Minimum-Parameter Representations of N-Dimensional Principal Rotations

Abstract

Classic techniques have been established to characterize N × N proper orthogonal matrices using the N-dimensional Euler’s theorem and the Cayley transform. These techniques provide separate descriptions of N-dimensional orientation in terms of the constituent principal rotations or a minimum-parameter representation. The two descriptions can be linked by the canonical form of the extended Rodrigues parameters. This form is developed into a new minimum-parameter representation that directly links to the principal rotations. The new representation is solved using analytic and geometric approaches for N = 3 and N = 4, and numerical solutions are found for N= 5. In fact multiple solutions, which are related geometrically by different coordinatizations of the principal planes, have been found. The new parameters represent a projection of the principal rotations onto the planes formed by the body coordinates.