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We introduce a method for approximating the signed distance function (SDF) of geometry corrupted by holes, noise, or self-intersections. The method implicitly defines a completed version of the shape, rather than explicitly repairing the given input. Our starting point is a modified version of theheat methodfor geodesic distance, which diffuses normal vectors rather than a scalar distribution. This formulation provides robustness akin togeneralized winding numbers (GWN), but provides distance function rather than just an inside/outside classification. Our formulation also offers several features not common to classic distance algorithms, such as the ability to simultaneously fit multiple level sets, a notion of distance for geometry that does not topologically bound any region, and the ability to mix and match signed and unsigned distance. The method can be applied in any dimension and to any spatial discretization, including triangle meshes, tet meshes, point clouds, polygonal meshes, voxelized surfaces, and regular grids. We evaluate the method on several challenging examples, implementing normal offsets and other morphological operations directly on imperfect curve and surface data. In many cases we also obtain an inside/outside classification dramatically more robust than the one obtained provided by GWN. 

In the plane, thewinding numberis the number of times a curve wraps around a given point. Winding numbers are a basic component of geometric algorithms such as point-in-polygon tests, and their generalization to data with noise or topological errors has proven valuable for geometry processing tasks ranging from surface reconstruction to mesh booleans. However, standard definitions do not immediately apply on surfaces, where not all curves bound regions. We develop a meaningful generalization, starting with the well-known relationship between winding numbers and harmonic functions. By processing the derivatives of such functions, we can robustly filter out components of the input that do not bound any region. Ultimately, our algorithm yields (i) a closed, completed version of the input curves, (ii) integer labels for regions that are meaningfully bounded by these curves, and (iii) the complementary curves that do not bound any region. The main computational cost is solving a standard Poisson equation, or for surfaces with nontrivial topology, a sparse linear program. We also introduce special basis functions to represent singularities that naturally occur at endpoints of open curves. 

Posing high-contact interactions is challenging and time-consuming, with hand-object interactions being especially difficult due to the large number of degrees of freedom (DOF) of the hand and the fact that humans are experts at judging hand poses. This paper addresses this challenge by elevating contact areas to first-class primitives. We provideend-to-end art-directable(EAD) tools to model interactions based on contact areas, directly manipulate contact areas, and compute corresponding poses automatically. To make these operations intuitive and fast, we present a novel axis-based contact model that supports real-time approximately isometry-preserving operations on triangulated surfaces, permits movement between surfaces, and is both robust and scalable to large areas. We show that use of our contact model facilitates high quality posing even for unconstrained, high-DOF custom rigs intended for traditional keyframe-based animation pipelines. We additionally evaluate our approach with comparisons to prior art, ablation studies, user studies, qualitative assessments, and extensions to full-body interaction.