Isometric Deformations of Discrete and Smooth T-surfaces

02/17/2023

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by   Ivan Izmestiev, et al.

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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.