Affine map equivalence versus critical set equivalence for quadratic maps of the plane

TL;DR

This paper classifies quadratic maps of the plane based on affine map equivalence and critical set equivalence, revealing a detailed relationship between these classifications and identifying eighteen distinct affine classes.

Contribution

It provides a complete enumeration of affine map equivalence classes for quadratic plane maps and clarifies their relationship with critical set classifications.

Findings

There are exactly eighteen affine map equivalence classes.

Twelve classes correspond one-to-one with critical set equivalence classes.

Three pairs of classes share critical sets but differ in other geometric properties.

Abstract

In recent work [Nien et al. 2016], the authors enumerated a classification of quadratic maps of the plane according to their critical sets and images. It is straightforward to show that quadratic maps which are affinely map equivalent are also equivalent in the critical set classification. The question remained whether maps that are equivalent in the critical set classification are also affinely map equivalent. This paper establishes a complete enumeration of the affine map equivalence classes. As a consequence, the relationship between affine map equivalence and critical set equivalence is established. In short, there are eighteen affine map equivalence classes. Three pairs of those classes have critical sets and images that match, but each pair has some other geometric property, preserved by affine map equivalence, that does not match. The other twelve affine map equivalence classes and the critical set equivalences are in one-to-one correspondence.