Affine spinor decomposition in three-dimensional affine geometry
type: Item
instance of: scholarly article
title: Affine spinor decomposition in three-dimensional affine geometry
zbMATH Open document ID: 1502.14148
author: Chengran Wu
author: Hongbo Li
published in: Acta Mathematica Scientia. Series B. (English Edition)
publication date: 2022-10-12T00:00:00Z
review text: 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 mat…
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 14R20
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 14L17
Mathematics Subject Classification ID: 51M30
zbMATH DE Number: 7600108
zbMATH Keywords: spin group
zbMATH Keywords: spinor decomposition
zbMATH Keywords: affine transformation
zbMATH Keywords: line geometry
zbMATH Keywords: affine line reflection
MaRDI profile type: Publication
full work available at URL: s10473-022-0607-9
OpenAlex ID: W4293560495
Wikidata QID: Q114227617
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DOI: 10.1007/S10473-022-0607-9
description: scientific article; zbMATH DE number 7600108