Enter parameterization data
Use explicit multiplication like 2*u. Supported functions include sin, cos, tan, sqrt, exp, log, abs, sinh, cosh, tanh, and constants pi, e.
Example data table
| Surface | x(u,v) | y(u,v) | z(u,v) | u range | v range |
|---|---|---|---|---|---|
| Sphere | sin(u)*cos(v) | sin(u)*sin(v) | cos(u) | 0.2 to 2.94 | 0 to 2π |
| Cylinder | cos(v) | sin(v) | u | -2 to 2 | 0 to 2π |
| Torus | (2+0.7*cos(u))*cos(v) | (2+0.7*cos(u))*sin(v) | 0.7*sin(u) | 0 to 2π | 0 to 2π |
| Saddle | u | v | u^2-v^2 | -2 to 2 | -2 to 2 |
Formula used
Surface parameterization
r(u,v) = (x(u,v), y(u,v), z(u,v))
Tangent vectors
ru = ∂r/∂u, rv = ∂r/∂v
Normal and area scaling
N = ru × rv, dS = |ru × rv| dudv
First fundamental form
E = ru·ru, F = ru·rv, G = rv·rv
Second fundamental form
e = n·ruu, f = n·ruv, g = n·rvv
Curvatures
K = (eg - f²)/(EG - F²), H = (Eg - 2Ff + Ge)/(2(EG - F²))
This implementation uses numerical differentiation, so it works for custom coordinate expressions without requiring symbolic algebra.
How to use this calculator
- Select a preset surface or choose custom mode.
- Enter the three coordinate functions x(u,v), y(u,v), and z(u,v).
- Set the u and v parameter ranges for the domain you want to inspect.
- Choose a sample point (u0, v0) inside that domain.
- Set grid resolution. Higher values create smoother plots but require more processing.
- Click Compute Surface Metrics to see tangents, normals, curvature estimates, area scaling, tangent plane, and the Plotly graph.
- Use the CSV button for numeric output and the PDF button for a visual report.
Frequently asked questions
1) What does a surface parameterization represent?
A surface parameterization maps two parameters, usually u and v, into a point in three-dimensional space. It describes shape, orientation, tangent directions, and local stretching behavior across a chosen domain.
2) Why are two parameters needed?
A regular surface is two-dimensional locally, even though it sits in three-dimensional space. Two parameters let you move independently in two tangent directions on that surface.
3) What does the vector ru × rv mean?
That cross product gives a normal direction to the surface and measures local area scaling. Its magnitude becomes zero at singular points or degenerate parameterizations.
4) What is the meaning of E, F, and G?
They form the first fundamental form. These values describe local metric behavior, including tangent lengths, angle distortion, and how the parameter grid stretches across the surface.
5) Why can curvature values look noisy?
Curvature depends on second derivatives, so it is more sensitive to sharp changes, singular points, and coarse resolution. Smoother domains and better sample locations usually improve stability.
6) Does the calculator compute exact symbolic derivatives?
No. It uses numerical finite differences. That makes it flexible for custom expressions while still providing practical tangent, normal, area, and curvature estimates.
7) How is total area over the domain estimated?
The calculator samples midpoint cells over the parameter grid and sums |ru × rv| dudv. This produces a numerical approximation to surface area on the selected domain.
8) What should I do if the result says singular or non-regular?
Choose a different sample point or revise the domain. Singular points occur where tangent directions collapse, making the normal vector and some geometric measurements unreliable.