Nurten GRSES, Mcahit AKBIYIK, Salim YCE


Kinematics is the branch of classical mechanics which describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without consideration of the causes of motion. It is often referred to as the geometry of pure motion-motion considered abstractly, without reference to force or mass. Obviously, kinematics studies the trajectories of points, lines and other geometric objects and their differential properties such as velocity and acceleration. Furthermore, it has a lot applications in astrophysics, mechanical engineering, robotics and biomechanics. In mathematics, kinematics is a fundamental part of geometric thinking and concepts of motion. The study of kinematics can be abstracted into purely mathematical functions. For instance, rotation can be represented by elements of the unit circle in the complex plane. Other planar algebras are used to represent the shear mapping of classical motion in absolute time and space and to represent the Lorentz transformations of relativistic space and time. Therefore, non-Euclidean geometries are appeared. In 1956, W. Blaschke and H.R. Mller introduced the one-parameter planar motions and obtained the relation between absolute, relative, sliding velocities and accelerations in the Euclidean plane \[{\mathbb{E}^2}\]. In 1983, the kinematics in the isotropic plane is studied by O. Rschel. In his work, the fundamental properties of the point-paths are investigated, and special motions: an isotropic elliptic motion and an isotropic four-bar-motion are studied. Also, one-parameter motions on Lorentzian plane \[{\mathbb{L}^2}\] and Galilean plane \[{\mathbb{G}^2}\] are introduced by A. A. Ergin and S. Yce, respectively. They also gave the relations between the velocities and accelerations. Moreover, one-parameter planar motions are given in affine Cayley-Klein planes (CK-planes) by generalizing of the motions in Euclidean, Galilean and Lorentzian planes. On the other hand, the ordinary (complex),, dual and double (perplex, split-complex or hyperbolic) numbers are significant members of a two-parameter family of complex number systems often called binary numbers or generalized complex numbers. In the literature; ordinary, dual and double numbers are usually denoted by different imaginary units, “\[i\] ”,“ \[\varepsilon \]” and “\[j\] ”, respectively. For generalizing this unit, we take \[J\] for three number systems. So the generalized complex numbers have the form \[z = x + Jy,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} (x,y \in \mathbb{R}){\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\text{where}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {J^2} = i\mathfrak{q} + \mathfrak{p},{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} (\mathfrak{p},\mathfrak{q} \in \mathbb{R}).\,\]. By taking \[{J^2} = \mathfrak{p};\mathfrak{q} = 0\] and \[ - \infty < \mathfrak{p} < \infty \], generalized complex number system can be presented as follows: \[{\mathbb{C}_\mathfrak{p}} = \{ x + Jy:{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} x,y \in \mathbb{R},{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {J^2} = \mathfrak{p}\} \]. \[{\mathbb{C}_\mathfrak{p}}\] is called generalized complex number plane ( \[\mathfrak{p}\]-complex plane). Moreover, the set \[{\mathbb{C}_J}\] is defined \[{\mathbb{C}_J} = \left\{ {x + Jy:{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} x,y \in \mathbb{R},{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {J^2} = \mathfrak{p},{\mkern 1mu} {\mkern 1mu} \mathfrak{p} \in \{ - 1,0,1\} } \right\}\] such that \[{\mathbb{C}_J} \subset {\mathbb{C}_\mathfrak{p}}\]. For \[\mathfrak{p} < 0\], \[{\mathbb{C}_\mathfrak{p}}\] is called elliptical complex, for \[\mathfrak{p} = 0\] , \[{\mathbb{C}_p}\] is called parabolic complex, and for \[\mathfrak{p} > 0\], \[{\mathbb{C}_\mathfrak{p}}\] is called hyperbolic complex number systems. In this study, we firstly give the basic notations of the \[\mathfrak{p}\]-complex plane \[{\mathbb{C}_\mathfrak{p}}\]. By using these concepts, we introduce one-parameter homothetic motions in \[\mathfrak{p}\]-complex plane \[{\mathbb{C}_J}\] such that \[{\mathbb{C}_J} \subset {\mathbb{C}_\mathfrak{p}}\]. Besides, we discuss the relations between absolute, relative, sliding velocities (also the relations between absolute, relative, sliding and corolois accelerations) and pole points under the homothetic motions denoted by \[{\mathbb{C}_J}/{\mathbb{C}'_J}\].

Anahtar Kelimeler: Generalized Complex Number Plane, One-Parameter Planar Homothetic Motion, Kinematics