Pohlke projections in the hyperbolic case

Let OP1, OP2, OP3 be three non-parallel segments in a plane ω. In this and the next article [9] we find the necessary and sufficient conditions for the existence of the common inscribed ellipse EI and the common circumscribing hyperbola HC (both with center O) of the three ellipses having as conjugate semi-diameters the pairs (OP1, OP2), (OP2, OP3) and (OP3, OP1). To this end we introduce a parallel projection (hyperbolic Pohlke’s projection) such that EI and HC are the outline of the image into ω of a suitable hyperboloid.