Rotation Parameters—a Survey of Techniques

Abstract

A number of parameterizations of rotations in three-dimensional space are compared with respect to their formulas and qualitative characteristics relevant for practical use. The number and uniqueness of parameters, singularities, visualization, and constraint equations within one parameter set are considered. Formulas are given for the transformation to and from rotation matrices and symmetric Euler parameters as well as for the rotation of a vector, the inverse rotation, the combination of two rotations, and the kinematic differential equations. The list of parameters includes the five-parametric Hopf parameters. Symmetric rotation matrices are treated separately. All properties are given in comprehensive tables with formulas ready for programming.