On Generalization of Helices in the Galilean and the Pseudo-Galilean Space

Abstract

In this paper a generalization of helices in the three-dimensional Galilean and the pseudo-Galilean space is proposed. The equiform general helices, which represent a generalization of "ordinary" helices, are defined and characterized. Particularly, all obtained results can be transferred to other Cayley-Klein spaces, including Euclidean.

On the Equiform Differential Geometry of AW(k)-Type Curves in Pseudo-Galilean 3-Space

“…In this section, we recall some basic notions from pseudo-Galilean geometry [11,12]. In the inhomogeneous affine coordinates for points and vectors (point pairs) the similarity group H 8 of G According to the motion group in the pseudo-Galilean space, there are non-isotropic vectors A(A 1 , A 2 , A 3 ) (for which holds A 1 = 0) and four types of isotropic vectors: spacelike (A 1 = 0, A 2 2 − A 2 3 > 0), timelike (A 1 = 0, A 2 2 − A 2 3 < 0) and two types of lightlike vectors (A 1 = 0, A 2 = ±A 3 ).…”

“…A curve γ(s) = (x(s), y(s), z(s)) is called an admissible curve if it has no inflection points (γ ×γ = 0) and no isotropic tangents (ẋ = 0) or normals whose projections on the absolute plane would be lightlike vectors (ẏ = ±ż). An admissible curve in G 1 3 is an analogue of a regular curve in Euclidean space [12]. For an admissible curve γ(s) : I ⊆ R → G 1 3 , the curvature κ(s) and torsion τ (s) are defined by…”

Spacelike and timelike admissible Smarandache curves in pseudo-Galilean space

“…In this section, let us first recall basic notions from pseudo-Galilean geometry [7][8][9][10][11] According to the motion group in the pseudo-Galilean space, there are non-isotropic vectors A(A 1 , A 2 , A 3 ) (for which holds A 1 = 0) and four types of isotropic vectors: spacelike (A 1 = 0,…”

Galilean geometry of corresponding surfaces to production models in economics

“…For later use, we provide a brief review of Galilean geometry from [7,18,19,21,25,[27][28][29][30]37].…”