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

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