Abstract We introduce a new extrinsic discretization of tangent vector fields on triangle meshes that is continuous, with bounded derivatives that are continuous almost everywhere, supporting pointwise evaluation and integration of differential operators. We achieve this by building a continuous normal field over the mesh via Phong interpolation and using minimal Rodrigues rotations to transport vertex‐based tangent vectors into triangle interiors. Unlike most existing discretizations, which typically sacrifice either continuity or the ability to evaluate derivatives pointwise, our approach supports both. Because it is pointwise evaluatable, and using the fact that the covariant derivative can be decomposed into its symmetric, antisymmetric, and scalar components, our discretization supports the construction of standard vector‐field processing operators including the connection and Hodge Laplacians, Killing energy, divergence, curl, and the Lie bracket. This framework provides a simple and practical finite‐element formulation for vector‐field processing on meshes, supporting both integration‐based operators and pointwise queries. To our knowledge, ours is the first discretization that jointly enables extrinsic continuous vector fields, bounded derivatives, and pointwise evaluation of this collection of operators.

Bounded bihramonic weights are a popular tool used to rig and deform characters for animation, to compute reduced-order simulations, and to define feature descriptors for geometry processing. They necessitate tetrahedralizing the volume bounded by the surface, introducing the possibility of meshing artifacts or tetrahedralization failure. We introduce a mesh-free and robust automatic skinning technique that generates weights comparable to the current state of the art, but works reliably even on open surfaces, triangle soups, and point clouds where current methods fail. We achieve this through the use of a specialized Lagrangian representation enabled by the advent of hardware ray-tracing, which circumvents the need for finite elements while optimizing the biharmonic energy and enforcing boundary conditions. The flexibility of our formulation allows us to integrate artistic control through weight painting during the optimization. We offer a thorough qualitative and quantitative evaluation of our method.

We present Geometry in Style, a new method for identity-preserving mesh stylization. Existing techniques either adhere to the original shape through overly restrictive deformations such as bump maps or significantly modify the input shape using expressive deformations that may introduce artifacts or alter the identity of the source shape. In contrast, we represent a deformation of a triangle mesh as a target normal vector for each vertex neighborhood. The deformations we recover from target normals are expressive enough to enable detailed stylizations yet restrictive enough to preserve the shape’s identity. We achieve such deformations using our novel differentiable As-Rigid-As-Possible (dARAP) layer, a neural-network-ready adaptation of the classical ARAP algorithm which we use to solve for per-vertex rotations and deformed vertices. As a differentiable layer, dARAP is paired with a visual loss from a text-to-image model to drive deformations toward style prompts, altogether giving us Geometry in Style. Our project page is at https://threedle.github.io/geometry-in-style.

We introduce mathematical tools to describe the geometric problem of sieves, two-dimensional holes that admit certain three-dimensional objects to pass through them, but block others. This is achieved by formulating the sieve design problem as a two-player game where both players (the one that wants to pass, and the one that wants to block) try to find a set of rigid transformations to achieve their objective. We also introduce an algorithm for solving this game by solving a global optimization problem employing both differentiable rendering with gradient-based optimization as well as particle swarm optimization. Our procedure accounts for real-world manufacturing concerns, and we fabricate a variety of examples demonstrating the practical viability of our sieves. Our implementation takes advantage of GPUs and does not rely on any clean or manifold input geometry as long as it is a triangle mesh. We can produce intricate sieves that block an arbitrary set of shapes \(\mathcal {B}\) but admit another arbitrary set of shapes \(\mathcal {A}\) (if finding a solution is possible for our method).

We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing.

Signed distance fields (SDFs) are a widely used implicit surface representation, with broad applications in computer graphics, computer vision, and applied mathematics. To reconstruct an explicit triangle mesh surface corresponding to an SDF, traditional isosurfacing methods, such as Marching Cubes and and its variants, are typically used. However, these methods overlook fundamental properties of SDFs, resulting in reconstructions that exhibit severe oversmoothing and feature loss. To address this shortcoming, we propose a novel method based on a key insight: each SDF sample corresponds to a spherical region that must lie fully inside or outside the surface, depending on its sign, and that must be tangent to the surface at some point. Leveraging this understanding, we formulate an energy that gauges the degree of violation of tangency constraints by a proposed surface. We then employ a gradient flow that minimizes our energy, starting from an initial triangle mesh that encapsulates the surface. This algorithm yields superior reconstructions to previous methods, even with sparsely sampled SDFs. Our approach provides a more nuanced understanding of SDFs and offers significant improvements in surface reconstruction.

Although shape correspondence is a central problem in geometry processing, most methods for this task apply only to two-dimensional surfaces. The neglected task of volumetric correspondence—a natural extension relevant to shapes extracted from simulation, medical imaging, and volume rendering—presents unique challenges that do not appear in the two-dimensional case. In this work, we propose a method for mapping between volumes represented as tetrahedral meshes. Our formulation minimizes a distortion energy designed to extract maps symmetrically, i.e., without dependence on the ordering of the source and target domains. We accompany our method with theoretical discussion describing the consequences of this symmetry assumption, leading us to select a symmetrized ARAP energy that favors isometric correspondences. Our final formulation optimizes for near-isometry while matching the boundary. We demonstrate our method on a diverse geometric dataset, producing low-distortion matchings that align closely to the boundary.

In theory, diffusion curves promise complex color gradations for infinite-resolution vector graphics. In practice, existing realizations suffer from poor scaling, discretization artifacts, or insufficient support for rich boundary conditions. Previous applications of the boundary element method to diffusion curves have relied on polygonal approximations, which either forfeit the high-order smoothness of Bézier curves, or, when the polygonal approximation is extremely detailed, result in large and costly systems of equations that must be solved. In this paper, we utilize the boundary integral equation method to accurately and efficiently solve the underlying partial differential equation. Given a desired resolution and viewport, we then interpolate this solution and use the boundary element method to render it. We couple this hybrid approach with the fast multipole method on a non-uniform quadtree for efficient computation. Furthermore, we introduce an adaptive strategy to enable truly scalable infinite-resolution diffusion curves.

We propose a variational technique to optimize for generalized barycentric coordinates that offers additional control compared to existing models. Prior work represents barycentric coordinates using meshes or closed-form formulae, in practice limiting the choice of objective function. In contrast, we directly parameterize the continuous function that maps any coordinate in a polytope's interior to its barycentric coordinates using a neural field. This formulation is enabled by our theoretical characterization of barycentric coordinates, which allows us to construct neural fields that parameterize the entire function class of valid coordinates. We demonstrate the flexibility of our model using a variety of objective functions, including multiple smoothness and deformation-aware energies; as a side contribution, we also present mathematically-justified means of measuring and minimizing objectives like total variation on discontinuous neural fields. We offer a practical acceleration strategy, present a thorough validation of our algorithm, and demonstrate several applications.

We introduce a robust optimization method for flip-free distortion energies used, for example, in parametrization, deformation, and volume correspondence. This method can minimize a variety of distortion energies, such as the symmetric Dirichlet energy and our new symmetric gradient energy. We identify and exploit the special structure of distortion energies to employ an operator splitting technique, leading us to propose a novel alternating direction method of multipliers (ADMM) algorithm to deal with the nonconvex, nonsmooth nature of distortion energies. The scheme results in an efficient method where the global step involves a single matrix multiplication and the local steps are closed-form per-triangle/per-tetrahedron expressions that are highly parallelizable. The resulting general-purpose optimization algorithm exhibits robustness to flipped triangles and tetrahedra in initial data as well as during the optimization. We establish the convergence of our proposed algorithm under certain conditions and demonstrate applications to parametrization, deformation, and volume correspondence.

Abstract Differential operators are widely used in geometry processing for problem domains like spectral shape analysis, data interpolation, parametrization and mapping, and meshing. In addition to the ubiquitous cotangent Laplacian, anisotropic second‐order operators, as well as higher‐order operators such as the Bilaplacian, have been discretized for specialized applications. In this paper, we study a class of operators that generalizes the fourth‐order Bilaplacian to support anisotropic behavior. The anisotropy is parametrized by a symmetric frame field, first studied in connection with quadrilateral and hexahedral meshing, which allows for fine‐grained control of local directions of variation. We discretize these operators using a mixed finite element scheme, verify convergence of the discretization, study the behavior of the operator under pullback, and present potential applications.

Abstract We present a simple and concise discretization of the covariant derivative vector Dirichlet energy for triangle meshes in 3D using Crouzeix‐Raviart finite elements. The discretization is based on linear discontinuous Galerkin elements, and is simple to implement, without compromising on quality: there are two degrees of freedom for each mesh edge, and the sparse Dirichlet energy matrix can be constructed in a single pass over all triangles using a short formula that only depends on the edge lengths, reminiscent of the scalar cotangent Laplacian. Our vector Dirichlet energy discretization can be used in a variety of applications, such as the calculation of Killing fields, parallel transport of vectors, and smooth vector field design. Experiments suggest convergence and suitability for applications similar to other discretizations of the vector Dirichlet energy.

Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions or they do not correctly account for intrinsic curvature, which leads to unnatural-looking behavior away from the boundary. This leads to an unfortunate trade-off: One can either have natural behavior in the interior or a distortion-free result at the boundary, but not both. We introduce a generalized Hessian energy for curved surfaces, expressed in terms of the covariant one-form Dirichlet energy, the Gaussian curvature, and the exterior derivative. Energy minimizers solve the Laplace-Beltrami biharmonic equation, correctly accounting for intrinsic curvature, leading to natural-looking isolines. On the boundary, minimizers are as-linear-as-possible, which reduces the distortion of isolines at the boundary. We discretize the covariant one-form Dirichlet energy using Crouzeix-Raviart finite elements, arriving at a discrete formulation of the Hessian energy for applications on curved surfaces. We observe convergence of the discretization in our experiments.

Developable surfaces are those that can be made by smoothly bending flat pieces without stretching or shearing. We introduce a definition of developability for triangle meshes which exactly captures two key properties of smooth developable surfaces, namely flattenability and presence of straight ruling lines. This definition provides a starting point for algorithms in developable surface modeling---we consider a variational approach that drives a given mesh toward developable pieces separated by regular seam curves. Computation amounts to gradient descent on an energy with support in the vertex star, without the need to explicitly cluster patches or identify seams. We briefly explore applications to developable design and manufacturing.

In geometry processing, smoothness energies are commonly used to model scattered data interpolation, dense data denoising, and regularization during shape optimization. The squared Laplacian energy is a popular choice of energy and has a corresponding standard implementation: squaring the discrete Laplacian matrix. For compact domains, when values along the boundary are not known in advance, this construction bakes in low-order boundary conditions. This causes the geometric shape of the boundary to strongly bias the solution. For many applications, this is undesirable. Instead, we propose using the squared Frobenius norm of the Hessian as a smoothness energy. Unlike the squared Laplacian energy, this energy’s natural boundary conditions (those that best minimize the energy) correspond to meaningful high-order boundary conditions. These boundary conditions model free boundaries where the shape of the boundary should not bias the solution locally. Our analysis begins in the smooth setting and concludes with discretizations using finite-differences on 2D grids or mixed finite elements for triangle meshes. We demonstrate the core behavior of the squared Hessian as a smoothness energy for various tasks.