Modeling Developable Surfaces with Discrete Orthogonal Geodesic Nets

Abstract

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 their convergence. We then study the DOG constraints, via looking at continous deformations on DOGs. We characterizae the shape space of DOGs for a given net connectivity. We show that generally, this space is locally a manifold of a fixed dimension, apart from a set of singularities, implying that DOGs are continuously deformable. Smooth flows can be constructed by a smooth choice of vectors on the manifold’s tangent spaces, selected to minimize a desired objective function under a given metric. The study of the shape space leads to a better understanding of the flexibility and rigidity of DOGs, and we devote an entire chapter to examining various notions of isometries on DOGs and a novel model termed ”discrete orthogonal 4Q geodesic net”. We further show how to extend the shape space of DOGs by supporting creases and curved folds. We derive a discrete binary charactarization for folds between discrete developable surfaces, accompanied by an algorithm to simultaneously fold creases and smoothly bend planar sheets. We complement our algorithm with essential building blocks for curved folding deformations: objectives to control dihedral angles and mountain-valley assignments. We apply our theory and resulting set of tools in the first interactive editing system for developable surfaces that supports arbitrary bending, stretching, cutting, (curved) folds, as well as smoothing and subdivision operations.