Investigation of curves in E^3 and E^4 and surfaces in E^3 via quaternions

The aim of this thesis is to re-formulate the well known Serret-Frenet equations for curves in three and four dimensional Euclidean spaces and Gauss-Weingarten equations for the surfaces in three space, which are analogous to Serret-Frenet equations, in an algebraic sense via quaternions. Basic concepts of the theory of complex numbers and quaternions are given in the first section.In the second section the well known Serret-Frenet equations for both unit and arbitrary speed curves in the complex plane are obtained. Corresponding equations for curves in E^3 and E^4are investigated by using quaternion algebra. In the final section, the first and second fundamental forms of a surface in E^3 and Gauss-Weingarten equations, are represented by quaternion multiplication.