From Triangle Polarity to Simplex Inversion

“Triangle polarity” acts in a projective plane \(\pi \) over a (commutative) field with characteristic \(\ne 2.\) It is a quadratic mapping of the point set of \(\pi \) to the line set with a triangle \(\Delta \) as singularity set. The image p of a point \(P\) is declared as the connection of the three collinear points on the sides of \(\Delta \) , which are harmonic to the “Ceva-projections” of \(P\) on these sides with respect to pairs of vertices of \(\Delta \) . In other words, it transforms the “Ceva configuration” into its dual, namely the “Menelaos configuration”. Obviously, this mapping can be generalized to higher dimensions. Furthermore, one can define an analogue to the classical inversion also based on a triangle or a simplex and an inversion center. For two given triangle polarities one can define the intersection \(P{^\prime}\) of the two polar lines of a point \(P\) as its “double polar point”. It turns out that this relationship is not symmetric.