Abstract
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.
Highlights
Differential Geometry of Ruled Surfaces in E3The parametric s-curve of this surface is a straight line of surface which is called ruling
Similar to the Bertrand curves, in [8] a new definition of special curve pairs has been given by Liu and Wang: Let C and C∗ be two space curves
An oriented line fixed in the moving system generates a closed ruled surface if the whole moving frame comes to its initial position and this surface is called closed trajectory ruled surface (CTRS) in E3 [7]
Summary
The parametric s-curve of this surface is a straight line of surface which is called ruling. A = q×h be a moving othonormal trihedron making a spatial motion along a closed space curve k(s), s ∈ R, in E3. In this motion, an oriented line fixed in the moving system generates a closed ruled surface if the whole moving frame comes to its initial position and this surface is called closed trajectory ruled surface (CTRS) in E3 [7]. The area of projection of a closed space curve x in direction of the generator of a CTRS -y(s, v) is (See [4]).
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