Conic section

Article By:

Blumenthal, Leonard M. Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.

Last reviewed:2014

DOI:https://doi.org/10.1036/1097-8542.156800

One of the class of curves in which a plane may cut a cone (surface) of revolution. They were extensively studied by the ancient Greeks. The section is a parabola if the plane is parallel to an element of the cone, an ellipse or circle if the plane cuts all elements of one nappe (but does not go through the apex), and a hyperbola if the plane cuts elements of both nappes (for example, the plane parallel to the cone's axis of revolution) and does not go through the apex (see illustration) If a line g intersects the cone's axis perpendicularly at a point distinct from the apex, a plane that revolves about g will cut all three kinds of conics from the cone. If a plane contains the cone's axis, it intersects the cone in a pair of lines (a degenerate hyperbola); if it goes through the apex but does not contain the axis, it has in common with the cone either the apex alone (degenerate ellipse) or an element of the cone (degenerate parabola).

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