Calculator
Choose one input method. The calculator supports Monge patch derivatives, fundamental forms, and direct principal curvatures.
Example Data Table
| Case | Input Style | Sample Inputs | Gaussian Curvature K | Classification |
|---|---|---|---|---|
| Sphere-like Local Patch | Principal Curvatures | k1 = 0.50, k2 = 0.50 | 0.25 | Elliptic |
| Saddle Surface | Principal Curvatures | k1 = 0.60, k2 = -0.40 | -0.24 | Hyperbolic |
| Parabolic Rim | Monge Patch | fx = 0, fy = 0, fxx = 2, fxy = 0, fyy = 0 | 0 | Parabolic |
| General Parametric Patch | Fundamental Forms | E = 1.04, F = 0.02, G = 1.01, e = 0.82, f = 0.06, g = 0.71 | Approximately 0.573 | Elliptic |
Formulas Used
1) From principal curvatures
K = k1 × k2
This is the most direct definition. Multiply the two principal curvatures at the same surface point.
2) From first and second fundamental forms
K = (e g − f²) / (E G − F²)
Use this when a surface parameterization gives the coefficients of the first and second fundamental forms.
3) For a Monge patch z = f(x, y)
K = (fxx fyy − fxy²) / (1 + fx² + fy²)²
This form is convenient when the surface is written locally as a graph over the xy-plane.
Helpful companion quantity
H = (k1 + k2) / 2
The calculator also reports mean curvature to help infer the local bending pattern and estimate principal curvatures when needed.
How to Use This Calculator
- Choose the input method that matches your known surface data.
- Enter either derivatives, fundamental form coefficients, or principal curvatures.
- Set decimal places, tolerance, and graph settings for your preferred output detail.
- Click the calculate button to place the result above the form.
- Review K, H, estimated principal curvatures, and the point classification.
- Use the Plotly surface preview to inspect local shape behavior.
- Download the summary as CSV or PDF for reports, assignments, or notes.
Frequently Asked Questions
1) What does Gaussian curvature measure?
It measures how a surface bends at one point by multiplying the two principal curvatures. Positive values indicate dome or bowl behavior, negative values indicate saddle behavior, and zero suggests a flat or parabolic direction.
2) What does a negative result mean?
A negative Gaussian curvature means the surface bends in opposite directions along its principal axes. Saddle points on hyperbolic surfaces are the standard example.
3) Why is zero curvature not always completely flat?
Zero Gaussian curvature only means the product of the principal curvatures is zero. One direction may still bend while the other remains straight, producing a parabolic point.
4) When should I use the Monge patch method?
Use the Monge patch method when your surface is written locally as z = f(x, y), and you know the first and second partial derivatives at a point.
5) What are E, F, G, e, f, and g?
E, F, and G are coefficients of the first fundamental form. e, f, and g belong to the second fundamental form. Together they describe the metric and curvature behavior of a parameterized surface.
6) Does the graph show the exact surface?
No. The graph is a local quadratic preview built from your curvature-related inputs. It is meant to illustrate nearby shape behavior, not reconstruct a complete global surface.
7) Why does the calculator also show mean curvature?
Mean curvature complements Gaussian curvature. Together they help recover or estimate principal curvatures and provide a richer description of the local surface geometry.
8) Can I use this for assignments and research checks?
Yes. It is useful for coursework, quick validation, and local surface analysis. Still, verify source formulas, coordinate conventions, and derivative values when preparing formal work.