MATHEMATICS LAB · GEOMETRY

Quadric Surfaces in 3D Space

Explore ellipsoids, paraboloids and hyperboloids. Rotate the surface and move a cutting plane to discover ellipses, hyperbolas, parabolas and degenerate sections.

The positive parameters a, b and c specify geometric scales. Only part of an unbounded surface is drawn. The cutting plane is parallel to a coordinate plane, and results are calculated analytically.

1. Define the surface

Type and parameters

Examples:

2. Interactive 3D graph

Rotatable orthographic view

EllipsoidThe plane highlights the cross-section of the surface.

Drag the graph to rotate the scene. Use the mouse wheel or + and − to zoom, and arrow keys to rotate the view. Move the plane to see how the intersection curve changes.

Surface Plane Section Point of tangency

3. Understanding the construction

4. Results

Live updates

5. Step-by-step calculations

Equations and numerical substitutions

What you can study

Closed and open surfaces

Ellipsoids, elliptic and hyperbolic paraboloids, and one- and two-sheet hyperboloids with their geometric parameters.

Sections by coordinate planes

Ellipses, circles, parabolas, hyperbolas, lines, points, and sections with no real intersection.

Graphs and calculations

3D rotation, comparisons across XY/XZ/YZ planes, section classification, and ellipse dimensions.

The simulator studies unrotated quadrics centered at the origin. Paraboloids have a distinguished Z axis and positive c. Surfaces are truncated only for display, not in the mathematical model. Only cutting planes parallel to the three coordinate planes are supported.