  We provide a complete classification of the critical sets and their images
for quadratic maps of the real plane. Critical sets are always conic sections,
which provides a starting point for the classification. The generic cases, maps
whose critical sets are either ellipses or hyperbolas, was published in
Delgado, et al. (2013). This work completes the classification by including all
the nongeneric cases: the empty set, a single point, a single line, a parabola,
two parallel lines, two intersecting lines, or the whole plane. We describe all
possible images for each critical set case and illustrate the geometry of
representative maps for each case.
