Some Results and Characterizations for Mannheim Offsets of the Ruled Surfaces

In this study, we give the dual characterizations of Mannheim offsets of ruled surfaces in terms of their integral invariants and obtain a new characterization of the Mannheim offsets of developable surface, i.e., we show that the striction lines of developable Mannheim offset surfaces are Mannheim partner curves. Furthermore, we obtain the relationships between the area of projections of spherical images for Mannheim offsets of ruled surfaces and their integral invariants.

On the kinematic interpretation of timelike ruled surfaces

Timelike ruled surfaces are studied in dual Lorentzian space [Formula: see text] by considering E. Study Mapping and Blaschke frame. A reference timelike ruled surface is considered and associated surfaces are defined. First, it is shown that the surface generated by the instantaneous screw axis (ISA) is a Mannheim offset of reference surface. Later, the kinematic interpretations between these surfaces are introduced by means of Blaschke invariants.

Mannheim Offsets of Ruled Surfaces

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.