Accurately simulating existing 3D objects and a wide variety of materials often demands expert knowledge and time-consuming physical parameter tuning to achieve the desired dynamic behavior. We introduce MotionPhysics, an end‑to‑end differentiable framework that infers plausible physical parameters from a user-proved natural language prompt for a chosen 3D scene of interest, removing the need for guidance from ground‑truth trajectories or annotated videos. Our approach first utilizes a multimodal large language model to estimate material parameter values, which are constrained to be within plausible ranges. We further propose a learnable motion distillation loss, which extracts robust motion priors from pretrained video diffusion models while minimizing appearance and geometry inductive biases to guide the simulation. We evaluate MotionPhysics across more than thirty scenarios, including real-world, human-designed, and AI-generated 3D objects, spanning a wide range of materials such as elastic solids, metals, foams, sand, and both Newtonian and non-Newtonian fluids. We demonstrate that it produces visually realistic dynamic simulations guided by natural language, surpassing the state of the art, with physically plausible parameters that are automatically determined.

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.

Reconstructing complex structures from planar cross-sections is a challenging problem, with wide-reaching applications in medical imaging, manufacturing, and topography. Out-of-the-box point cloud reconstruction methods can often fail due to the data sparsity between slicing planes, while current bespoke methods struggle to reconstruct thin geometric structures and preserve topological continuity. This is important for medical applications where thin vessel structures are present in CT and MRI scans. This paper introduces CrossSDF, a novel approach for extracting a 3D signed distance field from 2D signed distances generated from planar contours. Our approach makes the training of neural SDFs contour-aware by using losses designed for the case where geometry is known within 2D slices. Our results demonstrate a significant improvement over existing methods, effectively reconstructing thin structures and producing accurate 3D models without the interpolation artifacts or over-smoothing of prior approaches.

Positional encodings are a common component of neural scene reconstruction methods, and provide a way to bias the learning of neural fields towards coarser or finer representations. Current neural surface reconstruction methods use a “one-size-fits-all” approach to encoding, choosing a fixed set of encoding functions, and therefore bias, across all scenes. Current state-of-the-art surface reconstruction approaches leverage grid-based multi-resolution hash encoding in order to recover high-detail geometry. We propose a learned approach which allows the network to choose its encoding basis as a function of space, by masking the contribution of features stored at separate grid resolutions. The resulting spatially adaptive approach allows the network to fit a wider range of frequencies without introducing noise. We test our approach on standard benchmark surface reconstruction datasets and achieve state-of-the-art performance on two benchmark datasets.

Strip-decomposable quadrilateral (SDQ) meshes, i.e., quad meshes that can be decomposed into two transversal strip networks, are vital in numerous fabrication processes; examples include woven structures, surfaces from sheets, custom rebar, or cable-net structures. However, their design is often challenging and includes tedious manual work, and there is a lack of methodologies for editing such meshes while preserving their strip decomposability. We present an interactive methodology to generate and edit SDQ meshes aligned to user-defined directions, while also incorporating desirable properties to the strips for fabrication. Our technique is based on the computation of two coupled transversal tangent direction fields, integrated into two overlapping networks of strips on the surface. As a case study, we consider the fabrication scenario of robotic non-planar 3D printing of free-form surfaces and apply the presented methodology to design and fabricate non-planar print paths.

We present a novel method for realizing freeform surfaces with pieces of flat fabric, where curvature is created by stitching together points on the fabric using a technique known as smocking. Smocking is renowned for producing intricate geometric textures with voluminous pleats. However, it has been mostly used to realize flat shapes or manually designed, limited classes of curved surfaces. Our method combines the computation of directional fields with continuous optimization of a Tangram graph in the plane, which together allow us to realize surfaces of arbitrary topology and curvature with smocking patterns of diverse symmetries. Given a target surface and the desired smocking pattern, our method outputs a corresponding 2D smocking pattern that can be fabricated by sewing specified points together. The resulting textile fabrication approximates the target shape and exhibits visually pleasing pleats. We validate our method through physical fabrication of various smocked examples.

Reconstructing 3D shapes from planar cross-sections is a challenge inspired by downstream applications like medical imaging and geographic informatics. The input is an in/out indicator function fully defined on a sparse collection of planes in space, and the output is an interpolation of the indicator function to the entire volume. Previous works addressing this sparse and ill-posed problem either produce low quality results, or rely on additional priors such as target topology, appearance information, or input normal directions. In this paper, we present OReX, a method for 3D shape reconstruction from slices alone, featuring a Neural Field as the interpolation prior. A modest neural network is trained on the input planes to return an inside/outside estimate for a given 3D coordinate, yielding a powerful prior that induces smoothness and self-similarities. The main challenge for this approach is high-frequency details, as the neural prior is overly smoothing. To alleviate this, we offer an iterative estimation architecture and a hierarchical input sampling scheme that encourage coarse-to-fine training, allowing the training process to focus on high frequencies at later stages. In addition, we identify and analyze a ripple-like effect stemming from the mesh extraction step. We mitigate it by regularizing the spatial gradients of the indicator function around input in/out boundaries during network training, tackling the problem at the root. Through extensive qualitative and quantitative experimentation, we demonstrate our method is robust, accurate, and scales well with the size of the input. We report state-of-the-art results compared to previous approaches and recent potential solutions, and demonstrate the benefit of our individual contributions through analysis and ablation studies.11Code and data available at https://github.com/haimsaw/OReX

Abstract We propose a novel Möbius interpolator that takes as an input a discrete map between the vertices of two planar triangle meshes, and outputs a continuous map on the input domain. The output map interpolates the discrete map, is continuous between triangles, and has low quasi‐conformal distortion when the input map is discrete conformal. Our map leads to considerably smoother texture transfer compared to the alternatives, even on very coarse triangulations. Furthermore, our approach has a closed‐form expression, is local, applicable to any discrete map, and leads to smooth results even for extreme deformations. Finally, by working with local intrinsic coordinates, our approach is easily generalizable to discrete maps between a surface triangle mesh and a planar mesh, i.e., a planar parameterization. We compare our method with existing approaches, and demonstrate better texture transfer results, and lower quasi‐conformal errors.

We present a numerically robust algorithm for computing the constrained Delaunay tetrahedrization (CDT) of a piecewise-linear complex, which has a 100% success rate on the 4408 valid models in the Thingi10k dataset. We build on the underlying theory of the well-known tetgen software, but use a floating-point implementation based on indirect geometric predicates to implicitly represent Steiner points: this new approach dramatically simplifies the implementation, removing the need for ad-hoc tolerances in geometric operations. Our approach leads to a robust and parameter-free implementation, with an empirically manageable number of added Steiner points. Furthermore, our algorithm addresses a major gap in tetgen's theory which may lead to algorithmic failure on valid models, even when assuming perfect precision in the calculations. Our output tetrahedrization conforms with the input geometry without approximations. We can further round our output to floating-point coordinates for downstream applications, which almost always results in valid floating-point meshes unless the input triangulation is very close to being degenerate.

We introduce a framework for representing face-based directional fields of an arbitrary piecewise-polynomial order. Our framework is based on a primal-dual decomposition of fields, where the exact component of a field is the gradient of piecewise-polynomial conforming function, and the coexact component is defined as the adjoint of a dimensionally-consistent discrete curl operator. Our novel formulation sidesteps the difficult problem of constructing high-order non-conforming function spaces, and makes it simple to harness the flexibility of higher-order finite elements for directional-field processing. Our representation is structure-preserving, and draws on principles from finite-element exterior calculus. We demonstrate its benefits for applications such as Helmholtz-Hodge decomposition, smooth PolyVector fields, the vector heat method, and seamless parameterization.

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input that connect the PQ strips. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling-based representations of developable surfaces while still obtaining developable, curvature-aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its isolines are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight isolines is non-linear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle-based interactive freeform modeling of developable shapes.

We present a unified method to meshing surfaces with unconventional patterns, both periodic and aperiodic. These patterns, which have so far been studied on the plane, are patterns comprising a small number of tiles, that do not necessarily exhibit translational periodicity. Our method generalizes the de Bruijn multigrid method to the discrete setting, and thus reduces the problem to the computation of N -Directional fields on triangle meshes. We work with all cases of directional symmetries that have been little studied, including odd and high N. We address the properties of such patterns on surfaces and the challenges in their construction, including order-preservation, seamlessness, duality, and singularities. We show how our method allows for the design of original and unconventional meshes that can be applied to architectural, industrial, and recreational design.

We present a new approach for computing planar hexagonal meshes that approximate a given surface, represented as a triangle mesh. Our method is based on two novel technical contributions. First, we introduce Coordinate Power Fields , which are a pair of tangent vector fields on the surface that fulfill a certain continuity constraint. We prove that the fulfillment of this constraint guarantees the existence of a seamless parameterization with quantized rotational jumps, which we then use to regularly remesh the surface. We additionally propose an optimization framework for finding Coordinate Power Fields, which also fulfill additional constraints, such as alignment, sizing and bijectivity. Second, we build upon this framework to address a challenging meshing problem: planar hexagonal meshing. To this end, we suggest a combination of conjugacy, scaling and alignment constraints, which together lead to planarizable hexagons. We demonstrate our approach on a variety of surfaces, automatically generating planar hexagonal meshes on complicated meshes, which were not achievable with existing methods.

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the mixed finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure preserving: it reproduces curl-free fields precisely and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover’s distance computation, for efficient and robust computation.

Abstract We introduce the generalized nonogram, an extension of the well‐known nonogram or Japanese picture puzzle. It is not based on a regular square grid but on a subdivision (arrangement) with differently shaped cells, bounded by straight lines or curves. To generate a good, clear puzzle from a filled line drawing, the arrangement that is formed for the puzzle must meet a number of criteria. Some of these relate to the puzzle and some to the geometry. We give an overview of these criteria and show that a puzzle can be generated by an optimization method like simulated annealing. Experimentally, we analyze the convergence of the method and the remaining penalty score on several input pictures along with various other design options.

Abstract We propose a novel approach for computing correspondences between subdivision surfaces with different control polygons. Our main observation is that the multi‐resolution spectral basis functions that are open used for computing a functional correspondence can be compactly represented on subdivision surfaces, and therefore can be efficiently computed. Furthermore, the reconstruction of a pointwise map from a functional correspondence also greatly benefits from the subdivision structure. Leveraging these observations, we suggest a hierarchical pipeline for functional map inference, allowing us to compute correspondences between surfaces at fine subdivision levels, with hundreds of thousands of polygons, an order of magnitude faster than existing correspondence methods. We demonstrate the applicability of our results by transferring high‐resolution sculpting displacement maps and textures between subdivision models.

We propose a method for computing global Chebyshev nets on triangular meshes. We formulate the corresponding global parameterization problem in terms of commuting PolyVector fields, and design an efficient optimization method to solve it. We compute, for the first time, Chebyshev nets with automatically-placed singularities, and demonstrate the realizability of our approach using real material.

We propose an algorithm that interpolates between vector and frame fields on triangulated surfaces, designed to complement field design methods in geometry processing and simulation. Our algorithm is based on apolarconstruction, leveraging a conservation law from the Hopf-Poincaré theorem to match singular points using ideas from optimal transport; the remaining detail of the field is interpolated using straightforward machinery. Our model is designed with topology in mind, sliding singular points along the surface rather than having them appear and disappear, and it caters to all surface topologies, including boundary and generator loops.

We present a novel framework for creating Möbius-invariant subdivision operators with a simple conversion of existing linear subdivision operators. By doing so, we create a wide variety of subdivision surfaces that have properties derived from Möbius geometry; namely, reproducing spheres, circular arcs, and Möbius regularity. Our method is based on establishing a canonical form for each 1-ring in the mesh, representing the class of all 1-rings that are Möbius equivalent to that 1-ring. We perform a chosen linear subdivision operation on these canonical forms, and blend the positions contributed from adjacent 1-rings, using two novel Möbius-invariant operators, into new face and edge points. The generality of the method allows for easy coarse-to-fine mesh editing with diverse polygonal patterns, and with exact reproduction of circular and spherical features. Our operators are in closed-form and their computation is as local as the computation of the linear operators they correspond to, allowing for efficient subdivision mesh editing and optimization.

We propose a novel framework for the computational design of tensegrity structures, which are constructions made of struts and cables, held rigid by continuous tension between the elements. Tensegrities are known to be difficult to design---existing design methods are often restricted to using symmetric or templated configurations, limiting the design space to simple constructions. We introduce an algorithm to automatically create free-form stable tensegrity designs that satisfy both fabrication and geometric constraints, and faithfully approximate input geometric shapes. Our approach sidesteps the usual force-based approach in favor of a geometric optimization on the positions of the elements. Equipped with this formulation, we provide a design framework to explore the highly constrained space of tensegrity structures. We validate our method with simulations and real-world constructions.

The computation of smooth fields of orthogonal directions within a volume is a critical step in hexahedral mesh generation, used to guide placement of edges and singularities. While this problem shares high-level structure with surface-based frame field problems, critical aspects are lost when extending to volumes, while new structure from the flat Euclidean metric emerges. Taking these considerations into account, this article presents an algorithm for computing such “octahedral” fields. Unlike existing approaches, our formulation achieves infinite resolution in the interior of the volume via the boundary element method (BEM), continuously assigning frames to points in the interior from only a triangle mesh discretization of the boundary. The end result is an orthogonal direction field that can be sampled anywhere inside the mesh, with smooth variation and singular structure in the interior, even with a coarse boundary. We illustrate our computed frames on a number of challenging test geometries. Since the octahedral frame field problem is relatively new, we also contribute a thorough discussion of theoretical and practical challenges unique to this problem.

We present a framework for designing shapes from diverse combinatorial patterns, where the vertex 1-rings and the faces are as rotationally symmetric as possible, and define such meshes as regular. Our algorithm computes the geometry that brings out the symmetries encoded in the combinatorics. We then allow designers and artists to envision and realize original meshes with great aesthetic qualities. Our method is general and applicable to meshes of arbitrary topology and connectivity, from triangle meshes to general polygonal meshes. The designer controls the result by manipulating and constraining vertex positions. We offer a novel characterization of regularity, using quaternionic ratios of mesh edges, and optimize meshes to be as regular as possible according to this characterization. Finally, we provide a mathematical analysis of these regular meshes, and show how they relate to concepts like the discrete Willmore energy and connectivity shapes.

Abstract Direction fields and vector fields play an increasingly important role in computer graphics and geometry processing. The synthesis of directional fields on surfaces, or other spatial domains, is a fundamental step in numerous applications, such as mesh generation, deformation, texture mapping, and many more. The wide range of applications resulted in definitions for many types of directional fields: from vector and tensor fields, over line and cross fields, to frame and vector‐set fields. Depending on the application at hand, researchers have used various notions of objectives and constraints to synthesize such fields. These notions are defined in terms of fairness, feature alignment, symmetry, or field topology, to mention just a few. To facilitate these objectives, various representations, discretizations, and optimization strategies have been developed. These choices come with varying strengths and weaknesses. This report provides a systematic overview of directional field synthesis for graphics applications, the challenges it poses, and the methods developed in recent years to address these challenges.

We study the design and optimization of polyhedral patterns, which are patterns of planar polygonal faces on freeform surfaces. Working with polyhedral patterns is desirable in architectural geometry and industrial design. However, the classical tiling patterns on the plane must take on various shapes in order to faithfully and feasibly approximate curved surfaces. We define and analyze the deformations these tiles must undertake to account for curvature, and discover the symmetries that remain invariant under such deformations. We propose a novel method to regularize polyhedral patterns while maintaining these symmetries into a plethora of aesthetic and feasible patterns.

We present a framework for designing curl-free tangent vector fields on discrete surfaces. Such vector fields are gradients of locally-defined scalar functions, and this property is beneficial for creating surface parameterizations, since the gradients of the parameterization coordinate functions are then exactly aligned with the designed fields. We introduce a novel definition for discrete curl between unordered sets of vectors (PolyVectors), and devise a curl-eliminating continuous optimization that is independent of the matchings between them. Our algorithm naturally places the singularities required to satisfy the user-provided alignment constraints, and our fields are the gradients of an inversion-free parameterization by design.

We establish a framework to design triangular and circular polygonal meshes by using face-based compatible Möbius transformations. Embracing the viewpoint of surfaces from circles, we characterize discrete conformality for such meshes, in which the invariants are circles, cross-ratios, and mutual intersection angles. Such transformations are important in practice for editing meshes without distortions or loss of details. In addition, they are of substantial theoretical interest in discrete differential geometry. Our framework allows for handle-based deformations, and interpolation between given meshes with controlled conformal error.

Abstract We introduce N‐PolyVector fields, a generalization of N‐RoSy fields for which the vectors are neither necessarily orthogonal nor rotationally symmetric. We formally define a novel representation for N‐PolyVectors as the root sets of complex polynomials and analyze their topological and geometric properties. A smooth N‐PolyVector field can be efficiently generated by solving a sparse linear system without integer variables. We exploit the flexibility of N‐PolyVector fields to design conjugate vector fields, offering an intuitive tool to generate planar quadrilateral meshes.

Abstract We present a novel framework for polyhedral mesh editing with face‐based projective maps that preserves planarity by definition. Such meshes are essential in the field of architectural design and rationalization. By using homogeneous coordinates to describe vertices, we can parametrize the entire shape space of planar‐preserving deformations with bilinear equations. The generality of this space allows for polyhedral geometric processing methods to be conducted with ease. We demonstrate its usefulness in planar‐quadrilateral mesh subdivision, a resulting multi‐resolution editing algorithm, and novel shape‐space exploration with prescribed transformations. Furthermore, we show that our shape space is a discretization of a continuous space of conjugate‐preserving projective transformation fields on surfaces. Our shape space directly addresses planar‐quad meshes, on which we put a focus, and we further show that our framework naturally extends to meshes with faces of more than four vertices as well.

Abstract We offer a framework for editing and modeling of planar meshes, focusing on planar quad, and hexagonal‐dominant meshes, which are held in high demand in the field of architectural design. Our framework manipulates these meshes by affine maps that are assigned per‐face, and which naturally ensure the planarity of these faces throughout the process, resulting in a linear subspace of compatible planar deformations for any given mesh. Our modeling metaphors include classical handle‐based editing, mesh interpolation, and shape‐space exploration, all of which allow for an intuitive way to produce new polyhedral and near‐polyhedral meshes by editing.

We describe a simple algorithm to reconstruct the surface of smooth three-dimensional multilabeled objects from sampled planar cross-sections of arbitrary orientation. The algorithm has the unique ability to handle cross-sections in which regions are classified as being inside the object, outside the object, or unknown. This is achieved by constructing a scalar function on R 3 , whose zero set is the desired surface. The function is constructed independently inside every cell of the arrangement of the cross-section planes using transfinite interpolation techniques based on barycentric coordinates. These guarantee that the function is smooth, and its zero set interpolates the cross-sections. The algorithm is highly parallelizable and may be implemented as an incremental update as each new cross-section is introduced. This leads to an efficient online version, performed on a GPU, which is suitable for interactive medical applications.

Studying the behavior of the heat diffusion process on a manifold is emerging as an important tool for analyzing the geometry of the manifold. Unfortunately, the high complexity of the computation of the heat kernel -- the key to the diffusion process - limits this type of analysis to 3D models of modest resolution. We show how to use the unique properties of the heat kernel of a discrete two dimensional manifold to overcome these limitations. Combining a multi-resolution approach with a novel approximation method for the heat kernel at short times results in an efficient and robust algorithm for computing the heat kernels of detailed models. We show experimentally that our method can achieve good approximations in a fraction of the time required by traditional algorithms. Finally, we demonstrate how these heat kernels can be used to improve a diffusion-based feature extraction algorithm.

In this video we present an algorithm for computing the straight skeleton of a polyhedron in three dimensions.

Abstract We present a novel algorithm for reconstructing a subdivision of the three‐dimensional space (given arbitrarily‐oriented slices of it) into labeled domains. The input to the algorithm is a collection of nonparallel planar cross‐sections of an unknown object, where the sections might cover only portions of the supporting planes. (The information in the rest of these planes is, thus, “unknown.”) Each cross‐section consists of a partition of the plane into closed labeled (“colored”) domains with no restrictions whatsoever on either their geometries or topologies, and without any assumptions about similarities between partitions of different sections. The problem is to reconstruct the original three‐dimensional partition by interpolating simultaneously all the cross‐sections, so that planar domains in the input are connected only to other domains of the same color, no two reconstructed spatial domains intersect, and no unnecessary gaps remain between the reconstructed colored domains. The problem of reconstructing multiple‐labeled domains arises, for example, in medical imaging, where different types of tissues are scanned and reconstructed at the same time. Partial slices are typical, for example, in ultrasound scanning. In this work we use the three‐dimensional straight‐skeleton of the arrangement of the cross‐sections. Since the sections might be partial, cells of the arrangement might be nonconvex. For this we use the unambiguous definition, as well as the implementation of the computation, of the straight skeleton of a three‐dimensional polyhedron that we presented in a recent work [ BEGV08 ]. First, we define these cells and compute their skeleton. Second, we compute overlays of portions of sampled contours in the cross‐sections, using the cell skeletons to guide the reconstruction of the mesh.

The topic of interpolation between slices has been an intriguing problem for many years, as it offers means to visualize and investigate a three-dimensional object given only by its level sets. A slice consists of multiple non-intersecting simple contours, each defined by a cyclic list of vertices. An interpolation solution matches between a number of such slices (two or more at a time), providing means to create a closed surface connecting these slices, or the equivalent morph from one slice to another.