On the Representation of Rigid Rotations

Abstract

This expository article gives a straight-forward derivation of the classical representation formulas of the three-dimensional rotation group in explicit (non-symbolical) form starting with the most elementary representation of a rotation by a linear homogeneous transformation with a proper orthogonal matrix. Apart from Euler's formula (3.4) (which is taken as starting point in Benedikt's previous paper on the same subject) other representations often used in mechanics and physics are derived and the various connections existing between these formulas are established hereby going beyond Benedikt's paper by taking into account the simplest spin representations of the rotation group. The spin representation by the special unitary group is derived in two ways; the second method (Section 6) is based on a general fact (in italics) which seems not to be generally known. In the footnotes some references to the literature are given.