Affine spinor decomposition in three-dimensional affine geometry
By the Theorem of Mozzi-Chasles, a rigid body displacement \(\varrho \in \operatorname{SE}(3)\) is a helical displacement and it is well known that it can be decomposed into two reflections in straight lines. This fundemantal result has been generalized to projective three-space in [\textit{H. Li}, ``Three-dimensional projective geometry with geometric algebra'', Preprint, \url{arXiv:1507:06634}]: A projective transformation \(\pi\) of positive determinant is the product of at most three harmonic projective line reflections. Here, a harmonic projective line reflection is a map defined by two skew lines \(\ell\), \(\ell'\). It maps the point \(p \in \mathbb{P}^3\) to its harmonic conjugate \(p'\) with respect to the two intersection points \(t \cap \ell\), \(t \cap \ell'\) of the unique transversal line \(t\) of \(\ell\) and \(\ell'\) through~\(p\). This article investigates the possibility of similar decompositions in affine geometry. For this purpose, consider the map defined by the skew lines \(\ell\), \(\ell'\) and a real number \(\mu\) that maps a point \(p\) to the point \(p'\) of cross-ratio \(\mu\) with respect to \(t \cap \ell\), \(t \cap \ell'\), and \(p\) where, again, \(t\) is the transversal of \(\ell\) and \(\ell'\) through \(p\). It is called a ``general affine line reflection'' if either \(\ell\) or \(\ell'\) is a line at infinity. This is an affine map but, in contrast to Euclidean or projective line reflections, it is not involutory. The main result states that any affine transformation of positive determinant is the composition of at most three general affine line reflections. In a spin group representation of the affine transformation group it corresponds to the factorization of an affine spinor into the product of at most three Minkowski bispinors. Here, ``bispinor'' refers to algebra elements that are products of two invertible vectors and ``Minkowski'' essentially ensures reality of the two lines \(\ell\), \(\ell'\). The proof is based on a matrix parametrization of the spin group and a discussion of its affine Jordan forms.
scientific article; zbMATH DE number 7600108
scientific article