The theory of axial collineations as an extension of Steiner's theory of conoprimas (Kegelsсhnittbüschel)
Abstract
According to Steiner's famous theory, two given involutions of pairs of points on lines in a plane determine an involution on any line in the plane, that is, the complete secund of involution. The determining factor of all these involutions is a linear prima of curves, namely a pencil of conics (Kegelschnittbüschel, according to Steiner), having two pairs of points in common, of which not only one, but both pairs can be imaginary. Each line intersects each curve of the pencil in a pair of points belonging to its involution. Specifically, we can define a collineation by two axes without any involutions. If we call the axes, whose points are the real double points of all involutions, real axes, and the axes of isotropic involutions imaginary axes, then we obtain that every axial collineation can be defined by a pair of axes, real or imaginary (see the article).
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