Surface coefficient input
Use the first and second fundamental form coefficients at one surface point.
Formula used
The asymptotic direction condition comes from setting the normal curvature numerator to zero. In differential form, the second fundamental form must satisfy e du² + 2f du dv + g dv² = 0.
When du ≠ 0, the tangent slope m = dv/du solves g m² + 2f m + e = 0. Real asymptotic directions exist only when the discriminant f² - eg ≥ 0.
The calculator also reports K = (eg - f²)/(EG - F²), H = (Eg - 2Ff + Ge)/(2(EG - F²)), and the optional test-direction normal curvature kn = (e + 2fm + gm²)/(E + 2Fm + Gm²).
How to use this calculator
- Enter the first fundamental form coefficients E, F, and G for the chosen surface point.
- Enter the second fundamental form coefficients e, f, and g from the same parameterization.
- Optionally enter a trial slope dv/du to inspect its normal curvature.
- Press Calculate directions to show the result above the form.
- Review the point type, discriminant, asymptotic directions, and curvature values.
- Use the export buttons to save the result summary or example table.
Example data table
These sample inputs illustrate elliptic, parabolic, and hyperbolic behavior.
| Surface case | E | F | G | e | f | g | Point type | Asymptotic slopes dv/du |
|---|---|---|---|---|---|---|---|---|
| Hyperbolic sample | 1 | 0 | 1 | 1 | 2 | 1 | Hyperbolic | -0.267949, -3.732051 |
| Parabolic sample | 1 | 0 | 1 | 1 | 1 | 1 | Parabolic | -1 |
| Elliptic sample | 1 | 0 | 1 | 2 | 1 | 2 | Elliptic | No real slopes |
| Mixed metric sample | 2 | 0.3 | 1.5 | 1.2 | 0.4 | -0.8 | Hyperbolic | -0.822876, 1.822876 |
Frequently asked questions
1. What is an asymptotic direction?
It is a tangent direction where normal curvature becomes zero. On many saddle-like points, two distinct asymptotic directions pass through the same surface point.
2. Why can the calculator return no real directions?
That happens when f² - eg is negative. The point is elliptic, so the quadratic condition for asymptotic directions has no real solution in the tangent plane.
3. What does a parabolic result mean?
A parabolic point has discriminant zero. The surface has one repeated asymptotic direction, often marking the transition between elliptic and hyperbolic behavior.
4. Why are E, F, and G included?
The asymptotic equation itself uses e, f, and g, but E, F, and G are needed for Gaussian curvature, mean curvature, metric validity, and tangent-plane angle calculations.
5. What if g equals zero?
The solver automatically switches to a reciprocal form using du/dv when needed. This avoids losing vertical parameter-space directions during the quadratic solve.
6. How should I interpret the test slope output?
It evaluates normal curvature for a user-chosen tangent slope. When that value reaches zero, the tested slope is an asymptotic direction for the entered surface coefficients.
7. Does a negative Gaussian curvature guarantee asymptotic directions?
Yes, for a regular surface point with valid metric coefficients, negative Gaussian curvature corresponds to a hyperbolic point and gives two real asymptotic directions.
8. Are these directions geometric or coordinate dependent?
The geometric directions are intrinsic to the surface point, but their displayed components depend on the chosen surface parameters and basis for du and dv.