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| A '''toric section''' is an intersection of a [[Plane (mathematics)|plane]] with a [[torus]], just as a [[conic section]] is the intersection of a [[Plane (mathematics)|plane]] with a [[cone (geometry)|cone]].
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| ==Mathematical formulae==
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| In general, toric sections are fourth-order ([[quartic curve|quartic]]) [[plane curve]]s of the form
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| :<math>
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| \left( x^2 + y^2 \right)^2 + a x^2 + b y^2 + cx + dy + e = 0.
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| </math> | |
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| ==Spiric sections==
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| A special case of a toric section is the [[spiric section]], in which the intersecting plane is parallel to the rotational symmetry axis of the [[torus]]. They were discovered by the ancient Greek geometer [[Perseus (geometer)|Perseus]] in roughly 150 BC. Well-known examples include the [[hippopede]] and the [[Cassini oval]] and their relatives, such as the [[lemniscate of Bernoulli]].
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| == Villarceau circles == | |
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| Another special case is the [[Villarceau circles]], in which the intersection is a circle despite the lack of any of the obvious sorts of symmetry that would entail a circular cross-section.
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| ==General toric sections==
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| More complicated figures such as an [[annulus (mathematics)|annulus]] can be created when the intersecting plane is [[perpendicular]] or [[:wikt:oblique|oblique]] to the rotational symmetry axis.
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| [[Category:Algebraic curves]]
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| [[Category:Toric sections]]
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| {{geometry-stub}}
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Latest revision as of 01:18, 30 October 2014
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