Kinematic Parameters of a Rigid Body with a Fixed Point

This paper offers a compact general theory of Euler's parameters, Keli-Klein's parameters and quaternions. The theory is based on the isomorphism between groups SO(3), SU(2) and H. This isomorphism is established with the help of specific base of skew-Hermitian matrices which was especially picked out to save the symmetry of bases. By means of this base of skew-Hermitian matrices in this paper the conformity of vector operations, the multiplication of skew-Hermitian matrices and the quaternion algebra is developed. The structural identity of motion transformations in SU(2) and Euler rotation formulas was found out. By means of comparison of objects in those groups simple methods were suggested for deducing kinematic equations and calculation formulas for transformation from one group of parameters to another including angular coordinates also.