I am reading a paper stating the following:
"In the Cayley-Klein model, hyperbolic ellipses are conics that lie in the interior of K. The ellipse center is the unique vertex c of the common polar triangle P of C and K. It is indeed a center in elementary sense, as it halves the (hyperbolic) distance between the ellipse points on any line incident with c. The axes of C are the two sides of P through c. Degenerate pole triangles characterize the circles among the ellipses. Their center is still well-defined but the axes are undetermined so that any line through c can be addressed as axis. Figure 2 below displays a hyperbolic ellipse, its center and axes in the Cayley-Klein model."
QUESTION: Could anybody explain how to get the aforementioned "common polar triangle P of C and K" (or how to obtain the hyperbolic ellipse C based on the polar triangle, for that matter)? A step by step construction and/ or general explanation would be super helpful!
Nota Bene: I looked up the definition for "polar triangle", but it didn't make sense in the context of hyperbolic ellipses. According to Wikipedia: The polar triangle associated with a triangle ABC is defined as follows. Consider the great circle that contains the side BC. This great circle is defined by the intersection of a diametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as A is (conventionally) termed the pole of A and it is denoted by A'. The points B' and C' are defined similarly.