# Toric section

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Toric_section
> Markdown URL: https://mediated.wiki/source/Toric_section.md
> Source: https://en.wikipedia.org/wiki/Toric_section
> Source revision: 1348687425
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

A '''toric section''' is an intersection of a [plane](/source/Plane_(mathematics)) with a [torus](/source/torus), just as a [conic section](/source/conic_section) is the intersection of a [plane](/source/Plane_(mathematics)) with a [cone](/source/cone_(geometry)). Special cases have been known since antiquity, and the general case was studied by [Jean Gaston Darboux](/source/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](/source/quartic_curve)) [plane curve](/source/plane_curve)s<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](/source/spiric_section), in which the intersecting plane is parallel to the rotational symmetry axis of the [torus](/source/torus).  They were discovered by the ancient Greek  geometer [Perseus](/source/Perseus_(geometer)) 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](/source/hippopede) and the [Cassini oval](/source/Cassini_oval) and their relatives, such as the [lemniscate of Bernoulli](/source/lemniscate_of_Bernoulli).

== Villarceau circles ==
Another special case is the [Villarceau circles](/source/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](/source/annulus_(mathematics)) can be created when the intersecting plane is [perpendicular](/source/perpendicular) or [oblique](/source/%3Awikt%3Aoblique) 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

{{geometry-stub}}

---
Adapted from the Wikipedia article [Toric section](https://en.wikipedia.org/wiki/Toric_section) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Toric_section?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.