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Isometric deformations of surfaces of translation

Hussein Nassar

Vol. 12 (2024), No. 1, 1–17
Abstract

A surface of translation is a sum (u,v)α(u) + β(v) of two space curves: a path α and a profile β. 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.

Keywords
surface of translation, isometric deformation, bending, rigidity, origami, surface geometry
Mathematical Subject Classification
Primary: 35Q74, 52C25, 53A05, 53Z30, 74K25
Secondary: 70B15, 74Q99
Milestones
Received: 23 March 2023
Revised: 5 October 2023
Accepted: 2 November 2023
Published: 20 December 2023

Communicated by Francesco dell'Isola
Authors
Hussein Nassar
Department of Mechanical and Aerospace Engineering
University of Missouri
Columbia, MO
United States