We have shown that parabolic projection is equivalent to stereographic projection, as well as being dual to perspective projection.
Stereographic projection is a map with several important properties.
First the projection of any circle on the sphere is a circle in the plane. In particular the projection of a great circle is a circle. What is also important is that the map is conformal. This is important because it means for one thing that if two circles are horizons of some planes, and they are orthogonal, then the planes are perpendicular.