Collinear systems in a perspective position, but not involution

Abstract

If you have collineations of two systems, then both systems are equal, because both are in the position of involution, and any point a of the system is collinear to the same point a of the collinear system, regardless of which of these two systems the given point belongs to. But now let’s replace one of the systems with a system similar to it, and take the combined centers of collineation of both systems as the similarity center E. It is clear that under this condition the systems can no longer be brought into the position of involution, and therefore the construction of homologous (collinear) points becomes more complicated, and in any case, for each given point in the collection we will obtain two different homologous points, depending on which of the systems this point applies.