A Spatial Version of Octoidal Gears Via the Generalized Camus Theorem
Bibliographic record
Abstract
Understanding the geometry of gears with skew axes is a highly demanding task, which can be eased by invoking Study's Principle of Transference. By means of this principle, spherical geometry can be readily ported into its spatial counterpart using dual algebra. This paper is based on Martin Disteli's work and on the authors' previous results, where Camus' auxiliary curve is extended to the case of skew gears. We focus on the spatial analog of one particular case of cycloid bevel gears: When the auxiliary curve is specified as a pole tangent, we obtain “pathologic” spherical involute gears; the profiles are always interpenetrating at the meshing point because of G2-contact. The spatial analog of the pole tangent, a skew orthogonal helicoid, leads to G2-contact at a single point combined with an interpenetration of the flanks. However, when instead of a line a plane is attached to the right helicoid, the envelopes of this plane under the roll-sliding of the auxiliary surface (AS) along the axodes are developable ruled surfaces. These serve as conjugate tooth flanks with a permanent line contact. Our results show that these flanks are geometrically sound, which should lead to a generalization of octoidal bevel gears, or even of bevel gears carrying teeth designed with the exact spherical involute, to the spatial case, i.e., for gears with skew axes.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".