Extracting shape from curvature evolution in moving surfaces
Bibliographic record
Abstract
Shape is a crucial geometric property of surfaces, interfaces, and membranes in biology, colloidal and interface science, and many areas of physics. This paper presents theory, simulation and scaling of local shape and curvedness changes in moving surfaces and interfaces, under uniform normal motion, as in phase ordering transitions in liquid crystals. Previously presented measures of shape and curvedness are introduced in quantities and equations used in colloidal science and interfacial transport phenomena to separate shape effects from those of curvedness. Considering in parallel the new shape formalism with the classical curvature formalism, this paper sheds new light on what effects originate only from shape. The new shape evolution equations are solved under uniform normal surface flow. It is found that the solutions obey the so-called "astigmatism equation" fixing the linear relation between the radii of curvature. Astigmatic trajectories in the shape-curvedness phase plane, can be clearly classified into two modes: (i) constant shape evolution, and (ii) variable shape-variable curvedness. Shapes between spheres and cylinders follow the former mode for large curvedness and transition at smaller curvedness into the latter. Shapes' transitions between cylinder and saddles only follow the second mode. Under geometry-driven stagnation (i.e. zero normal velocity) shapes can be frozen. Evolving spheres and cylinders freeze into the same original shape, but perturbed cylinders can freeze into a variety of shapes including saddles. The results provide a useful complementary view on how to describe and control shape evolution in surfaces and interfaces, of wide interest in soft matter materials.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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".