Affine configurations of 4 lines in \(\mathbb R^3\)
If \(\ell_1,\dots,\ell_4\) are four lines in \(3\)-dimensional space \(\mathbb{R}^3\), then their affine configuration is their equivalence class under the natural action of the affine group \(\text{ Aff}(3)\). Their directions are in general position if the four corresponding points at infinity are in general position in the projective plane. They fix \(\mathbb{R}^3\) if the only affine isomorphism which fixes them as sets is the identity; when their directions are in general position, this happens just when they are not concurrent. This paper describes the space \(\mathbb{A}_{4,1}^3\) of four lines with the latter property. It turns out that, topologically, it is the \(4\)-dimensional projective space \(\mathbb{P}^4\). Furthermore, it has the polyhedral structure of the space \(\mathbb{A}_{6,0}^4\) of affine configurations of \(6\) points in \(\mathbb{R}^4\).
scientific article; zbMATH DE number 5014000