Collinear transformation of imaginary ray pairs

Abstract

Involutory-collinear transformations are among the most elementary operations of new geometry. But this always means the transformation of real geometric images. The problem of transforming imaginary images, for example, imaginary circles, as it seems, has not been posed and seems incomprehensible. When studying a system of pairs of rays, this problem was presented in all its reality in the following form. If two pairs of rays are given, we take them to be two pairs of tangent parabolas, which is easy to construct, and thus we find a linear primе of pairs of rays whose centers form a straight line, and the pairs of rays themselves are pairs of tangents to the parabola.