Conic Sections Explorer with Equations

Plane Orientation Plane angle α (0° = horizontal, 90° = vertical): 30°

α is the angle between the plane's normal and the cone axis (vertical). Angle between plane and axis is β = 90° − α.

Plane Position Offset (moves plane up/down): 0.00 Show plane & intersection

View Rotate around Y: Rotate around X: Zoom:

Mouse

Drag on the big canvas to rotate the view around X and Y axes.

Concept: Conic Sections

A double cone is formed by rotating a line (generator) around an axis. When a plane cuts this cone, the intersection curve is a conic section:

Circle: plane ⟂ axis (horizontal slice).
Ellipse: plane cuts one nappe at an angle (not parallel to a generator).
Parabola: plane parallel to a generator line.
Hyperbola: plane cuts both nappes of the cone.

The equations shown are in a local coordinate system inside the plane of the slice, the usual way conics are studied in analytic geometry.

Conic Sections Explorer – Double Cone, Plane, and Equation

Conic Info & Equation

Concept: Conic Sections