On the kinematic-geometry of one-parameter Lorentzian spatial movement

One-Parameter Hyperbolic Spatial Locomotions and Invariants of the Axode

“…In this section, we give a short synopsis of the dual numbers theory, and dual Lorentzian vectors [11][12][13][14][15]. If a and a * are real numbers, the term a = a + εa * is named a dual number.…”

“…The center and radius of this circle can be specified by the Euler-Savary formula, if the place of the point is specified in the movable plane. In different types of geometry, the Euler-Savary formula had been generalized for a line trajectory, i.e., the construction of the Disteli formulae [4][5][6][7][11][12][13] . Therefore, we now shall look to the Euler-Savary and Disteli formulae for the timelike axodes by utilizing the equipment just acquired above.…”

“…In the Minkowski 3-space E 3 1 , Lorentzian metric can have one of three Lorentzian causal characters; it can be positive, negative, or zero, while the metric in the Euclidean 3-space E 3 can only be positive definite. Therefore, the kinematic and geometric clarifications can be further meaningful in E 3 1 [11][12][13]. This study is as follows.…”

A Study on a Spacelike Line Trajectory in Lorentzian Locomotions

“…An outline of dual numbers theory and the dual Lorentzian vectors is specified in [23][24][25][26][27][28]. If ξ, and ξ * are real numbers, then a dual number can be designed as: ξ = ξ + εξ * , such that ε = 0 and ε 2 = 0.…”

“…Thus, it is necessary to differentiate oriented lines based on the value of distance , determining whether it is positive, negative, or zero. Oriented lines with , < 0 ( , > 0) are named timelike (T like) (spacelike (S like)) oriented lines and oriented lines with , = 0 are named null lines [23][24][25][26].…”