Advanced Differential Geometry Calculator

Calculator Inputs

Model the surface as z = ax² + bxy + cy² + dx + ey + f. The page computes local differential geometry at point (x₀, y₀).

Coefficient a

Coefficient b

Coefficient c

Coefficient d

Coefficient e

Constant f

Point x₀

Point y₀

Plot range

Grid resolution

Reset Values

Example Data Table

Example item	Value	Meaning
a, b, c, d, e, f	0.5, 0.2, 0.4, 0.1, -0.2, 1	Quadratic graph coefficients
Point (x₀, y₀)	(1, 0.5)	Evaluation point on the parameter domain
Surface point P	(1, 0.5, 1.7)	Point on the graph surface
E, F, G	2.44, 0.48, 1.16	First fundamental form coefficients
e, f, g	0.620174, 0.124035, 0.496139	Second fundamental form coefficients
Gaussian curvature K	0.112426	Intrinsic local curvature
Mean curvature H	0.348251	Average bending measure
Principal curvatures	0.442342 and 0.254161	Extreme normal curvatures

Formula Used

Surface model: z = f(x, y) = ax² + bxy + cy² + dx + ey + f

First derivatives: f_x = 2ax + by + d, f_y = bx + 2cy + e

Second derivatives: f_xx = 2a, f_xy = b, f_yy = 2c

First fundamental form: E = 1 + f_x², F = f_xf_y, G = 1 + f_y²

Area factor: W = √(1 + f_x² + f_y²) = √(EG − F²)

Unit normal: n = (−f_x, −f_y, 1) / W

Second fundamental form: e = f_xx/W, f = f_xy/W, g = f_yy/W

Gaussian curvature: K = (eg − f²) / (EG − F²)

Mean curvature: H = (Eg − 2Ff + Ge) / 2(EG − F²)

Principal curvatures: k₁,₂ = H ± √(H² − K)

Shape operator: S = I⁻¹II

Christoffel symbols: Γ^k_ij = ½ g^kℓ(∂_ig_jℓ + ∂_jg_iℓ − ∂_ℓg_ij)

These formulas let the calculator move from local derivatives to metric quantities, curvature invariants, connection coefficients, tangent geometry, and shape classification.

How to Use This Calculator

Enter the six coefficients for the quadratic surface graph.
Choose the evaluation point coordinates x₀ and y₀.
Set the plot range and graph resolution.
Press Calculate Geometry.
Review the result panel above the form.
Study the metric tensor, curvatures, normal, shape operator, and Christoffel symbols.
Inspect the Plotly graph to see the surface and local normal.
Use CSV or PDF export to save the computed report.

FAQs

1. What does this calculator measure?

It measures local differential geometry for a quadratic surface patch. It returns first and second fundamental forms, Gaussian curvature, mean curvature, principal curvatures, a unit normal, the shape operator, and Christoffel symbols at a chosen point.

2. Why is a quadratic surface used?

A quadratic graph is flexible, smooth, and simple enough for exact local formulas. It captures bowls, saddles, tilted surfaces, and mixed bending while keeping the calculator fast and stable.

3. What is the first fundamental form?

The first fundamental form describes local metric behavior on the surface. Its coefficients E, F, and G measure lengths, angles, and area distortion on the tangent plane.

4. What is the second fundamental form?

The second fundamental form measures how the surface bends relative to its normal direction. Its coefficients e, f, and g are used directly in Gaussian and mean curvature formulas.

5. What does Gaussian curvature tell me?

Gaussian curvature identifies intrinsic local shape. Positive values indicate elliptic behavior, negative values indicate saddle behavior, and values near zero suggest flat or parabolic behavior.

6. What do principal curvatures represent?

Principal curvatures are the largest and smallest normal curvatures at the point. They describe the strongest bending directions and summarize local shape more directly than raw derivatives.

7. Why are Christoffel symbols included?

Christoffel symbols describe how tangent basis directions vary across the surface. They are useful in advanced work involving geodesics, covariant derivatives, and local coordinate behavior.

8. Can I export my results?

Yes. After calculation, the page provides CSV and PDF download buttons. The exported report includes the main geometric quantities and equations computed for the selected point.