Calculator Input
This page models a surface in Monge form: z = ax² + bxy + cy² + dx + ey + g.
Example Data Table
| a | b | c | d | e | g | x₀ | y₀ | dx | dy | Det(H) | II(v) | K | H |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.00 | 0.50 | 2.00 | -1.00 | 1.00 | 0.00 | 1.00 | 0.50 | 0.30 | -0.20 | 7.750000 | 0.072752 | 0.035322 | 0.283947 |
Formula Used
Let the surface be z = f(x,y) = ax² + bxy + cy² + dx + ey + g.
First derivatives:
f_x = 2ax + by + d and f_y = bx + 2cy + e.
Hessian matrix:
Hess(f) = [[f_xx, f_xy], [f_xy, f_yy]] = [[2a, b], [b, 2c]].
For the Monge patch r(x,y) = (x, y, f(x,y)), the unit normal is:
n = (-f_x, -f_y, 1) / sqrt(1 + f_x² + f_y²).
First fundamental form coefficients:
E = 1 + f_x², F = f_x f_y, G = 1 + f_y².
Second fundamental form coefficients:
L = f_xx / sqrt(1 + f_x² + f_y²), M = f_xy / sqrt(1 + f_x² + f_y²), N = f_yy / sqrt(1 + f_x² + f_y²).
Directional second fundamental form for v = (dx, dy):
II(v) = Ldx² + 2Mdxdy + Ndy².
Curvatures:
K = (LN - M²) / (EG - F²) and H = (EN - 2FM + GL) / (2(EG - F²)).
How to Use This Calculator
- Enter coefficients
a, b, c, d, e, gfor the quadratic surface. - Enter the evaluation point
(x₀, y₀). - Enter direction components
(dx, dy)for a tangent direction. - Set the graph range for the local surface window.
- Click Calculate to show results above the form.
- Review Hessian entries, form coefficients, curvatures, and classification.
- Use the CSV button for tabular export.
- Use the PDF button to save the visible results panel.
FAQs
1. What does this calculator evaluate?
It evaluates the Hessian matrix, second fundamental form, directional normal curvature, and local curvature measures for a quadratic surface written as z = f(x,y).
2. Why is the Hessian matrix important?
The Hessian stores second derivatives. It reveals local bending, convexity, saddle behavior, and how the graph changes near the chosen point.
3. What are L, M, and N?
They are the coefficients of the second fundamental form. They measure how the surface bends relative to the unit normal direction.
4. What does II(v) mean?
II(v) is the second fundamental form evaluated on a tangent direction v = (dx, dy). It measures bending along that chosen direction.
5. What does the Gaussian curvature sign indicate?
Positive values indicate elliptic behavior, negative values indicate hyperbolic behavior, and values near zero indicate parabolic or nearly flat behavior.
6. Does this work for any surface equation?
This version is designed for quadratic Monge patch surfaces. You can adapt the formulas for broader symbolic surfaces with extra parsing logic.
7. What is the difference between I(v) and II(v)?
I(v) measures tangent metric length behavior. II(v) measures normal bending behavior. Their ratio gives normal curvature for nonzero tangent directions.
8. Why include a tangent plane in the graph?
The tangent plane gives a local linear reference. Comparing it with the surface helps you visually understand curvature and second-order deviation.