1-parameter planar motions in affine Cayley-Klein plane

This thesis consists of ten fundamental chapters and two appendix chapters. First chapter is the introduction chapter includes literature review, objective of the thesis and the original contribution. In the second chapter, the basic concepts required for the whole thesis are given. In the third, fourth and fifth chapters, 1-parameter planar motions in Euclidean, Lorentzian and Galilean planes are expressed with respect to the time parameter , respectively. Furthermore, the derivative formulae, velocities, pole points and accelerations of these motions are discussed. In the sixth chapter, the nine Cayley-Klein plane geometry defined by the concepts "measure of length between two points on a straight line" and "measure of angle between between two lines" is presented in a detailed manner. These concepts are elliptic, parabolic, hyperbolic angle and distance measurements and the classification of these nine plane geometries as Cayley-Klein sense based on I. M. Yagloms' book: "A simple non-Euclidean Geometry and Its Physical Basis" [16] is given. Then, the concept of affine Cayley-Klein planes and some linear algebra and differential geometry concepts of these planes are clarified. These planes are called in general. The original parts of this thesis are composed of the seventh chapter and the following fundamental chapters. In the seventh chapter, 1-parameter planar motion in Affine Cayley-Klein plane are examined and the derivative formulae, velocities-the composition of velocities, pole points, accelerations-the composition of accelerations and acceleration poles are obtained with respect to this movement. In the eighth chapter, the moving coordinate system under the 1-parameter planar motions in Affine Cayley-Klein planes is discussed. In the nineth section, the canonical relative system is given with the aim of the notions of 1-parameter planar motions and moving coordinate system given in seventh and eighth chapters. By the help of this canonical relative system, the Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, is calculated. In the tenth chapter, which is the last chapter of thesis, the original obtained results are mentioned and some suggestions are made for future studies. In the appendix parts of thesis, the basic concepts of Lorentzian and Galilean planes needed to whole thesis are given.