Abstract
In this study, using Darboux frame {T,g,n} of ruled surface ϕ(s,v), the evolute offsets ϕ^{∗}(s,v) with Darboux frame {T^{∗},g^{∗},n^{∗}} of ϕ(s,v) are defined. Characteristic properties of ϕ^{∗}(s,v) as a striction curve, distribution parameter and orthogonal trajectory are investigated using the Darboux frame. The distribution parameters of ruled surfaces ϕ_{T^{∗}}^{∗},ϕ_{g^{∗}}^{∗} and ϕ_{n^{∗}}^{∗} are given. By using Darboux frame of the surfaces we have given the relations between the instantaneous Pfaffian vectors of motions H/H′ and H^{∗}/H^{∗′}, where H={T,g,n} be the moving space along the base curve of ϕ(s,v), H^{∗}={T^{∗},g^{∗},n^{∗}} be the moving space along the base curve of ϕ^{∗}(s,v), H′ and H^{∗′} be fixed Euclidean spaces.
Highlights
Di¤erential geometry of the ruled surface is a important subject of the geometry
An evolute o¤sets of a given curve is Received by the editors: October 19, 2017; Accepted: January 08, 2019. 2010 Mathematics Subject Classi...cation
Bertrand o¤sets of the ruled surfaces with Darboux frame (RSDF) and their properties are studied by Sentürk and Yüce, [8]
Summary
Di¤erential geometry of the ruled surface is a important subject of the geometry. A ruled surface can always be parameterized. The ruled surfaces are very useful in many areas of sciences for instance Computer-Aided Manufacturing (CAM), Computer-Aided Geometric Design (CAGD), geometric modeling and kinematics Another special subject of geometry is di¤erential geometry of the curves. Bertrand o¤sets of the ruled surfaces with Darboux frame (RSDF) and their properties are studied by Sentürk and Yüce, [8]. The involute-evolute curves which lying on the surfaces have studied by Bektas and Yüce by using the Darboux frame of curves They obtained the relations between g; g; n and n for a curve to be the special involute partner D-curves. The evolute o¤sets of ruled surfaces with Darboux frame are de...ned.
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