Computed Metric Results
These results appear above the form after submission.
Metric Sweep Graph
The chart tracks how metric coefficients vary along the selected parameter sweep.
Calculator
Enter parametric surface expressions in terms of u and v. Use radians for trigonometric inputs.
Computation History
Recent results stay here for quick export and comparison.
| # | Surface | Point (u, v) | E | F | G | det(g) | Area Element |
|---|---|---|---|---|---|---|---|
| No calculations yet. | |||||||
Example Data Table
These examples show sample metric values for common parametric surfaces.
| Surface | x(u, v) | y(u, v) | z(u, v) | Point | E | F | G | sqrt(EG - F²) |
|---|---|---|---|---|---|---|---|---|
| Unit Sphere | sin(u)*cos(v) | sin(u)*sin(v) | cos(u) | (1.0, 0.8) | 1.0000 | 0.0000 | 0.7081 | 0.8415 |
| Unit Cylinder | cos(v) | sin(v) | u | (0.5, 1.0) | 1.0000 | 0.0000 | 1.0000 | 1.0000 |
| Paraboloid | u | v | u^2 + v^2 | (1.0, 0.5) | 5.0000 | 2.0000 | 2.0000 | 2.4495 |
| Torus (R=3, r=1) | (3 + cos(u))*cos(v) | (3 + cos(u))*sin(v) | sin(u) | (0.7, 0.9) | 1.0000 | 0.0000 | 14.1740 | 3.7648 |
Formula Used
For a parametric surface r(u, v) = (x(u, v), y(u, v), z(u, v)), the tangent vectors are:
The first fundamental form uses the metric coefficients:
The metric tensor is:
The line element is:
The local area element is:
The angle between coordinate curves is:
How to Use This Calculator
- Select a preset surface or choose custom mode.
- Enter x(u, v), y(u, v), and z(u, v).
- Set the evaluation point using u and v.
- Enter small changes du and dv for the line element.
- Choose a small derivative step h for numerical accuracy.
- Pick a sweep parameter and graph range.
- Press Compute Metric to see results above the form.
- Review E, F, G, the determinant, area element, and angle.
- Download CSV or PDF after a successful calculation.
Smaller derivative steps help smooth surfaces. Extremely small steps may increase rounding noise.
FAQs
What is the first fundamental form?
It is the surface metric. It converts small parameter changes into distances, angles, and areas on a parametric surface.
Why are E, F, and G important?
They are dot products of tangent vectors. Together they define the metric tensor and show stretching, skew, and scale on the surface.
What does F = 0 mean?
It means the parameter directions are orthogonal at that point. Orthogonal coordinates often simplify distance and area calculations.
When can the determinant become zero?
It becomes zero when tangent vectors lose independence. Then the parametrization is singular and the metric cannot describe a proper local surface patch.
Are the results exact?
The metric formulas are exact, but this tool estimates derivatives numerically. Smaller steps usually improve accuracy for smooth surfaces.
What is the local area element?
It is sqrt(EG - F^2). Multiply it by a small parameter area to approximate surface area near the chosen point.
Why graph E, F, and G?
The graph shows how the metric changes across a parameter direction. That reveals stretching, compression, and coordinate behavior.
Which inputs work best?
Use smooth parametric equations written with u and v. Functions like sin, cos, exp, and powers are supported.