apply geometric modeling

This study addresses the generalization bottleneck in point cloud geometric representation by proposing transferable Geometric Neural Operators (GNOs) as foundational models. Methodologically, it introduces the first pretraining framework for unordered, unstructured point clouds: leveraging mesh-free, coordinate-agnostic functional mappings; embedding differential-geometric priors—such as covariant derivatives and curvature constraints; and employing self-supervised geometric losses to enable robust representation learning across shapes, topologies, and noise levels. Contributions include: (1) unified support for curvature estimation, geometric PDE solving on manifolds, and curvature-driven deformation modeling; (2) state-of-the-art performance across multiple benchmarks, significantly outperforming existing methods; and (3) open-sourced code and pretrained weights enabling plug-and-play integration.

This work proposes the first differentiable geometry processing system that seamlessly integrates with modern machine learning frameworks, addressing the longstanding challenge of combining geometric algorithms—typically non-differentiable and reliant on complex control flow—with gradient-based optimization. By unifying the adjoint method with a scatter-gather mesh processing paradigm, the system enables efficient gradient computation for existing geometric algorithms without requiring algorithmic reimplementation. It supports state-of-the-art solvers such as local-global and ADMM schemes and provides native differentiability for classical operations including curvature flows and conformal parameterizations. Evaluated on multiple inverse geometry problems, the approach significantly reduces both memory consumption and computational overhead, outperforming general-purpose differentiable optimization tools in runtime efficiency while dramatically lowering implementation effort.

This work addresses the lack of effective support for Kendall’s 3D shape space in existing Python libraries—such as Geomstats—which has hindered the practical application of Riemannian geometry in three-dimensional shape analysis. We present the first systematic implementation of an efficient and user-friendly Python toolkit tailored specifically to Kendall’s 3D shape space, enabling scale-, position-, and orientation-invariant shape modeling and statistical analysis. By providing a ready-to-use, open-source solution for shape statistics on manifolds, this contribution fills a critical software gap in advanced 3D shape analysis, substantially lowering the barrier to entry for researchers, improving computational efficiency, and enhancing the reproducibility of results.

This paper addresses topological ambiguities and numerical robustness issues in Constructive Solid Geometry (CSG) Boolean operations and mesh repair—arising from non-manifold intersections, multi-operand expressions, and degenerate geometries (e.g., coplanar or collinear features). We present the first algorithm to construct an exact Weiler spatial decomposition model. Our method integrates exact geometric predicates (via multi-precision arithmetic), co-refinement, radial sorting, constrained Delaunay triangulation, and symbolic perturbation to achieve precise intersection localization, unambiguous face classification, and consistent regional subdivision. Key contributions include: (1) the first complete, exact implementation of the Weiler model; and (2) a unified geometric kernel architecture that systematically handles all degenerate cases, eliminating duplicate faces and topological inconsistencies. Evaluated on the Thingi10K and ThingiCSG benchmarks, our approach demonstrates significantly higher robustness than state-of-the-art methods.

This survey addresses the growing need for systematic understanding of geometric deep learning (GDL) in computer-aided design (CAD). We focus on three core challenges: design similarity analysis, 2D/3D model synthesis, and CAD reconstruction from point clouds or single/multi-view inputs. Methodologically, we propose the first task taxonomy tailored to CAD-specific GDL, rigorously delineating paradigm boundaries; consolidate over 12 benchmark datasets and 50+ open-source implementations; and unify diverse techniques—including graph neural networks, point cloud processing, mesh learning, generative modeling, and multimodal representation—into a coherent methodological framework. Our analysis identifies critical bottlenecks and open problems, while providing an extensible research roadmap and practical implementation guidelines. The work significantly advances the efficiency, generalizability, and real-world applicability of intelligent CAD design systems.

This work addresses the limitations of traditional geometric processing methods, which rely on manifold assumptions and struggle with non-manifold geometries featuring singular structures such as sharp features, self-intersections, or branches. The authors propose a novel “tangent blow-up” representation, introducing for the first time the algebraic geometry concept of blow-up into geometric processing. By jointly embedding each spatial point together with its tangent plane into the product space of Euclidean space and a Grassmannian manifold, the method iteratively disambiguates coincident points that differ in tangential or higher-order contact. Within this lifted domain, discrete gradient, divergence, and Laplace operators are rigorously defined. This structured representation enables a natural extension of classical differential operators to singular points, demonstrating effectiveness in tasks including geodesic computation, segmentation, parameterization, and curvature estimation.

Existing CAD learning approaches discretize B-Rep models into triangle meshes, thereby discarding the analytical surface representations and topological information essential for consistent instance-level analysis. This work proposes STEP-Parts, a deterministic pipeline that directly extracts geometric instance partitions from native STEP B-Rep data. The method defines partitions based on intrinsic B-Rep topology, merges faces using analytical surface types and near-tangent plane continuity criteria, and transfers labels to triangulated meshes via face-to-mesh correspondence mapping. STEP-Parts ensures boundary consistency across varying triangulations and processes the DeepCAD subset of the ABC dataset—comprising approximately 180,000 models—in under six hours. The resulting labels significantly enhance performance in implicit reconstruction-segmentation tasks and point cloud networks. Code and precomputed labels are publicly released.

Recovering editable, parameterized CAD construction sequences from geometric inputs such as meshes remains a fundamental challenge in design and manufacturing. This work proposes an IoU-driven hybrid optimization framework that, for the first time, formulates the reconstruction problem as structured CAD program optimization. By leveraging geometric feedback, the method iteratively fits and validates a rich set of parametric operations—including fillets and chamfers—within the procedural representation. The approach enables end-to-end image-to-CAD reconstruction across multiple modalities and significantly outperforms existing methods on established benchmarks, achieving superior performance in both volumetric IoU and Chamfer distance metrics. Moreover, it substantially reduces redundancy in the reconstructed programs, enabling efficient and high-fidelity recovery of complex CAD models.

This work addresses the challenges of large-scale, multi-scale microstructure modeling, which suffers from high geometric storage overhead and difficulties in maintaining cross-scale consistency. The authors propose an isogeometric generative modeling framework based on extended volumetric Catmull-Clark splines (ExVCC), leveraging hierarchical local refinement and compact shape encoding to enable on-demand generation and efficient evaluation of geometric details. By unifying geometric representation across scales, the method supports cross-scale associations and automatic propagation of modifications, significantly reducing memory consumption while preserving geometric consistency. Experimental results demonstrate the superiority of the approach in terms of modeling scale, accuracy, and computational efficiency.