Translation surface

This article defines a property that makes sense for a surface embedded in $\mathbb {R} ^{3}$, viz three-dimensional Euclidean space. The property is invariant under orthogonal transformations and scaling, i.e., under all similarity transformations.
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Definition

A surface in $\mathbb {R} ^{3}$ is termed a translation surface if it has the following Cartesian equation:

$z=f(x)+g(y)$

where $f,g$ are smooth functions from open sets in $\mathbb {R}$, to $\mathbb {R}$.