∑ Mathematics LabParametric Surfaces · v4.0.4

MATHEMATICS LAB · GEOMETRY

Parametric Surfaces in 3D Space

Adjust the surface parameters, explore its generating curves and cross-sections, and learn how to calculate surface areas and volumes.

Coordinates use arbitrary units (u). Open surfaces are displayed over finite parameter ranges: the lateral surface does not include any closing disks. Sections are parallel to the XY plane.

2. Interactive 3D graph

Rotatable orthographic view
The graph shows the surface and its horizontal cross-section.
This graph requires Canvas 2D. Results are still available below.

Drag the graph to rotate the view; use the mouse wheel or + and − to zoom. Adjust u and v to move P across the surface. Numerical results update with the graph.

Surface Plane Cross-section P and tangents

3. How to read the construction

4. Results

Live updates

5. Step-by-step calculations

Parametrizations and numerical substitutions

What you can study

Parametrizations

Each point of the surface is determined by two independent parameters u and v spanning a planar domain.

Ruled surfaces and surfaces of revolution

Compare translating a directrix curve with rotating a profile around an axis.

Sections and local properties

Examine level curves, tangent vectors, the normal vector and the local area element given by the cross product.

Open surfaces do not enclose a solid by themselves; reported volumes assume the explicitly described closures. The displayed surface is a geometric discretization.