Calculator Inputs
This page supports two approaches. Use the quadratic model for an analytic surface z = ax² + by² + cxy + dx + ey + f, or use the discrete mode for sampled grid data.
Plotly Graph
The graph updates after submission. Quadratic mode shows the full modeled surface. Discrete mode shows the sampled cross-shaped neighborhood as a small surface patch.
Example Data Table
| Mode | Key Inputs | Expected Laplacian | Interpretation |
|---|---|---|---|
| Quadratic surface | z = 1.2x² + 0.8y² + 0.4xy + 1.5x - 0.5y + 2 | 4 | Because fxx = 2.4 and fyy = 1.6, the Laplacian is 4. |
| Discrete sampled surface | zc = 10, zl = 9.6, zr = 10.8, zu = 10.9, zd = 9.7, hx = hy = 0.5 | 4 | Positive value means the center is below the nearby average. |
Formula Used
1) Quadratic height model
The analytic surface is defined as:
z(x, y) = ax2 + by2 + cxy + dx + ey + f
For this model:
fxx = 2a
fyy = 2b
fxy = c
∇2f = fxx + fyy = 2a + 2b
2) Central-difference surface Laplacian
When you only know neighboring samples, the discrete Laplacian is:
∇2f ≈ (zright - 2zcenter + zleft) / hx2
+ (zup - 2zcenter + zdown) / hy2
3) Curvature support metrics
The calculator also reports mean curvature and Gaussian curvature for the quadratic graph surface. These help describe how the surface bends locally.
The mixed derivative fxy affects curvature shape, but it does not appear directly in fxx + fyy.
How to Use This Calculator
- Choose Quadratic surface model when you know the coefficients of z(x, y).
- Choose Discrete sampled surface when you only know nearby measured surface values.
- Enter the x and y spacing carefully. These control the finite-difference scale.
- Click Calculate Surface Laplacian to place the result below the header and above the form.
- Review the summary cards, detailed table, and 3D Plotly graph.
- Use the CSV button for spreadsheets or the PDF button for printable reports.
- Compare analytic and discrete values in quadratic mode to validate step-size quality.
Frequently Asked Questions
1) What does the surface Laplacian measure?
It measures how the surface value at a point compares with nearby values. Positive values suggest a bowl-up tendency, while negative values suggest a bowl-down tendency around the chosen point.
2) Why does the mixed term cxy not appear in the Laplacian?
The Laplacian in Cartesian coordinates is fxx + fyy. The mixed derivative fxy affects local twisting and curvature behavior, but it is not part of the standard scalar Laplacian sum.
3) When should I use discrete mode?
Use discrete mode when your surface comes from measurements, image data, finite-difference grids, or simulation samples rather than a known symbolic formula.
4) What do hx and hy represent?
They represent spacing between neighboring points in the x and y directions. Smaller spacing often improves local resolution, but noisy data can still affect the estimate.
5) Why compare analytic and discrete Laplacians?
In quadratic mode, both values should match very closely. That comparison helps confirm your step sizes and demonstrates how central differences reproduce second derivatives on smooth surfaces.
6) What does a zero Laplacian mean?
A zero value means the point is locally balanced with nearby values in the chosen model or sample. It does not always mean the surface is flat everywhere.
7) Are mean curvature and Gaussian curvature required?
No. They are extra geometric indicators included for advanced analysis. They help describe bending and local shape beyond the Laplacian alone.
8) Can I use this for noisy experimental data?
Yes, but interpret results carefully. Noise can distort second differences strongly, so smoother data or preprocessing often gives more reliable surface Laplacian estimates.