GENERALIZING MONOTONICITY: ON RECOGNIZING SPECIAL CLASSES OF POLYGONS AND POLYHEDRA
Bibliographic record
Abstract
A simple polyhedron is weakly-monotonic in direction [Formula: see text] provided that the intersection of the polyhedron and any plane with normal [Formula: see text] is simply-connected (i.e. empty, a point, a line-segment or a simple polygon). Furthermore, if the intersection is a convex set, then the polyhedron is said to be weakly-monotonic in the convex sense. Toussaint10 introduced these types of polyhedra as generalizations of the 2-dimensional notion of monotonicity. We study the following recognition problems: Given a simple n-vertex polyhedron in 3-dimensions, we present an O(n log n) time algorithm to determine if there exists a direction [Formula: see text] such that when sweeping over the polyhedron with a plane in direction [Formula: see text], the cross-section (or intersection) is a convex set. If we allow multiple convex polygons in the cross-section as opposed to a single convex polygon, then we provide a linear-time recognition algorithm. For simply-connected cross-sections (i.e. the cross-section is empty, a point, a line-segment or a simple polygon), we derive an O(n2) time recognition algorithm to determine if a direction [Formula: see text] exists. We then study variations of monotonicity in 2-dimensions, i.e. for simple polygons. Given a simple n-vertex polygon P, we can determine whether or not a line ℓ can be swept over P in a continuous manner but with varying direction, such that any position of ℓ intersects P in at most two edges. We study two variants of the problem: one where the line is allowed to sweep over a portion of the polygon multiple times and one where it can sweep any portion of the polygon only once. Although the latter problem is slightly more complicated than the former since the line movements are restricted, our solutions to both problems run in O(n2) time.
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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 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.008 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".