Bibliographic record
Abstract
Polyhedra are an important basic structure in computational geometry. One of the most beautiful results concerning polyhedra is Cauchy’s theorem, which states that a convex polyhedron is uniquely defined by its graph, edge lengths and facial angles. (See Section 2 for definitions.) The proof of Cauchy’s theorem (see e.g. [2]) unfortunately is nonconstructive, and the only known algorithm to reconstruct the convex polyhedron is very slow (see also [5].) In this paper, we study similar topics for orthogonal polyhedra. Thus, given a graph, edge lengths and facial angles, when is this the graph of an orthogonally convex polyhedron? We give an algorithm that answers this question in polynomial time, and reconstructs the polyhedron if one exists. In particular, our algorithm implies a Cauchy-type theorem for orthogonally convex polyhedra: they are determined by their graph, edge lengths and facial angles alone. We also study general orthogonal polyhedra, and show that it is NPhard to decide whether a graph (with edge lengths and facial angles) is the graph of an orthogonal polyhedron. Our research was motived by the question how to represent polyhedra (and especially orthogonal polyhedra) efficiently. One common way is the vertex based model, where one stores the graph and the coordinates of each vertex. For orthogonal polyhedra, it suffices to store coordinates for vertices of odd degree, see [4, 1]. The vertex based model is rather cumbersome for manipulation of polyhedra, since every translation or rotation requires an update of all coordinates. A more versatile approach is to store edge lengths, facial angles and dihedral angles only. The polyhedron is then uniquely determined by the coordinates of three vertices. The results in our paper show that for orthogonally convex polyhedra, we can omit the dihedral angles, since they are uniquely determined from the other parameters.
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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 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.012 | 0.003 |
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".