A '''toric section''' is an intersection of a plane with a torus, just as a conic section is the intersection of a plane with a cone. Special cases have been known since antiquity, and the general case was studied by Jean Gaston Darboux.<ref name="sym">{{citation | last = Sym | first = Antoni | doi = 10.1088/1751-8113/42/40/404001 | issue = 40 | journal = Journal of Physics A: Mathematical and Theoretical | article-number = 404001 | title = Darboux's greatest love | volume = 42 | year = 2009}}.</ref>
==Mathematical formulae== In general, toric sections are fourth-order (quartic) plane curves<ref name="sym"/> of the form
:<math> \left( x^2 + y^2 \right)^2 + a x^2 + b y^2 + cx + dy + e = 0. </math>
==Spiric sections== 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 in roughly 150 BC.<ref>{{citation | last1 = Brieskorn | first1 = Egbert |author-link=Egbert Brieskorn | last2 = Knörrer | first2 = Horst |author2-link=Horst Knörrer | contribution = Origin and generation of curves | doi = 10.1007/978-3-0348-5097-1 | isbn = 3-7643-1769-8 | location = Basel | mr = 886476 | pages = 2–65 | publisher = Birkhäuser Verlag | title = Plane algebraic curves | year = 1986}}.</ref> Well-known examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli.
== Villarceau circles == 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.<ref>{{citation | last = Schoenberg | first = I. J. | issue = 4 | journal = Simon Stevin | mr = 840858 | pages = 365–372 | title = A direct approach to the Villarceau circles of a torus | volume = 59 | year = 1985}}.</ref>
==General toric sections== More complicated figures such as an annulus can be created when the intersecting plane is perpendicular or oblique to the rotational symmetry axis.
==References== {{reflist}}
== External links == * [https://www.lucamoroni.it/toric-sections/ "The toric section: intersection of a torus with a plane"] at ''"worlds of math and physics"''
Category:Toric sections Category:Quartic curves
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