On the eliptic inversions in l_1 and l_∞-normed planes

This thesis focuses on the inversion with respect to the ellipse, which is a generalization of the inversion with respect to the circle in the taxicab and maximum planes, which are planes equipped with l_1-norm and l_∞-norm, respectively.The elliptic inversion is defined and its properties are studied in this thesis. The thesis consists of eight chapters. In the first chapter, a literature review is conducted to determine the importance of the problem studied in this thesis, its place in current research, and the motivation behind the study. In the second chapter, some important definitions and theorems that are used throughout the thesis are given. In the third chapter, the properties of elliptic inversion in the taxicab plane are discussed and the geometric structures in the plane and their relationship to inversion are explored.. As of the fourth chapter, the original results of the thesis are given. In the fourth chapter, the inverses of the lines under elliptic inversions in the taxicab plane are examined. In the fifth chapter, elliptic inversions in the maximum plane are defined and their basic properties are determined. In the sixth chapter, the inverses of the lines under elliptic inversions in the maximum plane are examined and classified, and in the seventh chapter, the cross ratio and harmonic conjugates under elliptic inversions in the taxicab and maximum planes are examined. Finally, the eighth chapter is devoted to conclusions and suggestions, and in this section, the findings obtained in the thesis are evaluated and suggestions are made for further studies in the light of the thesis's contributions to the inversive geometry literature.