Efficient Representation and Derivation of Fundamental Transformation of Relationships using Euler Angles and Quaternions

This paper introduces and defines two principal rotational methods;the Euler angles and the quaternions theories with a brief insight into their definitions and algebraic properties. These methods are widely used in various scientific fields, only marginally in the aircraft industry, the robotics, the quantum mechanics, the electro mechanics, the cameras systems, the computer graphics, the heavy industry and other. The main part of this paper is devoted to the derivation of basic equations of the vector rotation around each rotational x, y, z axis using both rotational methods. Then, the general three-dimensional rotation matrix and the general operator of the quaternion rotation are derived. Finally the utilization of the matrices and quaternion equations are demonstrated on a simple example.