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We introduce a characterization for affine equivalence of two surfaces oftranslation defined by either rational or meromorphic generators. In turn, thisinduces a similar characterization for minimal surfaces. In the rational case,our results provide algorithms for detecting affine equivalence of thesesurfaces, and therefore, in particular, the symmetries of a surface oftranslation or a minimal surface of the considered types. Additionally, weapply our results to designing surfaces of translation and minimal surfaceswith symmetries, and to computing the symmetries of the higher-order Ennepersurfaces.

Affine equivalences of surfaces of translation and minimal surfaces, and applications to symmetry detection and design