Calculator Inputs
Choose an input mode, enter your values, then calculate the mean curvature.
Example Data Table
| Surface Example | k1 | k2 | Mean Curvature H | Quick Reading |
|---|---|---|---|---|
| Sphere with radius 2 | 0.5 | 0.5 | 0.5 | Equal positive bending in every tangent direction. |
| Cylinder with radius 3 | 0.333333 | 0 | 0.166667 | One curved direction and one flat direction. |
| Plane | 0 | 0 | 0 | No local bending at the chosen point. |
| Saddle sample | 1 | -1 | 0 | Bending cancels, giving zero mean curvature. |
| Elliptic sample point | 0.8 | 0.3 | 0.55 | Both principal curvatures bend the same way. |
Formula Used
1) From principal curvatures
H = (k1 + k2) / 2
2) From principal radii
H = 1/2 × (1/R1 + 1/R2)
3) From fundamental form coefficients
H = (eG - 2fF + gE) / (2(EG - F²))
K = (eg - f²) / (EG - F²)
The sign of mean curvature depends on surface orientation. Reversing the chosen unit normal changes the sign of H.
How to Use This Calculator
- Select the input method that matches your geometry data.
- Enter principal curvatures, principal radii, or both form coefficients.
- Press the calculate button to compute H and K.
- Review the result card, classification, and curvature chart.
- Use the CSV or PDF buttons to export the output.
FAQs
1) What does mean curvature measure?
It measures average bending at a point on a surface. It combines the two principal curvatures into one signed value.
2) Why can mean curvature be negative?
The sign depends on the chosen surface normal. Reversing that normal flips the sign while keeping the bending magnitude unchanged.
3) What is the difference between mean and Gaussian curvature?
Mean curvature averages principal bending. Gaussian curvature multiplies the principal curvatures and highlights whether directions bend the same way.
4) When is mean curvature zero?
It becomes zero when the principal curvatures cancel each other. Planes and many minimal surfaces satisfy this condition locally.
5) Can I use radii instead of curvatures?
Yes. The calculator converts each principal radius into curvature using the reciprocal relation before averaging them.
6) Why does the form coefficient method need EG - F²?
That quantity is the metric determinant for the surface patch. It must stay nonzero so the local parameterization remains regular.
7) What does the chart show?
It shows how mean curvature changes when one selected input varies around your entered values. This helps visualize local sensitivity.
8) Are the exported CSV and PDF values exact?
They export the displayed numerical results and sampled chart data. Precision follows the calculator output formatting used on the page.