Metric properties of the Stephanos bijection (Q1903354)

The Stephanos bijection \(\sigma\) maps the collineations of the real projective line \(P^1\) onto the points of the real 3-dimensional projective space \(P^3\); a collineation \(\Pi_A\) being represented by its homogeneous transformation matrix \(A = (a_{ik}) \in \mathbb{R}^{2,2}\) is mapped onto its Stephanos image point \(P_A \in P^3\) having the projective co-ordinates \(x_0 : x_1 : x_2 : x_3 = a_{11} : a_{12} : a_{21} : a_{22}\). If the ruled quadric \(Q^2_{42} : x_0 x_3 - x_1 x_2 = 0\) which corresponds to the singular collineations of the projective line \(P^1\) is regarded as an absolute quadric, the projective image space \(P^3\) of \(\sigma\) becomes a hyperbolic space \(P^3_{|2}\) with index 2. The author now examines metric properties of \(\sigma\) being connected with the Cayley/Klein metric defined in \(P^3_{|2}\). We mention two of the various results obtained: The locus of the Stephanos image points \(P_A\) of all elliptic or all directly hyperbolic projectivities \(\Pi_A\) with a given characteristic cross ratio \(c_A\) is a distance sphere with the Stephanos image point \(P_E\) if the identity as centre (Remark 5). For the Stephanos image points \(P_A\) and \(P_B\) of two hyperbolic projectivities \(\Pi_A\) and \(\Pi_B\) with altogether four pairwise distinct fixed points \(Q_A\), \(R_A\), \(Q_B\) and \(R_B\), the angle of the lines \(P_E + P_A\) and \(P_E + P_B\) can be given as a function of the cross ratio \(DV(Q_A, R_A, Q_B, R_B)\), if \(Q_A\), \(Q_B\) are not separated by \(R_A\), \(R_B\) on the projective line \(P^1\) (Propos. 5).