MATHEMATICS LAB · GEOMETRY

Differential Geometry of Surfaces

Move a point along the surface to study tangent planes, normals, principal curvatures, Gaussian curvature and mean curvature, with interactive constructions and explained calculations.

Coordinates use arbitrary units (u). Input angles use degrees, but derivatives use radians. The sign of curvature depends on normal orientation: this tool uses n = (rᵤ × rᵥ)/|rᵤ × rᵥ| e II = −n · d²r. At umbilic points the principal directions are not unique.

2. Interactive 3D construction

Rotatable orthographic view

Movable point and curvatures.

Drag to rotate the surface; use the mouse wheel or +/− to zoom. Change u and v, or press Animate P: the graph automatically scrolls into view.

Surface Tangent plane Normal n Direction k₁ Direction k₂

3. Understanding the construction

4. Results

Live updates

5. Step-by-step calculations

First and second fundamental forms

What you can explore

Tangent plane and normal

Partial derivatives, tangent vectors, cross products and tangent planes to regular surfaces.

Fundamental forms

Surface metric, second fundamental form, area element and curvature of normal sections.

Surface curvature

Principal, Gaussian and mean curvature; elliptic, hyperbolic and parabolic points; minimal surfaces.

Curvatures depend on the point. The sign of H and principal curvatures depends on the chosen normal, while K does not. At umbilic points no two principal directions are preferred. Open surfaces are shown only over bounded domains.