Inversions with respect to circles in the generalized taxicab plane

In the thesis study, the inversion with respect to generalized taxicab circles was defined in the generalized taxicab plane. The images of points, lines, line segments, the pencil of parallel lines, the inversion circle and concentric circles with the inversion circle under this inversion transformation were examined The thesis contains five chapters. In the first chapter, the introduction and purpose of the thesis are given. The second part consists of some studies in the literature on the subject of the thesis. In the third chapter, the basic definitions and concepts in the generalized taxicab plane are mentioned in order to make the thesis more understandable. In the fourth chapter, inversion transformation with respect to generalized taxicab circles in the generalized taxicab plane is defined and its properties are presented. The results obtained by examining the images of the points under inversions with respect to generalized taxicab circle are given. In addition, the properties of their images were determined by analytically examining the inverses of the lines under inversion with respect to the circle. In addition to these, the inverses of the line segments are examined according to the regions where the endpoints are located, and the properties of the inverses are obtained. Then, considering the pencil of parallel line, their inverses were investigated. In addition, at the end of this section, the directed generalized taxicab length of a line segment is defined, and accordingly the generalized taxicab cross ratio and generalized taxicab harmonic conjugate concepts are defined and some properties are given, and the effect of inversion in the generalized taxicab circle on these is investigated. In the fifth chapter, there are conclusions about the thesis and suggestions based on it.