Classification of critical sets and their images for quadratic maps of the plane

TL;DR

This paper completes the classification of critical sets and their images for quadratic maps of the plane, covering all generic and nongeneric cases, and describes their geometric properties.

Contribution

It provides a comprehensive classification of all possible critical sets and their images for quadratic plane maps, including nongeneric cases previously unclassified.

Findings

Critical sets are conic sections.

Classification includes all nongeneric cases.

Descriptions of images for each critical set case.

Abstract

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.