In this article, we examine the relationship between Darboux frames along parameter curves
and the Darboux frame of the base curve of the ruled surface. We derive the equations of the …

As is well-known, a surface is said to be “ruled” if it is generated by moving a straight line
continuously in Euclidean space E3 (O'Neill, 1997). Ruled surfaces are one of the simplest …

Bertrand curves are a pair of curves that have a common principal normal vector at any point
and are related to symmetry properties. In the present paper, we define the notion of 1, 3-V …

In this paper, a one-to-one correspondence is given between the tangent bundle of unit 2-
sphere, T 𝕊 2, and the unit dual sphere, 𝕊 𝔻 2. According to Study's map, to each curve on 𝕊 …

In this paper, firstly, the ruled surface is expressed as a spatial quaternionic. Also, the spatial
quaternionic definitions of the Striction curve, the distribution parameter, angle of pitch and …

In this paper, we analyzed surfaces family possessing a Mannheim partner of a given curve
as a asymptotic. Using the Frenet frame of the curve in Euclidean 3-space, we express the …

This work examines some classical results of Bertrand curves for timelike ruled and
developable surfaces using the E. Study map. This provides the ability to define two timelike …

In this paper, we introduce the dual geodesic trihedron (dual Darboux frame) of a timelike
ruled surface. By the aid of the E. Study Mapping, we consider timelike ruled surfaces as …

In this study, using Darboux frame\left {T, g, n\right\} T, g, n of ruled surface φ (s, v) φ (s, v),
Bertrand offset φ^*(s, v) φ∗(s, v) of ruled surface φ (s, v) φ (s, v) is identified. Characteristic …

In this paper, we investigate and specify the Bertrand offsets of slant ruled and developable
surfaces in Euclidean 3-space E 3. This is accomplished by utilizing the symmetry of slant …