Isometric deformations of surfaces of translation

A \emph{surface of translation} is a sum $(u,v)\mapsto\gt\alpha(u)+\gt\beta(v)$ of two space curves: a \emph{path} $\gt\alpha$ and a \emph{profile} $\gt\beta$. A fundamental problem of differential geometry and shell theory is to determine the ways in which surfaces deform isometrically, i.e., by bending without stretching. Here, we explore how surfaces of translation bend. Existence conditions and closed-form expressions for special bendings of the infinitesimal and finite kinds are provided. In particular, all surfaces of translation admit a purely torsional infinitesimal bending. Surfaces of translation whose path and profile belong to an elliptic cone or to two planes but never to their intersection further admit a torsion-free infinitesimal bending. Should the planes be orthogonal, the infinitesimal bending can be integrated into a torsion-free (finite) bending. Surfaces of translation also admit a torsion-free bending if the path or profile has exactly two tangency directions. Throughout, smooth and piecewise smooth surfaces, i.e., surfaces with straight or curved creases, are invariably dealt with and some extra care is given to situations where the bendings cause new creases to emerge.