Solid Representations

• Solid representations are mathematical or computational models that encode the geometry and topology of 3D objects, crucial for simulation, design, and analysis.
• They encompass methods such as B-Reps, CSG, mesh-based, implicit, and skeletal approaches, each offering distinct trade-offs in precision and application.
• Recent advances like neural generative models and hybrid schemes enhance efficiency, editability, and robustness in solid modeling for complex architectures.

A solid representation encodes the geometry and topology of a 3D object in a mathematical or computational form suitable for querying, simulation, design, and manufacturing. Several canonical representations exist, each with distinctive trade-offs in terms of precision, expressivity, robustness, and suitability for downstream applications. This article surveys the principal paradigms for solid representation—including boundary representations (B-Reps), Constructive Solid Geometry (CSG), mesh-based volumetric and chain-based representations, implicit and neural approaches, hybrid schemes, and advanced representations for analysis and learning—drawing on foundational and contemporary research.

1. Boundary Representation (B-Rep)

Boundary representation (B-Rep) encodes a solid as a 2-dimensional cell complex corresponding to its closed surface boundary. A B-Rep is composed of faces (parametric surface patches, typically NURBS or trimmed quadrics), edges (parameterized curve segments, such as lines, arcs, B-splines), and vertices (points), together with incidence and orientation data that encode topological loops and incidences (Adsul et al., 2014).

B-Reps possess several crucial properties:

• Exactness: Capable of representing analytic and free-form surfaces with high parametric fidelity, particularly when based on NURBS (Non-Uniform Rational B-Splines) (Fan et al., 2024).
• Topological richness: Supports explicit encoding and traversal of cells, co-edges, and adjacency maps, enabling robust operations like boundary traversal and co-edge orientation (Adsul et al., 2014).
• Procedural and geometric flexibility: Directly supports surface/edge/vertex blending, filleting, shelling, and trimming operations.

Recent advances include autoregressive generative models such as SolidGen (Jayaraman et al., 2022), which synthesize B-Reps via neural transformers that directly predict vertex, edge, and face objects in a consistent index-based format, supporting class, image, or voxel conditioning.

NURBS-based B-Rep Encoding

NURBS surfaces, the industrial standard for B-Reps, are defined by control points, non-uniform knot vectors, and positive weights: $$S(u,v) = \frac{\sum_{i=1}^n \sum_{j=1}^m N_{i,p}(u) N_{j,q}(v) w_{ij} P_{ij}} {\sum_{i=1}^n \sum_{j=1}^m N_{i,p}(u) N_{j,q}(v) w_{ij}}$$ where $N_{i,p}$ are $p$-degree B-spline basis functions in $u$, and $N_{j,q}$ in $v$ (Fan et al., 2024). NeuroNURBS proposes a direct encoding of NURBS parameters (control points, weights, knot vectors) via a transformer-based variational autoencoder, offering dramatic reductions in storage and training cost compared to grid-based surface sampling and mitigating approximation artifacts observed in UV-grid approaches.

B-Rep for Solid Sweeps and Sharp Features

Algorithms for swept-volume B-Rep (solid sweeps) construct the boundary of a solid moved along a rigid trajectory by associating each face/edge/vertex with a contact surface traced by its motion, employing geometric “funnel” conditions and adjacency-lifting to reconstruct the output B-Rep (Adsul et al., 2014, Adsul et al., 2014). Extensions for solids with sharp (G⁰) features handle discontinuities in the unit normal by parametrizing the cone of normals along edges and providing precise trimming and singularity analysis (Adsul et al., 2014).

2. Constructive Solid Geometry (CSG)

Constructive Solid Geometry represents solids as rooted trees (construction trees) with leaf nodes corresponding to bounded primitive solids (cuboids, spheres, cylinders, parametrized sweeps, etc.) and internal nodes encoding Boolean operations (union $\cup$, intersection $\cap$, difference $\setminus$) (Wassermann et al., 2018).

CSG offers:

• Intrinsic watertightness: By construction, the result of Boolean combinations of exact primitive solids is watertight.
• Parametric editability: Modifications to primitive parameters or the construction tree structure are immediately reflected in the solid, supporting design ‘by construction’ workflows.
• Seamless design-through-analysis: In finite cell (embedded domain) methodologies (e.g., Finite Cell Method, FCM), CSG models enable direct simulation without requiring boundary meshing—the only geometric query required is a robust point-in-membership (PIM) test, efficiently computable by tree traversal (Wassermann et al., 2018).

Limitations include challenges in representing free-form surfaces, fillets, or non-constructible geometry, and potential inefficiency for highly complex or detailed models.

3. Volumetric and Mesh-Based Representations

Solid representations can also be realized by volumetric discretization or explicit meshings, often favored in physical simulation, rendering, and numerical PDE frameworks.

Convex Polyhedral Meshes

A robust volume mesh can represent a solid by partitioning its interior into convex polyhedral cells, such that the union of a distinguished subset of facets (constrained facets) recovers the input surface (Diazzi et al., 2021). Modern convex polyhedral meshing algorithms:

• Tolerate input defects (self-intersections, gaps, non-manifolds),
• Guarantee exactness through the use of indirect predicates (symbolic LPI/TPI types) for geometric computation,
• Support robust Boolean and Minkowski sum operations by operating on the conformal convex-cell mesh,
• Enable efficient inside/outside labeling via min-cut on the dual cell graph.

Chain-Based and Algebraic-Topological Representations

The (co)chain complex associated with a mesh (cell complex) encodes both geometry and topology. The structure is assembled into the Hasse matrix—a block-bidiagonal matrix collecting all boundary and coboundary operators across cell dimensions (0812.3249). Topology-preserving refinements (Euler operators) correspond to multilinear transformations (block congruences) of the Hasse matrix, enabling structured, exact, and simultaneously topological and geometric representations for mesh generation, adaptive refinement, and field discretization.

4. Implicit and Neural Representations

Implicit representations encode a solid by a scalar field $F(x)$, typically such that the solid is $N_{i,p}$0 and the boundary is the zero-level set $N_{i,p}$1. Recent neural approaches generalize this paradigm:

Neural Halfspace/Binary Tree Representations

The Neural Halfspace Representation (NH-Rep) (Guo et al., 2022) represents a solid by a set of locally-tuned implicit functions $N_{i,p}$2 (MLP outputs trained as SDF approximants for surface patches) assembled into a Boolean tree (with $N_{i,p}$3, $N_{i,p}$4) that reconstructs the solid by logical composition. This architecture:

• Preserves sharp features and patchwise boundaries,
• Achieves state-of-the-art metrics on reconstruction and sharp-feature preservation,
• Admits differentiable Boolean, offsetting, and blending operations via continuous $N_{i,p}$5-functions.

Limitations involve interactive editability and the prerequisite of a manifold, correctly-oriented input segmentation.

Sparse Implicit Layered Images

Layered Depth-Normal Images (LDNIs) encode a solid’s boundary as a sparse multi-layered array of intersection depths and normals for each pixel in three principal directions (Wang et al., 2010). This achieves $N_{i,p}$6 storage for practical solids (for sampling resolution $N_{i,p}$7), enables robust segment-wise Boolean and offsetting operations, and provides accurate reconstruction of thin features via dual-contouring algorithms. The approach is well-suited for GPU implementation and geometry processing.

Stochastic Volumetric Formulations

Stochastic geometry models represent a solid by a random indicator field $N_{i,p}$8, generalizing the classical deterministic inside/outside function. The expected visibility along rays is governed by exponential transport law $N_{i,p}$9, with the attenuation coefficient $p$0 derived as a functional of the local gradient of the vacancy probability $p$1 (Miller et al., 2023). The theory clarifies the relation of this formalism to neural implicit-surface models, microfacet transport, and the correct specification of reciprocal and physically-meaningful attenuation in learning-based volumetric rendering.

5. Skeleton-Based (Medial and Curve Skeleton) Representations

Skeletal representations capture the volume and topology of a solid via the loci of maximal inscribed balls (Medial Axis Transform, MAT) or their lower-dimensional abstractions (curve skeletons) (Tagliasacchi, 2013). The MAT encodes both geometry (local thickness/radius) and topology (homotopy equivalence, branching) but suffers from instability and high dimensionality in 3D (medial sheets/seams). Curve skeletons reduce the MAT to a robust, part-aware 1D graph, suitable for applications in deformation, animation, segmentation, and shape analysis. Algorithms for computing MATs/curve skeletons draw on Voronoi diagrams, distance transforms, Morse theory, and contraction paradigms.

6. Advanced Hybrid and Isogeometric Representations

Gregory Solids and Volume Parameterization

Trivariate Gregory solids generalize Gregory surface patches to solid volumes, interpolating non-four-sided and trimmed surface patches meeting at each corner of a polyhedral parametric domain (Hu et al., 2018). Robust polyhedral volume parametrization is achieved via sparse regularization of the mapping Jacobian and ADMM optimization, guaranteeing accurate boundary interpolation and mesh validity.

Analysis and Design “Solid” Categories

The categorical perspective on solid representations, especially in the context of $p$2-adic Lie groups and condensed mathematics, produces the theory of solid vector spaces and their analytic and locally analytic representation categories (Jacinto et al., 2021, Jacinto et al., 2023, Porat, 15 Oct 2025). These frameworks support derived, stratified, and stacky approaches to the representation of solid objects and their symmetry actions, enabling applications to $p$3-adic Hodge theory, eigenvarieties, and the Langlands program.

7. Comparative Summary

Representation	Primary Use/Strength	Limitation / Pitfall
B-Rep (NURBS)	CAD, exact geometry/topology	Non-watertight/dirty geometry, meshing required
CSG	Parameterization, watertightness	Lacks free-form surfaces, large for detailed models
Volumetric Mesh	Simulation, defect tolerance	High memory, post-hoc boundary reconstruction
Chain/Hasse	Topology, multiresolution, cochains	Abstract, relies on cell complex decomposition
Neural Implicit	Learning, shape fitting, feature-preserving	Editability, training time, segmented input
LDNI	Sparse, efficient implicit, CSG/offsets	Storage grows in pathological cases
Skeletal	Shape abstraction, topology	Instability (MAT), non-invertibility (curveskel)

Each approach is best-suited to a specific regime of geometry complexity, application, and interoperability constraints, and hybridizations are increasingly common in advanced design and learning systems.

• (Adsul et al., 2014, Adsul et al., 2014): Foundational frameworks for B-Rep and solid sweeps.
• (Fan et al., 2024): NURBS-based B-Reps and neural parameter representations (NeuroNURBS).
• (Wassermann et al., 2018): Design-through-analysis pipeline with CSG and Finite Cell Method.
• (Diazzi et al., 2021): Convex polyhedral meshing and robust combinatorial solid modeling.
• (Jayaraman et al., 2022): Neural autoregressive direct B-Rep synthesis (SolidGen).
• (Hu et al., 2018): Trivariate Gregory solids for isogeometric volume parametrization.
• (Guo et al., 2022): Neural halfspace implicit representations (NH-Rep).
• (Miller et al., 2023): Stochastic geometry and probabilistic volumetric modeling.
• (Wang et al., 2010): Sparse implicit (LDNI) solid models and feature-preserving algorithms.
• (Tagliasacchi, 2013): Medial/skeletal representations and applications.
• (0812.3249): Chain/cochain and Hasse matrix representations.
• (Jacinto et al., 2021, Jacinto et al., 2023, Porat, 15 Oct 2025): Solid representation theory in condensed mathematics and analytic $p$4-adic settings.