In this article, firstly, the isomorphism between the subset of the tangent bundle of Lorentzian unit sphere, TM, and Lorentzian unit sphere, S12 is represented. Secondly, the isomorphism between the subset of hyperbolic unit sphere, TM, and hyperbolic unit sphere, H2 is given. According to E. Study mapping, any curve on S12 or H2 corresponds to a ruled surface in R13. By constructing these isomorphisms, we correspond to any natural lift curve on TM or TM a unique ruled surface in R13. Then we calculate striction curve, shape operator, Gaussian curvature and mean curvature of these ruled surfaces. We give developability condition of these ruled surfaces. Finally, we give examples to support the main results.

Highlights• A unique ruled surface is corresponded to the natural lift curve.• Properties of ruled surfaces generated by natural lift curves are examined. • A method is given for modelling motions on ̅ instead of 2 .

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It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the 4×4 matrices associated with hybrid numbers by using trigonometric identities. Also, we give the roots of the matrices of hybrid numbers. Moreover, we give some illustrative examples to support the main formulas.

This paper aims to present, in a unified manner, results which are valid on both split quaternions with quaternion coefficients and quaternions with dual coefficients, simultaneously, calling the attention to the main differences between these two quaternions. Taking into account some results obtained by Karaca, E. et al., 2020, each of these quaternions is studied and some important differences are remarked on.

In this study, the ruled surface generated by the natural lift curve in IR^3 is obtained by using the isomorphism between unit dual sphere, DS^2 and the subset of the tangent bundle of unit 2-sphere, T\bar{M}. Then, exploitting E. Study mapping and the isomorphism mentioned below, each natural lift curve on T\bar{M} is corresponded to the ruled surface in IR^3. Moreover, the singularities of this ruled surface are examined according to RM vectors and these ruled surfaces have been classified. Some examples are given to support the main results.

In this study, firstly, each natural lift curve of the main curve is corresponded to the ruled surface by exploitting E. Study mapping and the relation among the subset of the tangent bundle of unit 2-sphere, T\bar{M} and ruled surfaces in IR^{3}. Secondly, the intersection of two ruled surfaces, which are obtained by using the relation given above, is examined for the condition of the zero-set of λ(u, v) = 0. Then, all redundant and non-redundant solutions of the zero-set are investigated. Furthermore, the degenerate situations (u, v) = 0, where the whole plane is degenerated by the zero-set,, are denoted. Finally, some examples are given to verify the results.

In this paper, an isomorphism between unit dual sphere, DS2, and the subset of tangent bundle of unit 2-sphere, T?M, is represented. According to E. Study mapping, a ruled surface in IR3 corresponds to each curve on DS2. Through this isomorphism, new forms of ruled surfaces called slant ruled surfaces in IR3 were introduced. Moreover, conditions for these surfaces to be slant ruled surfaces were given. Finally, a unique ?q?,?h? and ?? slant ruled surfaces in IR3 were corresponded to each striction curve of natural lift curve on T?M.