Difference between revisions of "Conic section"
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| − | A '''conic section''' is any of | + | A '''conic section''' is any of the geometric figures that can arise when a [[plane]] [[intersect]]s a [[cone]]. (In fact, one usually considers a "two-ended cone," that is, two [[congruent]] right circular cones placed tip to tip so that their axes align.) As is clear from their definition, the conic sections are all [[plane curve]]s, and every conic section can be described in [[Cartesian coordinates]] by a [[polynomial]] [[equation]] of degree two or less. |
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| + | == Classification of conic sections == | ||
| + | All conic sections fall into the following categories: | ||
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| + | === Nondegenerate conic sections === | ||
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| + | * A [[circle]] is the conic section formed when the cutting plane is [[parallel]] to the [[base (geometry) | base]] of the cone or equivalently [[perpendicular]] to the axis. | ||
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| + | * An [[ellipse]] is formed if the cutting plane makes an [[angle]] with the axis that is larger than the angle between the [[cone element | element of the cone]] and the axis. | ||
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| + | * A [[parabola]] is formed when the cutting plane makes an angle with the axis that is equal to the angle between the element of the cone and the axis. | ||
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| + | * An [[hyperbola]] is formed when the cutting plane makes an angle with the axis that is smaller than the angle between the element of the cone and the axis. | ||
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| + | === Degenerate conic sections === | ||
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| + | If the cutting plane passes through the vertex of the cone, the result is a degenerate conic section. | ||
== See Also == | == See Also == | ||
Revision as of 10:48, 27 May 2008
A conic section is any of the geometric figures that can arise when a plane intersects a cone. (In fact, one usually considers a "two-ended cone," that is, two congruent right circular cones placed tip to tip so that their axes align.) As is clear from their definition, the conic sections are all plane curves, and every conic section can be described in Cartesian coordinates by a polynomial equation of degree two or less.
Contents
Classification of conic sections
All conic sections fall into the following categories:
Nondegenerate conic sections
- A circle is the conic section formed when the cutting plane is parallel to the base of the cone or equivalently perpendicular to the axis.
- An ellipse is formed if the cutting plane makes an angle with the axis that is larger than the angle between the element of the cone and the axis.
- A parabola is formed when the cutting plane makes an angle with the axis that is equal to the angle between the element of the cone and the axis.
- An hyperbola is formed when the cutting plane makes an angle with the axis that is smaller than the angle between the element of the cone and the axis.
Degenerate conic sections
If the cutting plane passes through the vertex of the cone, the result is a degenerate conic section.
