B-REpresentation · Mechanical engineering · ai supported CAD modeling
Holistic Geometry and Topology Representation for Training Local Neural Models for Functional B-Rep Reconstruction
Peter Glova, Eduard Remetei · Valta Engineering s.r.o. — July 2026
Abstract
We formulate the theoretical foundation for training local neural models designed to understand, recognize, and reconstruct the geometric and functional characteristics of a component from a defined manufacturing domain (sheet metal parts) based on text or image input from the user. All three operations—understanding, recognizing, and constructing—are defined based on the boundary representation (B-Rep) of the 3D geometry. The central working hypothesis, following HoLa (Liu et al., 2025), is to treat geometry and topology not as separate carriers of information, but as a single unit. We explain why the STEP format (ISO 10303-242) is optimal for the geometry and feature carrier layer, as it encodes not only faces, edges, and points but also datum points, datum lines, datum planes, and local coordinate systems (LCS) for a deterministic functional definition. For the topology representation, we discuss three methods pursued in parallel: (i) the continuous Voronoi/Dual-Field method according to DualBrep (Liu et al., 2026), (ii) the vertex-based representation analogous to the PicoGK kernel from LEAP 71, and (iii) the UV-grid-graph representation based on UV-Net (Jayaraman et al., 2021). In addition, we describe the automated, LCS-anchored generation of functional and conceptual descriptions of individual component regions and the entire component using a controlled vocabulary of deterministic tokens.
1 Problem Statement and Objectives
We adopt the representation logic of our training from HoLa (Liu et al., 2025). The core insight is this: The topological connection between two surfaces is intrinsically linked to the geometry of their intersection curve . Every curve in a B-rep is the intersection of two surfaces. This allows us to reformulate the learning of topology as a geometric reconstruction problem in Euclidean space, rather than treating it as a separate discrete classification problem.
Formally, the latent space is defined exclusively via surface primitives. An encoder E maps the surface geometry to a holistic latent vector z_s, and a neural intersection module I derives from each pair of surfaces their intersection curve and the existence of the topological connection:
I : (z_s^(i), z_s^(j)) ⟶ (z_c^(ij), a_ij), a_ij ∈ {0,1}, (2)
where z_c^(ij) is the reconstructed curve feature and a_ij is the binary adjacency decision (“do surfaces i and j intersect?”). The training loss combines a reconstruction component based on the sampled points of surfaces and curves, a binary classification component for the intersection, and KL regularization (Liu et al., 2025):
L = w₁·L_recon + w₂·L_inter + w₃·L_reg, (3)
where L_recon = Σᵢ ‖Sᵢ − Ŝᵢ‖₁ + Σⱼ ‖Cⱼ − Ĉⱼ‖₁ is the L1 reconstruction loss and L_reg = Σᵢ KL(N(0,1) ‖ N(μᵢ, σᵢ)). The weights used by Liu et al. (2025) are w₁ = 1, w₂ = 10⁻¹, w₃ = 10⁻⁶. The half-curve structure is essential for robust training: curves are learned with a specific orientation (counterclockwise as the outer edge, clockwise as the inner hole), relative to the first surface of the surface pair; if the order of the pair is swapped, the orientation is inverted.
2 The Holistic Principle: Topology as a Subset of Geometry
We adopt the representation logic of our training from HoLa (Liu et al., 2025). The core insight is this: The topological connection between two surfaces is intrinsically linked to the geometry of their intersection curve . Every curve in a B-rep is the intersection of two surfaces. This allows us to reformulate the learning of topology as a geometric reconstruction problem in Euclidean space, rather than treating it as a separate discrete classification problem.
Formally, the latent space is defined exclusively via surface primitives. An encoder E maps the surface geometry to a holistic latent vector z_s, and a neural intersection module I derives from each pair of surfaces their intersection curve and the existence of the topological connection:
I : (z_s^(i), z_s^(j)) ⟶ (z_c^(ij), a_ij), a_ij ∈ {0,1}, (2)
where z_c^(ij) is the reconstructed curve feature and a_ij is the binary adjacency decision (“do surfaces i and j intersect?”). The training loss combines a reconstruction component based on the sampled points of surfaces and curves, a binary classification component for the intersection, and KL regularization (Liu et al., 2025):
L = w₁·L_recon + w₂·L_inter + w₃·L_reg, (3)
where L_recon = Σᵢ ‖Sᵢ − Ŝᵢ‖₁ + Σⱼ ‖Cⱼ − Ĉⱼ‖₁ is the L1 reconstruction loss and L_reg = Σᵢ KL(N(0,1) ‖ N(μᵢ, σᵢ)). The weights used by Liu et al. (2025) are w₁ = 1, w₂ = 10⁻¹, w₃ = 10⁻⁶. The half-curve structure is essential for robust training: curves are learned with a specific orientation (counterclockwise as the outer edge, clockwise as the inner hole), relative to the first surface of the surface pair; if the order of the pair is swapped, the orientation is inverted.
2.1 Critical Analysis and Architectural Consistency
HoLa achieves a validity rate of about 82% compared to ~50% for the previous state of the art (Liu et al., 2025)—impressive for a generative diffusion model, but a deal-breaker for a manufacturing-ready workflow: having one in five components geometrically invalid is unacceptable. We therefore adopt the representation logic (topology ⊂ geometry, surfaces as primary carriers), but not the probabilistic generation architecture. This leads to the fundamental decision of the GCI axis:
The AI does not generate geometry—it generates design decisions. A deterministic system generates geometry.
The AI interprets the intent and generates geometric constraints, design parameters, and feature sequences; a deterministic B-Rep design pipeline based on Open CASCADE (OCCT) executes these and guarantees topological validity by design. HoLa’s holistic principle serves as conceptual proof that our LCS-based derivation of edges and bends from surface pairs is well-founded: What HoLa learns neurally and implicitly, we do explicitly via geometric constraints.
3 Geometry and Feature Layer: Why STEP (AP242)
For the geometry layer, we choose the STEP format according to ISO 10303-242 (AP242 ed3). The reason goes beyond the mere definition of geometry: In addition to surfaces, edges, and points, STEP also encodes construction auxiliary geometry—datum points, datum lines, datum planes—as well as named local coordinate systems. These entities serve as the basis for the deterministic functional definition.
In our pipeline, every functional feature in CATIA is annotated with an LCS. During STEP export (with the “Export as Geometrical Sets” option enabled), the LCS are not exported as axis systems but as six systematically named construction lines that survive as COMPOSITE_CURVE entities. The parser reconstructs the LCS origin using least-squares intersection of the three functional lines and the axes derived from their directions; the line lengths define the bounding box extent in the local frame. The convention distinguishes between:
• F-lines → functional bounding box (the feature itself), mandatory (three lines).
• C-lines → contextual bounding box (the carrier geometry, larger, enclosing F), optional; if absent, the contextual box inherits the functional dimensions.
A deliberately chosen convention is the asymmetric origin with the Z+ convention: The positive Z-axis always points from the primary contact surface into the material (functional contact direction). For fastening features, this is evident, for example, in Z-extensions ranging from [−1.0, +3.0]—1 mm above the surface, 3 mm into the material. This regularity is visible in the data and can potentially be learned. The Z+ convention deliberately replaces an earlier “Z out of the material” rule because it directly encodes the penetration semantics.
Important for separating training and generation time: Datum lines are exclusively training-time construction geometry and are filtered out by a dedicated geometry extractor before rendering in MENTA CAD—the B-Rep output at generation time does not contain them.
3 Geometry and Feature Layer: Why STEP (AP242)
For the geometry layer, we choose the STEP format according to ISO 10303-242 (AP242 ed3). The reason goes beyond the mere definition of geometry: In addition to surfaces, edges, and points, STEP also encodes construction auxiliary geometry—datum points, datum lines, datum planes—as well as named local coordinate systems. These entities serve as the basis for the deterministic functional definition.
In our pipeline, every functional feature in CATIA is annotated with an LCS. During STEP export (with the “Export as Geometrical Sets” option enabled), the LCS are not exported as axis systems but as six systematically named construction lines that survive as COMPOSITE_CURVE entities. The parser reconstructs the LCS origin using least-squares intersection of the three functional lines and the axes derived from their directions; the line lengths define the bounding box extent in the local frame. The convention distinguishes between:
• F-lines → functional bounding box (the feature itself), mandatory (three lines).
• C-lines → contextual bounding box (the carrier geometry, larger, enclosing F), optional; if absent, the contextual box inherits the functional dimensions.
A deliberately chosen convention is the asymmetric origin with the Z+ convention: The positive Z-axis always points from the primary contact surface into the material (functional contact direction). For fastening features, this is evident, for example, in Z-extensions ranging from [−1.0, +3.0]—1 mm above the surface, 3 mm into the material. This regularity is visible in the data and can potentially be learned. The Z+ convention deliberately replaces an earlier “Z out of the material” rule because it directly encodes the penetration semantics.
Important for separating training and generation time: Datum lines are exclusively training-time construction geometry and are filtered out by a dedicated geometry extractor before rendering in MENTA CAD—the B-Rep output at generation time does not contain them.
4 Representation of Topology: Three Methods Being Pursued in Parallel
Because B-Rep topology is discrete and non-differentiable, we are currently testing several approaches to convert it into a form suitable for neural learning. All three convert the discrete adjacency into a continuous or graph-based structure without sacrificing the holistic connection to the geometry.
4.1 Continuous Dual-Field / Voronoi Diagram (DualBrep)
DualBrep (Liu et al., 2026) encodes a CAD model using two scalar fields: a Signed Distance Function (SDF) S : ℝ³ → ℝ for the global shape and an Unsigned Distance Field (UDF) U : ℝ³ → ℝ over a generalized Voronoi diagram (GVD) for the topology. Conceptually, the GVD partitions the surrounding space into Voronoi cells; the cell boundaries form continuous “medial sheets” exactly where two surfaces meet. The superposition of both fields provides a complete B-Rep definition:
S(p) ≈ 0 ⟺ p lies on the object surface; (4)
U(p) ≈ 0 ⟺ p is equidistant from multiple surfaces (edge). (5)
The appeal of GCI lies in the fact that the SDF defines where the surface exists, while the UDF/GVD defines where the surface identity changes—the topology thus becomes continuous and differentiable without having to predict discrete adjacency matrices. DualBrep reports a validity rate of 76.3% with high geometric accuracy (chamfer distance) on the ABC dataset. For us, this is primarily confirmation of the foundation: continuous distance fields on a grid provide a viable foundation for CAD machine learning—consistent with our voxel-based fusion pipeline (aligned channels on a common voxel grid).
4.2 Vertex-Based Representation Analogous to LEAP 71 / PicoGK
As a second approach, we investigate the extraction of vertices from the B-Rep—parsed together with the STEP B-Rep—as a training representation of the topology. The analogy is the PicoGK kernel from LEAP 71, which generates geometry via implicit fields and a representation built from vertices/voxels (LEAP 71, 2024). We share the observation that a vertex-/field-based intermediate representation is natively tensorial and thus ML-compatible, whereas B-Rep is not.
Our difference from LEAP 71 lies in our approach: PicoGK remains in the voxel/field space (a direct path to the additive slicer), whereas we explicitly use the vertex representation only as an intermediate layer for training and topology and ultimately still generate an exact, industrially interchangeable B-Rep via OCCT. The path described in our documentation—STEP → voxel grid (Occupancy, SDF, Shell, Depth) → aligned label grid → neural network—implements precisely this intermediate layer and has been validated end-to-end on 15 components (85 LCS, 0 errors).
4.3 UV Grid Graph Representation (UV-Net)
As a third approach, we pursue the UV-Net method (Jayaraman et al., 2021). A solid model is represented by regular UV grids that capture the geometry of each face and edge by discretizing the parameter space, along with a graph that encodes the topology via face adjacency. Faces are sampled as 10×10 UV grids with typically 4 or 7 channels (3 coordinates, 3 normals, 1 trimming mask), and edges as 1D UV grids.
2D and 1D CNNs operate on these grids, and their outputs serve as node and edge features of a graph neural network. Node updates in the graph follow a message-passing rule of the form
hᵥ^(k) = φ^(k)( (1+ε^(k))·hᵥ^(k−1) + Σ_{u∈N(v)} f_Θ(h_uv^(k−1)) ⊙ h_u^(k−1) ), (6)
where φ^(k) is an MLP, N(v) is the one-hop neighborhood of surface v, and h_uv is the edge feature. The advantage for GCI: The UV-grid representation is largely invariant with respect to the exact parameterization (the grid does not change when a planar surface is converted into a NURBS patch or when vertices are inserted), and local neighborhoods in parameter space correspond to local neighborhoods in the geometry (Jayaraman et al., 2021). UV-Net achieves a per-face accuracy of 99.95% on the surface segmentation of the MFCAD dataset and significantly outperforms point cloud and mesh baselines.
The three methods are not mutually exclusive. DualBrep provides a continuous field substrate, the vertex representation a tensorial intermediate training layer, and UV-Net a parameterization-invariant graph encoding of face adjacency. Their common denominator is the holistic premise from Section 2: topology is never represented in isolation from geometry.
5 Automated Functional and Conceptual Description
Training on (text, image, B-Rep) tuples requires a functional description for each component. We generate these automatically and deterministically using code that derives the functional and conceptual descriptions—both of individual component regions and of the component as a whole—from the LCS annotations. Determinism is ensured by a controlled vocabulary and a formal naming grammar.
5.1 Naming Grammar and Token Vocabulary
Each LCS has a name according to the following grammar
PRIMARY_FUNCTION [_MECHANISM] _COMPONENT(s) [_FORM] _INDEX [__PATTERN]
Example: FASTENING_SCREW_INTO_THREAD_HOLE_FLAT_TAB_01 breaks down into PRIMARY = FASTENING, MECHANISM = SCREW_INTO_THREAD, COMPONENT = HOLE, FORM = FLAT_TAB, INDEX = 01. The vocabulary (v1.1) comprises 18 primary functions, ~48 mechanisms, ~34 geometric components, and 11 forms. Five GC subcategories—SUBTRACTIVE, ADDITIVE, EDGE, FORM, MARKER—determine the LCS placement convention (not the name). Newly added tokens from the component analysis include DART, GUSSET_RIB, WELD_POINT, WELD_SEAM, ADHESIVE_AREA, CONTACT_AREA, BLIND_RIVET_NUT, SINGLE_SHEAR_PIVOT, RAISED_BOSS, C_CLAMP, RIM, and KEYHOLE_SLOT. Atomic tokens such as KEYHOLE_SLOT are deliberately not composed of HOLE + SLOT because the geometric connection is constitutive of the function.
5.2 From Token to Description
The description generator reads the parsed LCS records and the associated bounding boxes and generates a structured entry for each feature (function, mechanism, component, shape, position in the global frame, dimensions). At the component level, these entries are aggregated into a coherent overall description . Each derived value carries a provenance tag (deterministically measured / from sidecar / from default rule), making the training pairs bit-for-bit reproducible. Designers need only place LCS and name them correctly. The primary output is a `description.yaml` file per component; the pipeline supports single, batch, and regression test modes.
This completes the theoretical framework: The LCS-anchored, token-based description is the textual side of the training stack; the STEP-B Rep (with the topology representations discussed in Section 4) is the geometric side; and the holistic principle from Section 2 guarantees that both sides describe the same unit of geometry and topology.
6 Summary
The GCI theory combines four building blocks: (1) the holistic representation principle from HoLa, which treats topology as geometrically derivable; (2) STEP/AP242 as a carrier of geometry and features with an LCS-based, deterministic functional definition; (3) three complementary topology representations (DualBrep Voronoi fields, LEAP-71-analog vertex representation, UV-Net graph); and (4) automated, vocabulary-based description generation. The model learns design decisions, not geometry tensors—the exact, valid B-rep generation is reserved for a deterministic OCCT pipeline. This approach combines the learnability of continuous representations with the validity guarantee of deterministic geometry generation, which is indispensable for industrial applications in the sheet metal bending domain.
References
Jayaraman, P. K., Sanghi, A., Lambourne, J. G., Willis, K. D. D., Davies, T., Shayani, H., & Morris, N. (2021). UV-Net: Learning from Boundary Representations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 11703–11712.
LEAP 71 (2024). PicoGK — Computational Geometry Kernel: Documentation and Blog. Available at https://picogk.org/doc/ (accessed July 2026).
Liu, Y., Jayaraman, P., Reddy, C., Xu, X., & Shayani, H. (2026). DualBrep: A Dual-Field Continuous Representation for B-rep Modeling. Autodesk Research. arXiv:2606.31579. Code: https://github.com/AutodeskAILab/DualBrep.
Liu, Y., et al. (2025). HoLa: B-Rep Generation using a Holistic Latent Representation. ACM Transactions on Graphics, 44(4), Article 1. Publication date: August 2025.
ISO 10303-242 (AP242 ed3). Industrial automation systems and integration — Product data representation and exchange — Part 242: Managed model-based 3D engineering. International Organization for Standardization.
Internal project documentation, Valta Engineering (2026): Vocabulary_v1.x.yaml; lcs_placement_convention.yaml; LCS_Vocabulary_v1.x_SK.docx; STEP_Description_PRD.md.