This is an erratum for the article “Lifted Surfacing of Generalized Sweep Volumes” published in ACM Trans. Graph. 44, 6, Article 249 (December 2025), 17 pages.

This paper explores the analysis and design of the resting configurations of a rigid body, without the use of physical simulation. In particular, given a rigid body in R 3 , we identify all possible stationary points, as well as the probability that the body will stop at these points, assuming a random initial orientation and negligible momentum. The forward version of our method can hence be used to automatically orient models, to provide feedback about object stability during the design process, and to furnish plausible distributions of shape orientation for natural scene modeling. Moreover, a differentiable inverse version of our method lets us design shapes with target resting behavior, such as dice with target, nonuniform probabilities. Here we find solutions that would be nearly impossible to find using classical techniques, such as dice with additional unstable faces that provide more natural overall geometry. From a technical point of view, our key observation is that rolling equilibria can be extracted from the Morse-Smale complex of the support function over the Gauss map. Our method is hence purely geometric, and does not make use of random sampling, or numerical time integration. Yet surprisingly, this purely geometric model makes extremely accurate predictions of rest behavior, which we validate both numerically, and via physical experiments. Moreover, for computing rest statistics, it is orders of magnitude faster than state of the art rigid body simulation, opening the door to inverse design—rather than just forward analysis.

Computing the boundary surface of the 3D volume swept by a rigid or deforming solid remains a challenging problem in geometric modeling. Existing approaches are often limited to sweeping rigid shapes, cannot guarantee a watertight surface, or struggle with modeling the intricate geometric features (e.g., sharp creases and narrow gaps) and topological features (e.g., interior voids). We make the observation that the sweep boundary is a subset of the projection of the intersection of two implicit surfaces in a higher dimension, and we derive a characterization of the subset using winding numbers. These insights lead to a general algorithm for any sweep represented as a smooth time-varying implicit function satisfying a genericity assumption, and it produces a watertight and intersection-free surface that better approximates the geometric and topological features than existing methods.

We present a method for generating a simplicial (e.g., triangular or tetrahedral) grid to enable adaptive discretization of implicit shapes defined by a vector function. Such shapes, which we call implicit complexes, are generalizations of implicit surfaces and useful for representing non-smooth and non-manifold structures. While adaptive grid generation has been extensively studied for polygonizing implicit surfaces, few methods are designed for implicit complexes. Our method can generate adaptive grids for several implicit complexes, including arrangements of implicit surfaces, CSG shapes, material interfaces, and curve networks. Importantly, our method adapts the grid to the geometry of not only the implicit surfaces but also their lower-dimensional intersections. We demonstrate how our method enables efficient and detail-preserving discretization of non-trivial implicit shapes.

Computing injective maps with low distortions is a long-standing problem in computer graphics. Such maps are particularly challenging to obtain in the presence of positional constraints, because an injective initial map is often not available. Recently, several energies were proposed and shown to be highly successful in optimizing injectivity from non-injective initial maps while satisfying positional constraints. However, minimizing these energies tends to produce elements with significant isometric distortions. This paper presents simple variants of these energies that retain their desirable traits while promoting isometry. While our method is not guaranteed to provide an injective map, we observe that, on large-scale 2D and 3D data sets, minimizing the proposed isometric variants results in a similar level of success in recovering injectivity as the original energies but a significantly lower isometric distortion.

Implicit surface networks, such as arrangements of implicit surfaces and materials interfaces, are used for modeling piecewise smooth or partitioned shapes. However, accurate and numerically robust algorithms for discretizing either structure on a grid are still lacking. We present a unified approach for computing both types of surface networks for piecewise linear functions defined on a tetrahedral grid. Both algorithms are guaranteed to produce a correct combinatorial structure for any number of functions. Our main contribution is an exact and efficient method for partitioning a tetrahedron using the level sets of linear functions defined by barycentric interpolation. To further improve performance, we designed look-up tables to speed up processing of tetrahedra involving few functions and introduced an efficient algorithm for identifying nested 3D regions.

We introduce a novel approach to describe mesh generation, mesh adaptation, and geometric modeling algorithms relying on changing mesh connectivity using a high-level abstraction. The main motivation is to enable easy customization and development of these algorithms via a declarative specification consisting of a set of per-element invariants, operation scheduling, and attribute transfer for each editing operation. We demonstrate that widely used algorithms editing surfaces and volumes can be compactly expressed with our abstraction, and their implementation within our framework is simple, automatically parallelizable on shared-memory architectures, and with guaranteed satisfaction of the prescribed invariants. These algorithms are readable and easy to customize for specific use cases. We introduce a software library implementing this abstraction and providing automatic shared-memory parallelization.

We port Boolean set operations between 2D shapes to surfaces of any genus, with any number of open boundaries. We combine shapes bounded by sets of freely intersecting loops, consisting of geodesic lines and cubic Bézier splines lying on a surface. We compute the arrangement of shapes directly on the surface and assign integer labels to the cells of such arrangement. Differently from the Euclidean case, some arrangements on a manifold may be inconsistent. We detect inconsistent arrangements and help the user to resolve them. Also, we extend to the manifold setting recent work on Boundary-Sampled Halfspaces, thus supporting operations more general than standard Booleans, which are well defined on inconsistent arrangements, too. Our implementation discretizes the input shapes into polylines at an arbitrary resolution, independent of the level of resolution of the underlying mesh. We resolve the arrangement inside each triangle of the mesh independently and combine the results to reconstruct both the boundaries and the interior of each cell in the arrangement. We reconstruct the control points of curves bounding cells, in order to free the result from discretization and provide an output in vector format. We support interactive usage, editing shapes consisting up to 100k line segments on meshes of up to 1M triangles.

Softness is one of the most important factors in human tactile perception. With recent advances in 3Dprinting, there has been significant progress in fabricating compliant objects. However, existing methods typically remain inaccessible to end-users, mainly due to the separation between designing shapes and setting printing parameters to achieve desired softness, resulting in the exclusion of its customization in early design processes. In this work, we contribute an end-to-end design tool that takes a design-by-example approach: given a 3D model, a user can specify the region of interest and a level of softness, by shopping everyday objects as a reference. The tool then generates both geometry and 3D printing parameters to reproduce the desired softness, which can be fabricated using low-cost FDM 3D printing and materials for it. We also provide a data-driven pipeline to enable other compliance modeling methods to be generalized within our design tool. In two user studies, we demonstrated that users could easily locate existing reference objects’ softness to a 3D printed object. In a design session, end-users successfully used OmniSoft to design augmented functions.

Injective parameterizations of triangulated meshes are critical across applications but remain challenging to compute. Existing algorithms to find injectivity either require initialization from an injective starting state, which is currently only possible without positional constraints, or else can only prevent triangle inversion, which is insufficient to ensure injectivity. Here we present, to our knowledge, the first algorithm for recovering a globally injective parameterization from an arbitrary non-injective initial mesh subject to stationary constraints. These initial meshes can be inverted, wound about interior vertices and/or overlapping. Our algorithm in turn enables globally injective mapping for meshes with arbitrary positional constraints. Our key contribution is a new energy, called smooth excess area (SEA), that measures non-injectivity in a map. This energy is well-defined across both injective and non-injective maps and is smooth almost everywhere, making it readily minimizable using standard gradient-based solvers starting from a non-injective initial state. Importantly, we show that maps minimizing SEA are guaranteed to be locally injective and almost globally injective, in the sense that the overlapping area can be made arbitrarily small. Analyzing SEA's behavior over a new benchmark set designed to test injective mapping, we find that optimizing SEA successfully recovers globally injective maps for 85% of the benchmark and obtains locally injective maps for 90%. In contrast, state-of-the-art methods for removing triangle inversion obtain locally injective maps for less than 6% of the benchmark, and achieve global injectivity (largely by chance as prior methods are not designed to recover it) on less than 4%.

We present a novel representation of solid models for shape design. Like Constructive Solid Geometry (CSG), the solid shape is constructed from a set of halfspaces without the need for an explicit boundary structure. Instead of using Boolean expressions as in CSG, the shape is defined by sparsely placed samples on the boundary of each halfspace. This representation, called Boundary-Sampled Halfspaces (BSH), affords greater agility and expressiveness than CSG while simplifying the reverse engineering process. We discuss theoretical properties of the representation and present practical algorithms for boundary extraction and conversion from other representations. Our algorithms are demonstrated on both 2D and 3D examples.

Mapping a source mesh into a target domain while preserving local injectivity is an important but highly non-trivial task. Existing methods either require an already-injective starting configuration, which is often not available, or rely on sophisticated solving schemes. We propose a novel energy form, called Total Lifted Content (TLC), that is equipped with theoretical properties desirable for injectivity optimization. By lifting the simplices of the mesh into a higher dimension and measuring their contents (2D area or 3D volume) there, TLC is smooth over the entire embedding space and its global minima are always injective. The energy is simple to minimize using standard gradient-based solvers. Our method achieved 100% success rate on an extensive benchmark of embedding problems for triangular and tetrahedral meshes, on which existing methods only have varied success.

We propose a robust 2D meshing algorithm, TriWild , to generate curved triangles reproducing smooth feature curves, leading to coarse meshes designed to match the simulation requirements necessary by applications and avoiding the geometrical errors introduced by linear meshes. The robustness and effectiveness of our technique are demonstrated by batch processing an SVG collection of 20k images, and by comparing our results against state of the art linear and curvilinear meshing algorithms. We demonstrate for our algorithm the practical utility of computing diffusion curves, fluid simulations, elastic deformations, and shape inflation on complex 2D geometries.

We propose a novel tetrahedral meshing technique that is unconditionally robust, requires no user interaction, and can directly convert a triangle soup into an analysis-ready volumetric mesh. The approach is based on several core principles: (1) initial mesh construction based on a fully robust, yet efficient, filtered exact computation (2) explicit (automatic or user-defined) tolerancing of the mesh relative to the surface input (3) iterative mesh improvement with guarantees, at every step, of the output validity. The quality of the resulting mesh is a direct function of the target mesh size and allowed tolerance: increasing allowed deviation from the initial mesh and decreasing the target edge length both lead to higher mesh quality. Our approach enables "black-box" analysis, i.e. it allows to automatically solve partial differential equations on geometrical models available in the wild, offering a robustness and reliability comparable to, e.g., image processing algorithms, opening the door to automatic, large scale processing of real-world geometric data.

Many high-level geometry processing tasks rely on low-level constructive solid geometry operations. Though trivial for implicit representations, boolean operations are notoriously difficult to execute robustly for explicit boundary representations. Existing methods for 3D triangle meshes fall short in one way or another. Some methods are fast but fail to produce closed, self-intersection free output. Other methods are robust but place prohibitively strict assumptions on the input, e.g., no hollow cavities, non-manifold edges or self-intersections. We propose a systematic recipe for conducting a family of exact constructive solid geometry operations. The two-stage method makes no general position assumptions and does not resort to numerical perturbation. The method is variadic , operating on any number of input meshes. This generalizes unary mesh-repair operations, classic binary boolean differencing, and n -ary operations such as finding all regions inside at least k out of n inputs. We demonstrate the superior effectiveness and robustness of our method on a dataset of 10,000 "real-world" meshes from a popular online repository. To encourage development, validation, and comparison, we release both our code and dataset to the public.

We introduce elastic textures: a set of parametric, tileable, printable, cubic patterns achieving a broad range of isotropic elastic material properties: the softest pattern is over a thousand times softer than the stiffest, and the Poisson's ratios range from below zero to nearly 0.5. Using a combinatorial search over topologies followed by shape optimization, we explore a wide space of truss-like, symmetric 3D patterns to obtain a small family. This pattern family can be printed without internal support structure on a single-material 3D printer and can be used to fabricate objects with prescribed mechanical behavior. The family can be extended easily to create anisotropic patterns with target orthotropic properties. We demonstrate that our elastic textures are able to achieve a user-supplied varying material property distribution. We also present a material optimization algorithm to choose material properties at each point within an object to best fit a target deformation under a prescribed scenario. We show that, by fabricating these spatially varying materials with elastic textures, the desired behavior is achieved.

Direct digital manufacturing is a set of rapidly evolving technologies that provide easy ways to manufacture highly customized and unique products. The development pipeline for such products is radically different from the conventional manufacturing pipeline: 3D geometric models are designed by users often with little or no manufacturing experience, and sent directly to the printer. Structural analysis on the user side with conventional tools is often unfeasible as it requires specialized training and software. Trial-and-error, the most common approach, is time-consuming and expensive. We present a method that would identify structural problems in objects designed for 3D printing based on geometry and material properties only , without specific assumptions on loads and manual load setup. We solve a constrained optimization problem to determine the "worst" load distribution for a shape that will cause high local stress or large deformations. While in its general form this optimization has a prohibitively high computational cost, we demonstrate that an approximate method makes it possible to solve the problem rapidly for a broad range of printed models. We validate our method both computationally and experimentally and demonstrate that it has good predictive power for a number of diverse 3D printed shapes.

Man-made objects are ubiquitous in the real world and in virtual environments. While such objects can be very detailed, capturing every small feature, they are often identified and characterized by a small set of defining curves. Compact, abstracted shape descriptions based on such curves are often visually more appealing than the original models, which can appear to be visually cluttered. We introduce a novel algorithm for abstracting three-dimensional geometric models using characteristic curves or contours as building blocks for the abstraction. Our method robustly handles models with poor connectivity, including the extreme cases of polygon soups, common in models of man-made objects taken from online repositories. In our algorithm, we use a two-step procedure that first approximates the input model using a manifold, closedenvelopesurface and then extracts from it a hierarchical abstraction curve network along with suitable normal information. The constructed curve networks form a compact, yet powerful, representation for the input shapes, retaining their key shape characteristics while discarding minor details and irregularities.