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arxiv: 2504.11435 · v3 · pith:7ESYRZH5new · submitted 2025-04-15 · 💻 cs.GR · cs.CG· cs.NA· math.NA

Robust Containment Queries over Collections of Trimmed NURBS Surfaces via Generalized Winding Numbers

Jacob Spainhour , Kenneth Weiss This is my paper

Pith reviewed 2026-05-22 20:17 UTC · model grok-4.3

classification 💻 cs.GR cs.CGcs.NAmath.NA
keywords generalized winding numbertrimmed NURBScontainment querysolid angleadaptive quadratureCAD modelsnon-watertight surfaces
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The pith

Containment queries on trimmed NURBS surfaces use a boundary formulation of the solid angle to compute generalized winding numbers without discretization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a containment test for shapes defined by collections of trimmed NURBS surfaces that stays reliable even when those surfaces fail to meet perfectly along their edges. It determines whether a point lies inside the shape by computing the generalized winding number, which measures the net winding of the surfaces around the point. The key step replaces the usual surface integral of solid angle with an equivalent integral along the trimming curves alone, which is then evaluated by adaptive quadrature that refines only where needed. Because the calculation never approximates the curved patches with flat pieces, the query respects the exact geometry of the input and remains accurate at any distance from the surface. The same quadrature data is reused across batches of points to reduce repeated work.

Core claim

The generalized winding number relative to trimmed NURBS surfaces can be obtained by converting the surface integral of the 3D solid angle into a boundary integral over the trimming curves and solving that integral with rapidly converging adaptive quadrature; memoization of node positions and tangents further accelerates batches of queries. This formulation is indifferent to gaps or overlaps between surfaces and preserves all curved features of the original NURBS patches, yielding a containment classification that is robust to the non-watertight character of typical CAD models.

What carries the argument

Boundary formulation of the 3D solid angle integral, solved by adaptive quadrature along NURBS trimming curves.

If this is right

  • Containment classification remains correct for points arbitrarily close to the surface without discretization-induced errors.
  • All original curved geometry of the NURBS patches is respected rather than approximated.
  • Batches of queries run faster once quadrature node positions and tangents are cached and reused.
  • The method applies directly to complex trimming geometry found in real CAD models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same boundary reduction might be applied to other surface integrals that appear in graphics or simulation pipelines that ingest CAD data.
  • Direct use on parametric surfaces could reduce the need for intermediate meshing steps when testing containment inside simulation loops.
  • Extending the quadrature caching scheme to neighboring points or to time-varying queries could yield further speedups in interactive settings.

Load-bearing premise

The boundary formulation of the solid angle integral can be evaluated accurately and stably for arbitrary trimming curves on NURBS surfaces without introducing numerical instabilities or requiring special handling of trimming topology.

What would settle it

A sequence of query points placed arbitrarily close to trimming curves of increasing curvature or with complex intersections, where the adaptive quadrature either fails to converge to a consistent winding number or produces values that flip the inside-outside classification.

Figures

Figures reproduced from arXiv: 2504.11435 by Jacob Spainhour, Kenneth Weiss.

Figure 1
Figure 1. Figure 1: CAD models composed of trimmed NURBS surfaces are ubiquitous in engineering and design, but robustly defining an interior for such objects is [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: In the exaggerated example (left) we see open and non-manifold [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: In 2D (top) and 3D (bottom), the GWN of watertight shapes (left) [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparing the difference (right, log scale) between the GWN field of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 1
Figure 1. Figure 1: These cases can be categorized as: • Far-field: For points sufficiently far from the surface, i.e. out￾side a bounding box encompassing the surface, the number of intersections is known to be zero. As such, we evaluate the GWN directly with our unadjusted boundary formulation. • Near-field: For points closer to the surface, we use the same boundary formulation, but must also perform a line-surface intersec… view at source ↗
Figure 5
Figure 5. Figure 5: Far- and near-field points are classified according to whether the line [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: We illustrate a property of the GWN in both 2D (top) and 3D (bottom). [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Edge-Case Evaluation: For query points within a surface bounding box, we consider a line containing the point. If the line intersects the surface near a [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: GWN fields and evaluation times along an [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparing surface-based 2D tensor product Gaussian quadrature [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: We demonstrate our calculation of the GWN field on non-watertight CAD models of varying size and complexity. Each field is viewed through two 2D [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparing our direct GWN method to discretized alternatives derived from triangulations [Jacobson et al [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Sensitivity study for the line-surface intersection threshold [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Sensitivity study for extracted disk radius and quadrature tolerance parameter [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Disk extraction applied to the edge case in Figure 8 (a). [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: We apply Algorithm 1 to a shape with a patch with complex, but defective trimming curves. The original collection of trimming curves (top left) [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: This “Spring” shape is derived from the ABC dataset model with index 86. We modified this originally watertight shape by removing the caps at each [PITH_FULL_IMAGE:figures/full_fig_p022_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: This “Bobbin” shape is derived from the ABC dataset model with index 933. We modified this originally watertight shape by removing the interior [PITH_FULL_IMAGE:figures/full_fig_p023_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: This “Trailer” shape is derived from the ABC dataset model with index 4192. We modified this originally watertight shape by removing the bottom [PITH_FULL_IMAGE:figures/full_fig_p023_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: This “Pipe” shape is derived from the ABC dataset model with index 9992. We modified this originally watertight shape by removing the front face of [PITH_FULL_IMAGE:figures/full_fig_p024_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: This watertight “Gear” shape is derived from the ABC dataset model with index 9979. [PITH_FULL_IMAGE:figures/full_fig_p024_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: This “Joint” shape is derived from the ABC dataset model with index 13. We modified this originally watertight shape by adding trimming curves to [PITH_FULL_IMAGE:figures/full_fig_p025_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: We modeled this open “Sliced-Cylinder” shape after an example from [Marussig and Hughes 2017] using Rhino3D. [PITH_FULL_IMAGE:figures/full_fig_p025_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: We modeled this watertight “Box-Sphere” shape after an existing STL mesh using Rhino3D. Although the example looks relatively simple, the fileted [PITH_FULL_IMAGE:figures/full_fig_p026_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: This “Vase” shape is adapted from an example in [Martens and Bessmeltsev 2025], and was created as a surface of revolution from 7 cubic Bézier [PITH_FULL_IMAGE:figures/full_fig_p026_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: This “Connector” model is bundled with releases of OpenCascade [Open Cascade SAS 2011]. [PITH_FULL_IMAGE:figures/full_fig_p027_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: This original “Utah Teapot” [Johnson 2000] is composed of Bézier patches, many of which overlap. Note that the lid to the teapot is disconnected from [PITH_FULL_IMAGE:figures/full_fig_p027_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: This “Lamp” shape is taken from the ABC dataset model with index 3800. Note that the central sphere of the shape has several overlapping watertight [PITH_FULL_IMAGE:figures/full_fig_p028_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: This “Bearings” shape is derived from the ABC dataset model with index 7963. Note that there is a small section of the outer ring that intersects with [PITH_FULL_IMAGE:figures/full_fig_p028_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: This “Bolt” assembly is derived from the ABC dataset model with index 3450. Note that there is a small section where adjacent watertight parts [PITH_FULL_IMAGE:figures/full_fig_p029_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: This “Slide” shape is derived from the ABC dataset model with index 4237. [PITH_FULL_IMAGE:figures/full_fig_p029_31.png] view at source ↗
read the original abstract

We propose a containment query that is robust to the watertightness of regions bound by trimmed NURBS surfaces, as this property is difficult to guarantee for in-the-wild CAD models. Containment is determined through the generalized winding number (GWN), a mathematical construction that is indifferent to the arrangement of surfaces in the shape. Applying contemporary techniques for the 3D GWN to trimmed NURBS surfaces requires some form of geometric discretization, introducing computational inefficiency to the algorithm and even risking containment misclassifications near the surface. In contrast, our proposed method leverages properties of the 3D solid angle to solve the relevant surface integral using a boundary formulation with rapidly converging adaptive quadrature. Batches of queries are further accelerated by \textit{memoizing} (i.e. caching and reusing) quadrature node positions and tangents as they are evaluated. We demonstrate that our GWN method is robust to complex trimming geometry in a CAD model, and is accurate up to arbitrary precision at arbitrary distances from the surface. The derived containment query is therefore robust to model non-watertightness while respecting all curved features of the input shape.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a containment query for collections of trimmed NURBS surfaces based on the generalized winding number (GWN). It converts the surface solid-angle integral to a boundary integral over trimming curves, evaluates it via rapidly converging adaptive quadrature, and accelerates batches via memoization of quadrature nodes and tangents. The method is claimed to be robust to non-watertightness, to respect all curved features without discretization, and to achieve arbitrary precision at arbitrary distances from the surface.

Significance. If the numerical stability and accuracy claims hold, the work would be significant for CAD and geometry-processing applications, where trimmed NURBS models are routinely non-watertight. It offers a direct, discretization-free route to GWN queries that preserves exact curved geometry, addressing a practical pain point that current meshing-based approaches cannot reliably solve.

major comments (2)
  1. [Abstract] Abstract: the central claims of 'accurate up to arbitrary precision' and 'robust to complex trimming geometry' are asserted without any quantitative validation data, error analysis, convergence plots, or comparison against existing GWN or containment methods; this absence leaves the load-bearing robustness guarantee unsupported.
  2. [Method (boundary formulation)] The boundary formulation of the solid-angle integral is presented as the key technical step that enables stable adaptive quadrature on trimmed NURBS. However, no derivation, explicit line-integral expression, or handling of trimming-curve topology (intersections, open endpoints, high curvature) is supplied; without this, it is impossible to verify that the quadrature remains well-defined and singularity-free precisely where the method is advertised to succeed.
minor comments (1)
  1. [Implementation] The memoization strategy for batch queries is mentioned but its data structures, invalidation policy, and interaction with adaptive node placement are not described; a short algorithmic outline would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the potential significance of our work for CAD applications and for highlighting areas where the manuscript can be strengthened. We provide point-by-point responses to the major comments and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims of 'accurate up to arbitrary precision' and 'robust to complex trimming geometry' are asserted without any quantitative validation data, error analysis, convergence plots, or comparison against existing GWN or containment methods; this absence leaves the load-bearing robustness guarantee unsupported.

    Authors: We acknowledge that the abstract, as a concise summary, does not embed quantitative data or plots. The manuscript body includes demonstrations on complex CAD models that illustrate robustness to trimming geometry and accuracy at varying distances. To directly address the concern, we will revise the abstract to reference the supporting experimental observations and add dedicated error analysis, convergence plots, and a comparison subsection in the results. revision: yes

  2. Referee: [Method (boundary formulation)] The boundary formulation of the solid-angle integral is presented as the key technical step that enables stable adaptive quadrature on trimmed NURBS. However, no derivation, explicit line-integral expression, or handling of trimming-curve topology (intersections, open endpoints, high curvature) is supplied; without this, it is impossible to verify that the quadrature remains well-defined and singularity-free precisely where the method is advertised to succeed.

    Authors: We agree that a self-contained derivation and explicit treatment of curve topology would improve verifiability. The formulation is obtained via Stokes' theorem, reducing the solid-angle surface integral to a line integral over trimming curves whose explicit integrand depends on the position and tangent vectors. Topology is handled by splitting curves at intersections, treating open endpoints as boundary contributions, and relying on adaptive quadrature to refine near high-curvature or near-singular regions. We will insert the full derivation, the closed-form line-integral expression, and a dedicated paragraph on topology handling into the revised method section. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard solid-angle boundary reduction and quadrature to GWN

full rationale

The paper's central step converts the surface solid-angle integral for the generalized winding number into a boundary line integral over trimming curves, then evaluates it via adaptive quadrature. This reduction follows from classical vector calculus identities (Stokes' theorem applied to the solid-angle kernel) and does not redefine any quantity in terms of itself, fit parameters to the target output, or rely on a self-citation chain whose validity is internal to the present work. The abstract and described method treat the boundary formulation and quadrature convergence as externally established mathematical facts, with the novelty lying in their application to trimmed NURBS without discretization. No equation is shown to equal its own input by construction, and the robustness claim is presented as a consequence of these independent properties rather than a tautology. Therefore the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard mathematical properties of the solid angle and generalized winding numbers; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)
  • standard math Properties of the 3D solid angle permit an exact boundary formulation of the surface integral over trimmed patches.
    Invoked to justify the quadrature method without discretization.

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