Isometric Deformations of Discrete and Smooth T-surfaces

TL;DR

This paper explores the properties and deformations of T-hedra and T-surfaces, offering explicit parametrizations and descriptions of their isometric deformations, bridging discrete and smooth geometries.

Contribution

It provides the first explicit parametrization of isometric deformations for T-hedra and T-surfaces, linking discrete and smooth cases with synthetic and analytic methods.

Findings

Explicit parametrization of T-hedron deformations

Synthetic and analytic descriptions of T-surfaces

Discussion of deformability range of T-surfaces

Abstract

Quad-surfaces are polyhedral surfaces with quadrilateral faces and the combinatorics of the square grid. A generic quad-surface is rigid. T-hedra is a class of flexible quad-surfaces introduced by Graf and Sauer in 1931. Particular examples of T-hedra are the celebrated Miura fold, discrete surfaces of revolution, and discrete molding surfaces. We provide an explicit parametrization of the isometric deformation of a T-hedron. T-hedra have a smooth analog, T-surfaces. For these we provide a synthetic and an analytic description, both similar to the corresponding descriptions of T-hedra. We also parametrize the isometric deformations of T-surfaces and discuss their deformability range.