Some appli̇cati̇ons of hyper-dual spli̇t vectors

This thesis consists of 4 chapters. The first chapter includes the introduction part. The aim of the thesis is explained; the history and application areas of dual and hyper dual numbers are discussed. Dual and hyper dual number systems, the Lorentz-Minkowski space, and related fundamental concepts are introduced. In the second chapter, hyper dual split vectors are detailed, and their algebraic structures and geometric interpretations are explained. In the third chapter, by using hyper dual split vectors defined with the help of hyper dual numbers, the Euler-Rodrigues rotation formulas are extended and hyper dual split Euler-Rodrigues equations are introduced. These equations are derived for both spacelike and timelike unit axes, and hyper dual split rotation matrices are obtained. In the final chapter, the relationship between the E.Study transformation and hyper dual split structures is examined, and the classical fixed-point theorem of Euler is reformulated in this new framework. Within the scope of the thesis, the theoretical framework is supported with various examples and geometric interpretations are included.