable ruled surface have a Mannheim offset if and only if an equation should be satisfied between the geodesic curvature and the arc-length of spherical indicatrix of it. Moreover, we obtain that the Mannheim offset of developable ruled surface is constant distance from it. Finally, examples are also given.

One-Parameter Equations of Spherical Conics and Its Applications

“…Many building materials can be considered as straightlines (e.g. woodstick) and can be benefited for creating ruled surfaces in architectural practices (Orbay, Kasap, & Aydemir, 2009;Güngör & Tosun, 2010). …”

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

“…For example, the building materials such as wood are straight and they can be considered as straight lines. The result is that if engineers are planning to construct something with curvature, they can use a ruled surface since all the lines are straight [9].…”

“…Then, Orbay and et. al have given a generalization of the theory of Mannheim partner curves to the ruled surface and they have called this surface offset as Mannheim offset and shown that every developable ruled surface have a Mannheim offset if and only if an equation should be satisfied between the geodesic curvature and the arc-length of spherical indicatrix of the reference surface [9]. The corresponding characterizations of Mannheim offsets of ruled surfaces in Minkowski 3-space have been given in [10,14].…”

Properties of Integral Invariants of the Involute-Evolute Offsets of Ruled Surfaces

“…Using line geometry, it is shown that a theory similar to that of involute-evolute curves have been developed for a ruled surface. Mannheim offsets of ruled surface with respect to the geodesic Frenet frame is defined in [9].…”