One‐Shot Method for Computing Generalized Winding Numbers
Bibliographic record
Abstract
Abstract The generalized winding number is an essential part of the geometry processing toolkit, allowing to quantify how much a given point is inside a surface, even when the surface has boundaries and noise. We propose a new universal method to compute a generalized winding number, based only on the surface boundary and the intersections of a single ray with the surface, supporting any oriented surface representations that support a ray intersection query. Due to the focus on the boundary, our algorithm has a unique set of properties. For 2D parametric curves, on a regular grid of query points, our method is up to 4× faster than the current state of the art, maintaining the same precision. In 3D, our method can compute a winding number of a surface without discretizing it, including parametric surfaces. For some meshes with many triangles and a simple boundary, our method is faster than the hierarchical evaluation of the generalized winding number while still being precise. Similarly, on some parametric surfaces with a simple boundary, our method can be faster than adaptive quadrature. We validate our algorithms theoretically, numerically, and by demonstrating a gallery of results on a variety of parametric surfaces and meshes, as well uses in a variety of applications, including voxelizations and boolean operations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.008 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".