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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.

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When the plane passes through the apex of the cone, the intersection is a point, one line, or a double line.

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Contributed by: Phil Ramsden (March 2011)
Open content licensed under CC BY-NC-SA

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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

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