Projectivities and Collineations, a Constructive Approach

A projectivity within a line pencil \(L(x)\) is a linear mapping \(\alpha :x\mapsto x{^\prime}\) and defined by three pairs \((u,u{^\prime}), (v,v{^\prime}), (w,w{^\prime})\in L\times L\) . If we consider the Euclidean plane as plane of action, one can ask for orthogonal pairs \((r,r{^\prime})\in L\times L\) . It turns out that there exist, in algebraic sense, two such pairs, and they define an involutoric projectivity \(\iota \) connected to \(\alpha \) , and they may coincide. If \(\alpha \) is not parabolic, its (real or conjugate imaginary) fixed lines define a second involutoric projectivity \(\omega .\) For a collineation \(\kappa{:}x\mapsto x{^\prime}\) within a bundle \(L(x)\) of lines one can ask a similar question: Find orthogonal pairs \((r,r{^\prime})\in L\times L\) , and among the set \(\{\left(r,r{^\prime}\right)\}\) of such pairs a pair \((\) ( \({r}_{1}\) , \({r}_{1}{^\prime}\) ) \(,\left({r}_{2},{r}_{2}{^\prime}\right))\) such that these lines are edge lines of a regular octahedron. The presentation provides a revival of a constructive approach to this topic and it has a didactical aspect, too.