Geometry in the set of the generalized dual numbers

This thesis investigates the generalized dual numbers, generalized dual quaternions and generalized dual screw motion as an alternative to the standard dual number system. The study begins with a historical overview of dual numbers and explores the generalizations presented in the literature. The generalized dual numbers, introduced here for the first time, are constructed based on the parabolic number framework, inspired by generalizations of complex numbers proposed by mathematicians such as Yaglom and Catoni. In this context, the generalized dual numbers are defined as Dₖ = { z = x + yεₖ : x, y, k ∈ ℝ and (εₖ − k)² = 0, εₖ ∉ ℝ }, and their algebraic structure, associated generalized Galilean plane and geometric interpretation are examined. The polar form and rotation matrix for these numbers are derived, leading to the introduction of a three-dimensional generalized dual module. Additionally, the generalized Study theorem is proved and its implications are discussed. A new algebraic system combining generalized dual numbers with quaternions is introduced under the name generalized dual quaternions. Using this framework, a generalized screw motion is formulated and its mathematical properties are presented.