On the evolute offsets of ruled surfaces using the Darboux frame

Abstract

In this study, using Darboux frame fT; g; ng of ruled surface '(s; v), the evolute o¤sets ' (s; v) with Darboux frame fT ; g ; n g of '(s; v) are de…ned. Characteristic properties of ' (s; v) as a striction curve, distribution parameter and orthogonal trajectory are investigated using the Darboux frame. The distribution parameters of ruled surfaces ' T ; ' g and ' n are given. By using Darboux frame of the surfaces we have given the relations between the instantaneous Pfa¢ an vectors of motions H=H 0 and H =H 0 , where H = fT; g; ng be the moving space along the base curve of '(s; v), H = fT ; g ; n g be the moving space along the base curve of ' (s; v), H 0 and H 0 be …xed Euclidean spaces.

Bertrand Offsets of Ruled Surfaces with Blaschke Frame in Euclidean 3-Space

“…where ω = ω + εω * = fl x + g is the Darboux vector, and γ( s) := γ + εγ * is the dual geodesic curvature of x( s) ∈ K. The tangent of c(s) is specified by [16]:…”

“…Senturk and Yuce have considered the integral invariants of the offsets by the geodesic Frenet frame [13]. Important contributions to the Bertrand offsets of these ruled surfaces have been studied in [14][15][16].…”

Bertrand Offsets of Slant Ruled Surfaces in Euclidean 3-Space

“…M. Onder and O. Kaya in [18] obtained new characterizations for slant RS in the Euclidean 3-space. There exist a considerable number of written works on the topic of comprehensive diverse treatises; for instance, [19][20][21][22][23][24]. Nevertheless, to our knowledge, there is no work related to creating BO of slant RS via the geometrical properties of the striction curve (S C).…”

Smarandache Ruled Surfaces according to Darboux Frame in E 3

“…Besides, they studied characteristic properties of that ruled surface and gave the relationship between Darboux frame and Frenet frame. Moreover, in [11], the authors defined the evolute offsets of ruled surface with Darboux frame and studied its characteristic properties in E 3 .…”