Some special surfaces constructed by quaternions and their characterizations

This study, prepared as a doctoral thesis, consists of four main chapters. The first chapter is the introduction, providing a general framework by addressing the purpose, scope, and place of the study in the literature. The second chapter comprehensively explains the definitions and theorems that will serve as the foundation for the subsequent parts of the thesis. The third and fourth chapters contain the original parts of the work. In the third chapter, parametrizations of twisted surfaces obtained using different base curves such as planar conchoid curves, involute curves, and Bertrand curve pairs are presented, and various geometric properties of these surfaces are analyzed. The fourth chapter deals with special surfaces defined using quaternions in Euclidean and Minkowski spaces. First, canal and tube surfaces defined by quaternions in Euclidean space are examined; then, canal and tube surfaces are obtained and analyzed in Minkowski space through split quaternions. The parametrizations of these surfaces are expressed using the q-frame, and different geometric properties are revealed depending on whether the base curve is spacelike or timelike. Additionally, the geometry of twisted and conchoidal twisted surfaces generated by quaternions is discussed; it is shown that these structures can be diversified with different types of quaternions through twisted surfaces defined by elliptic quaternions in Euclidean space.