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

TL;DR

The derived containment query is therefore robust to model non-watertightness while respecting all curved features of the input shape, and is accurate up to arbitrary precision at arbitrary distances from the surface.

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.

Figures (30)

• Figure 1: In the exaggerated example (left) we see open and non-manifold edges, highlighted in red, that prohibit a strict definition of the shape's interior volume. Analogous issues are present in the in-the-wild example (right) which are numerically significant, but otherwise unnoticeable to a human observer and therefore difficult to predict or correct.
• Figure 2: In 2D (top) and 3D (bottom), the GWN of watertight shapes (left) is integer valued (middle), while the fractional GWN of a non-watertight shape (right) is equal to the signed angle subtended by its boundary. To more easily visualize the 3D GWN as a scalar field, we consider a 2D slice of the field as it intersects with the surface.
• Figure 3: Comparing the difference (right, log scale) between the GWN field of the original curved geometry (left) and its linearly discretized approximation (middle). While the induced approximation error in 2D (top) is integer-valued, the analogous distribution of errors in 3D (bottom, displayed on a 2D slice of the 3D surface shown in the inset) is non-zero almost everywhere.
• Figure 4: Far- and near-field points are classified according to whether the line of discontinuities for antiderivative $F$ does not intersect (a), can be rotated to avoid intersecting (b), or unavoidably intersects (c) the surface.
• Figure 5: We illustrate a property of the GWN for two surfaces which share a boundary, viewed from different angles. In general, any two surfaces which share a boundary generate the same GWN field, modulo some integer which depends on the orientation of the query point relative to the surface.