Isometric deformations of discrete and smooth T-surfaces

Quad-surfaces are polyhedral surfaces with quadrilateral faces and the combinatorics of a square grid. Isometric deformation of the quad-surfaces can be thought of as transformations that keep all the involved quadrilaterals rigid. Among quad-surfaces, those capable of non-trivial isometric deformations are identified as flexible, marking flexibility as a core topic in discrete differential geometry. The study of quad-surfaces and their flexibility is not only theoretically intriguing but also finds practical applications in fields like membrane theory, origami, architecture and robotics. A generic quad-surface is rigid, however, certain subclasses exhibit a 1-parameter family of flexibility. One of such subclasses is the T-hedra which are originally introduced by Graf and Sauer in 1931. This article provides a synthetic and an analytic description of T-hedra and their smooth counterparts namely, the T-surfaces. In the next step the parametrization of their isometric deformation is obtained and their deformability range is discussed. The given parametrizations and isometric deformations are provided for general T-hedra and T-surfaces. However, specific subclasses are extensively examined and explored, particularly those that encompass notable and well-known structures, including the Miura fold, surfaces of revolution and molding surfaces.