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| [[Image:Spiric section.svg|thumb|right|''a'' = 1, ''b'' = 2, ''c'' = 0, 0.8, 1]]
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| In [[geometry]], a '''spiric section''', sometimes called a '''spiric of Perseus''', is a quartic [[plane curve]] defined by equations of the form
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| :<math>(x^2+y^2)^2=dx^2+ey^2+f. \, </math>
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| Equivalently, spiric sections can be defined as [[Circular algebraic curve|bicircular]] quartic curves that are symmetric with respect to the ''x'' and ''y''-axes. Spiric sections are included in the family of [[toric section]]s and include the family of [[hippopede]]s and the family of [[Cassini oval]]s. The name is from σπειρα meaning torus in ancient Greek.
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| A spiric section is sometimes defined as the curve of intersection of a [[torus]] and a plane parallel to its rotational symmetry axis. However, this definition does not include all of the curves given by the previous definition unless [[Imaginary number|imaginary]] planes are allowed.
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| Spiric sections were first described by the ancient Greek geometer [[Perseus (geometer)|Perseus]] in roughly 150 BC, and are assumed to be the first toric sections to be described.
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| ==Equations==
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| Start with the usual equation for the torus:
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| :<math>(x^2+y^2+z^2+b^2-a^2)^2 = 4b^2(x^2+y^2). \, </math> | |
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| Interchanging ''y'' and ''z'' so that the axis of revolution is now on the ''xy''-plane, and setting ''z''=''c'' to find the curve of intersection gives
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| :<math>(x^2+y^2-a^2+b^2+c^2)^2 = 4b^2(x^2+c^2). \, </math>
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| In this formula, the [[torus]] is formed by rotating a circle of radius ''a'' with its center following another circle of radius ''b'' (not necessarily larger than ''a'', self-intersection is permitted). The parameter ''c'' is the distance from the intersecting plane to the axis of revolution. There are no spiric sections with ''c'' > ''b'' + ''a'', since there is no intersection; the plane is too far away from the torus to intersect it.
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| Expanding the equation gives the form seen in the definition
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| :<math>(x^2+y^2)^2=dx^2+ey^2+f \, </math>
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| where
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| :<math>d=2(a^2+b^2-c^2),\ e=2(a^2-b^2-c^2),\ f=-(a+b+c)(a+b-c)(a-b+c)(a-b-c). \, </math>
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| In [[polar coordinates]] this becomes
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| :<math>(r^2-a^2+b^2+c^2)^2 = 4b^2(r^2\cos^2\theta+c^2) \, </math>
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| or
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| :<math>r^4=dr^2\cos^2\theta+er^2\sin^2\theta+f. \, </math>
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| ==Examples of spiric sections== | |
| Examples include the [[hippopede]] and the [[Cassini oval]] and their relatives, such as the [[lemniscate of Bernoulli]]. The [[Cassini oval]] has the remarkable property that the ''product'' of distances to two foci are constant. For comparison, the sum is constant in [[ellipse]]s, the difference is constant in [[hyperbola]]e and the ratio is constant in [[circle]]s.
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| ==References==
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| *{{MathWorld|title=Spiric Section|urlname=SpiricSection}}
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| *[http://www-groups.dcs.st-and.ac.uk/~history/Curves/Spiric.html MacTutor history]
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| *[http://www.2dcurves.com/quartic/quartics.html 2Dcurves.com description]
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| *[http://www.mathcurve.com/courbes2d/spiricdeperseus/spiricdeperseus.shtml "Spirique de Persée" at Encyclopédie des Formes Mathématiques Remarquables]
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| *[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Perseus.html MacTutor biography of Perseus]
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| [[Category:Algebraic curves]]
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| [[Category:Spiric sections]]
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