Calculator
Example data table
| Surface | Coefficients (nonzero) | Test point | Expected behavior |
|---|---|---|---|
| Paraboloid | a3=0.5, a5=0.5 | (0,0) | Umbilic at the vertex for symmetric paraboloid. |
| Elliptic paraboloid | a3=1, a5=2 | (0,0) | Usually not umbilic unless curvatures match. |
| Saddle-like | a3=1, a5=-1 | (0,0) | Principal curvatures differ in sign. |
Formula used
For a graph surface z=f(x,y), define E=1+fx2, F=fxfy, G=1+fy2. Let W=√(1+fx2+fy2) and L=fxx/W, M=fxy/W, N=fyy/W.
Umbilic points satisfy proportionality of the fundamental forms: [L M; M N] = λ [E F; F G]. This yields residuals r1=L·F−M·E, r2=M·G−N·F, plus a consistency check r3=L·G−N·E.
Principal curvatures are eigenvalues of the shape operator S = I−1II, where I=[E F; F G] and II=[L M; M N].
How to use this calculator
- Enter polynomial coefficients for f(x,y) up to cubic terms.
- Provide a point (x,y) where you want to test umbilicity.
- Set a tolerance. Smaller tolerances demand stronger checks.
- Optionally enable scanning and choose range and step size.
- Press Submit to see results above the form.
- Download CSV or PDF for records or sharing.
FAQs
What is an umbilic point?
An umbilic point is where the two principal curvatures are equal. Locally, the surface bends the same way in every tangent direction.
Why does this tool use a graph surface model?
Graphs allow direct formulas from derivatives. Many smooth surfaces can be expressed as a local graph patch after rotating coordinates.
How should I choose the tolerance?
Start with 1e-6 for typical inputs. Reduce it for stricter checks. Increase it for approximate fitted surfaces or coarse scans.
Does k1 equals k2 always mean umbilic?
For regular points, equal principal curvatures indicate umbilicity. Numerically, the residual test adds stability when curvature calculations are sensitive.
What does scanning actually do?
The scan evaluates the umbilic residual norm on a grid. It reports local minima below a practical threshold, producing approximate candidate locations.
Why might I see no candidates in a scan?
Your surface may have no umbilics in the region, or the step is too large. Reduce step size or expand the search range.
Can I model spheres or ellipsoids here?
Not globally. Near a point you can approximate with a Taylor polynomial graph patch and test local umbilic behavior through derivatives.
What do E,F,G and L,M,N represent?
E,F,G describe the tangent metric. L,M,N describe how the surface bends normally. Umbilics occur when these forms are proportional.