Hybrid numbers and their some geometric applications

In this thesis, some geometric applications of hybrid numbers are investigated. In the first chapter, a brief information about the history and properties of numbers from complex numbers to quaternions and the outline of the thesis are given. In the second chapter, the general properties of complex, hyperbolic and dual numbers, which are 2-dimensional number systems, and the properties of hybrid numbers are given. Geometric interpretations of quaternions and split quaternions, which are 4-dimensional number systems, and rotation transformation matrices in the space E³ and E₁³ are given. Also in this section, the inversion of a point with respect to a conic and the Möbius transform are introduced. In the last section, the geometry of hybrid numbers is introduced. 3D hybrid space, vector product and plane definitions are given. Elliptic, hyperbolic and parabolic rotation transformations are studied. In addition, rotation matrices are obtained with the help of Rodrigues formula and Cayley transform. With the help of hybrid numbers, the inversion of a point with respect to a conic and the inversion of a conic with respect to a conic were examined. Finally, the determinant of 2×2 matrices with hybrid number elements and the Möbius transform on hybrid numbers are examined.