Highlights• This paper focuses on the theories of dual-generalized and hyperbolic-generalized complex numbers.• The algebraic structures of dual-generalized and hyperbolic-generalized complex numbers are given.• Dual-generalized complex and hyperbolic-generalized complex valued functions are defined.• The matrix representations of dual-generalized and hyperbolic-generalized complex numbers are stated.• An efficient classification includes complex-generalized complex numbers is examined.

This paper aims to develop dual-generalized complex Fibonacci and Lucas numbers and obtain recurrence relations. Fibonacci and Lucas's approach to dual-generalized complex numbers contains dual-complex, hyper-dual and dual-hyperbolic situations as special cases and allows general contributions to the literature for all real number p . For this purpose, Binet's formulas along with Tagiuri's, Hornsberger's, D'Ocagne's, Cassini's and Catalan's identities, are calculated for dual-generalized complex Fibonacci and Lucas numbers. Finally, the results are given, and the special cases for this unification are classified.

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In this paper, firstly, our purpose is to give the relationship among the densities of the sets of collinear points, the relationship among the densities of the sets of noncollinear points, and the relationship among the densities of the sets of the intersecting lines in Euclidean plane, respectively. In addition to that, we define the density formulas for the sets of points and lines under the two‐parameter planar Euclidean motion. By means of these results, we obtain essential properties that explain the connection among the densities of the sets of points and among the densities of the sets of intersecting lines under the two‐parameter planar Euclidean motion.

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.