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Description:
Surfaces that are locally isometric to a plane are called developable surfaces. In the physical world, these surfaces can be formed by bending thin flat sheets of material, which makes them particularly attractive in manufacturing, architecture and art. Consequently, the design of freeform developable surfaces has been an active research topic in computer graphics, computer aided design, architectural geometry and computational origami for several decades. This thesis presents a discrete theory and a set of computational tools for modeling developable surfaces. The basis of our theory is a discrete model termed discrete orthogonal geodesic nets (DOGs). DOGs are regular quadrilateral meshes satisfying local angle constraints, extending the rich theory of nets in discrete differential geometry. Our model is simple, local, and, unlike previous works, it does not directly encode the surface rulings. Thus, DOGs can be used to model continuous deformations of developable surfaces independently of their decomposition into torsal and planar patches or the surface topology. We start by examining the “locking” phenomena common in computational models for developable surfaces, which was the primary motivation behind our work. We then follow up with the derivation and definitions behind our solution - DOGs, while theoretically and empirically demonstrating the connection between our model and its smooth counterpart and its resilience to the locking problem. We prove that every sampling of the smooth counterpart satisfies our constraints up to second order and establish connections between DOGs and other nets in discrete differential geometry. We then develop a theoretical and computational framework for deforming DOGs. We first derive a variety of geometric attributes on DOGs, including notions of normals, curvatures, and a novel DOG Laplacian operator. These can be used as objectives for various modeling tasks. By utilizing the regular nature of our model, our discrete quantities are simple yet precise, and we discuss ...