This series of examples illustrates the rectangle loci for different pairings of four lines A, B, C, D. The rectangle loci are found as in Lemma 4.3 by intersecting the HR-cones generated by the different pairings. For example, the first snapshot involves the intersection of the HR-cones generated by the pairs AB and CD. The snapshot is taken from above, looking directly down into the two HR-cones. (The two cones are only shown below a fixed height.) The four lines are superimposed on the cross-section in order that their relationship to the two cones is more apparent. The small spheres mark the center of the cone of corresponding color as well as the paired lines that determine the cone. By Lemma 4.3, the projection to the xy-plane of the intersection of the cones is the rectangle locus for the two pairs. In the first three snapshots, this projection is a non-degenerate hyperbola.
AC, DC !