Projective Bivector Parametrization of Isometries in Low Dimensions

Abstract

The paper provides a pedagogical study on vectorial parameterizations first proposed by O. Rodrigues for the rotation group in $\mathbb{R}^3$ by means of the so-called Rodrigue’s vector. Although his technique yields significant advantages in both theoretical and applied context, the vectorial interpretation is easily seen to be completely wrong and in order to benefit most from this otherwise fruitful approach, we put it in the proper perspective, namely, that of Clifford’s geometric algebras, spin groups and projective geometry. This allows for a natural generalization and straightforward implementations in various physical models, some of which are pointed out below in the text.