About imaginary conoprimes and conoseconds

Abstract

In the very foundations of the new geometry lies the concept of involution and two cases are distinguished: involution with a pair of real (hyperbolic) and a pair of imaginary (elliptic) double elements. In particular, how for points on a line we can move from one type of involution to another if one from the systems of points that make up the involution, we turn it over so that the points that were conjugate to themselves (double) become conjugate to each other, so in the involution on the plane (polar system), the conoprime that determines the involution becomes imaginary (see article). The difference between the real and imaginary conoprime is clarified, as well as the real and imaginary conosecond. The example shows polar relations for all imaginary conoprimes and conoseconds.