Burmester theory in Lorentzian plane

This thesis consists of five chapters. The first chapter is the introduction chapter which includes a review of the literature and the scope of the research problem. In the second chapter, the algebraic and geometric properties of the complex numbers and hyperbolic numbers are summarized. In the third chapter, motion in the Euclidean plane are examined in detail. The fourth chapter is the original part of this study and it is organized as three subsections. In the first subsection of the fourth chapter, the basic concepts with Lorentzian plane motion, pole points, special systems of reference, Bottema's instantaneous invariants, curvature of orbits, canonical systems, path of the origin, inverse motion, curvatures of the fixed and the moving polode at the pole and curvature of the second fixed polode at the second pole are investigated. In the second subsection, the circling point curve, centering point curve, Ball points, Ball points of the inverse motion and Ball points with excess are examined in the Lorentzian plane. Moreover, the Lorentzian circles formed in the degenerate cases of the circling point curve and the centering point curves are analyzed and geometric interpretations of these curves are given. In the third subsection, by defining Burmester point in the Lorentzian plane, the equation of the curve which is the geometric locus of the Burmester points is obtained and the real intersection of this curve and circling point curve at infinity is investigated. As the result of this investigation, the number and geometric location of the Burmester points in Lorentz plane are represented. In the fifth chapter of this thesis, a brief summary of the study is given and a suggestion is proposed for further investigations.