Calculator Inputs
Use the form below to estimate radius decay, curvature growth, and extinction time for common symmetric cases of mean curvature flow.
Plotly Graph
The graph shows radius decay and mean curvature growth until the theoretical extinction time.
Formula Used
dr/dt = -k / r
r(t) = √(r₀² - 2kt), valid while t ≤ r₀² / (2k)
T = r₀² / (2k)
H(t) = k / r(t)
This calculator uses the classical exact solution for round and symmetric settings. It is especially useful for circles, spheres, cylinders, and custom test cases where the evolution reduces to a radius-only ordinary differential equation.
For fully general surfaces, true mean curvature flow is a geometric partial differential equation. This tool gives an accurate reduced model, not a full mesh-based numerical PDE solver.
How to Use This Calculator
- Select the geometric model that best matches your problem.
- Enter the initial radius and the time where you want evaluation.
- Use the custom factor for generalized symmetric test scenarios.
- Set cylinder length only when using the cylinder model.
- Choose the plotting steps and desired decimal precision.
- Click calculate to show the result above the form.
- Review the chart, geometry measures, and extinction time.
- Export the generated data as CSV or PDF if needed.
Answer to the Requested Concept
Is mean curvature flow the gradient flow of the L2 norm?
More precisely, mean curvature flow is the L²-gradient flow of the area functional. The area decreases most steeply when surface variations are measured using the L² inner product. So the evolving surface follows the steepest descent direction for area, not the gradient flow of the L² norm itself.
Example Data Table
Sample case: sphere in 3D, initial radius 5, curvature factor 2.
| Time t | Radius r(t) | Mean Curvature H(t) |
|---|---|---|
| 0.00 | 5.0000 | 0.4000 |
| 1.00 | 4.5826 | 0.4364 |
| 2.00 | 4.1231 | 0.4851 |
| 3.00 | 3.6056 | 0.5547 |
| 4.00 | 3.0000 | 0.6667 |
| 5.00 | 2.2361 | 0.8944 |
| 6.00 | 1.0000 | 2.0000 |
FAQs
1) What does this calculator compute?
It computes radius evolution, mean curvature, extinction time, shrink speed, and selected geometry measures for symmetric mean curvature flow models.
2) Is this a full PDE solver for arbitrary surfaces?
No. It is a reduced geometric model for symmetric cases such as circles, spheres, cylinders, and custom radius-based test problems.
3) Why does the radius shrink faster near extinction?
Because the model uses dr/dt = -k/r. As the radius gets smaller, the magnitude of the shrink rate increases.
4) What is extinction time?
Extinction time is the moment when the symmetric solution reaches zero radius. In this model, that occurs at T = r₀²/(2k).
5) How is the curvature factor k chosen?
For a circle, k = 1. For a round sphere in 3D, k = 2. For a round cylinder in 3D, k = 1. Custom mode lets you test other symmetric factors.
6) What happens if I enter a time after extinction?
The tool caps evaluation at the extinction time, shows zero radius, and marks the state as extinct so the result remains mathematically consistent.
7) Can I export the generated results?
Yes. The page includes one-click CSV and PDF export buttons for the computed time series and main summary values.
8) Why is mean curvature flow important?
It appears in geometry, image processing, materials science, and surface smoothing because it naturally reduces area while evolving shapes by curvature.