# Conic Sections: The Double Cone

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The quadratic curves are circles, ellipses, parabolas, and hyperbolas. They are called *conic sections* because each one is the intersection of a double cone and an inclined plane.
If the plane is perpendicular to the cone's axis, the intersection is a circle. If it is inclined at an angle greater than zero but less than the half-angle of the cone, it is an (eccentric) ellipse. If the plane's inclination is equal to this half-angle, the intersection is a parabola. If it exceeds the half-angle, it is a hyperbola.

Contributed by: Phil Ramsden (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Mathematical note: If the half-angle of the cone is then the eccentricity of the intersection is , where is the angle of inclination of the plane. Its projection in the plane is also a conic section, with focus at the origin and eccentricity . Note that the eccentricity of the intersection and of its projection are equal for precisely two values of , namely 0 and , corresponding to eccentricities of 0 and 1, and thus to the circle and parabola, respectively.

Snapshot 1: circle

Snapshot 2: ellipse

Snapshot 3: parabola

Snapshot 4: hyperbola

## Permanent Citation

"Conic Sections: The Double Cone"

http://demonstrations.wolfram.com/ConicSectionsTheDoubleCone/

Wolfram Demonstrations Project

Published: March 7 2011