On Pappusʹ configuration in non-commutative projective geometry

In non-commutative projective geometry there exist Pappusʹ configurations whose diagonal points are not collinear. In this paper, we consider two lines r,s in a projective plane and the points A,B on r,A′,B′,C′ on s, and we investigate which points X on r lead to collinear diagonal points in the corresponding Pappusʹ configuration. A geometric interpretation of this result is given, showing that these are exactly all the fixed points of a projectivity of the line r. Finally, we show that the system of fixed points of a large class of projectivities of the line r may be considered as the set of points X on r such that the diagonal points of a suitable Pappusʹ configuration defined by X and other points, are collinear.