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On the geometric generation of linear line mappings. (Zur geometrischen Erzeugung linearer Geradenabbildungen.) (German) Zbl 1114.51007

This paper can be seen as addition to H. Brauner [Monatsh. Math. 77, 10–20 (1973; Zbl 0256.50013)] (see also H. Havlicek [Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 192, 99–111 (1983; Zbl 0527.51027)]). Denote by \(\mathbb P^n\) the \(n\)-dimensional complex extended real projective space and by \(\mathcal L\) the line set of \(\mathbb P^3\). There exist four types of linear mappings \(\mu:\mathcal L\rightarrow\mathbb P^3\) according to the situation of the one-dimensional projection center \(Z\) with respect to the Klein quadric \(H_5\). For each non-singular (i.e., \(Z\not\subset\,H_5\)) linear mapping \(\mu:\mathcal L\rightarrow\mathbb P^3\) the author gives a construction which can be done completely within \(\mathbb P^3\); \(\mu\) is composed of a projection via a linear congruence of lines, a stereographic projection, and a polarity of \(\mathbb P^3\). The author establishes connections between the linear line mappings to \(\mathbb P^3\) and those to \(\mathbb P^2\) and shows how the kinematic mapping of W. Blaschke and J. Grünwald, the sperical kinematic mapping of W. K. Clifford, the line mappings due to L. Eckhart and those due to F. Rehbock can be integrated.
The considerations and computations of the paper are escorted by seven very attractive figures.

MSC:

51M15 Geometric constructions in real or complex geometry
51M30 Line geometries and their generalizations
51N15 Projective analytic geometry
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