Editing Torus (section)

== Geometry ==

{{Multiple image
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|direction=vertical
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|header=Bottom-halves and<br />vertical cross-sections
|image1=Standard_torus-ring.png
|alt1=ring
|caption1={{math|''R'' > ''r''}}: ring torus or anchor ring
|image2=Standard_torus-horn.png
|alt2=horn
|caption2={{math|1=''R''=''r''}}: horn torus
|image3=Standard_torus-spindle.png
|alt3=spindle
|caption3={{math|''R'' < ''r''}}: self-intersecting spindle torus
|footer =
}}
[[File:Toroidal coord.png|thumb|Poloidal direction (red arrow) and toroidal direction (blue arrow)]]

A torus of revolution in 3-space can be [[parametric equation|parametrized]] as:<ref>{{cite web |url=http://www.geom.uiuc.edu/zoo/toptype/torus/standard/eqns.html |title=Equations for the Standard Torus |publisher=Geom.uiuc.edu |date=6 July 1995 |access-date=21 July 2012 |url-status=live |archive-url=https://web.archive.org/web/20120429011957/http://www.geom.uiuc.edu/zoo/toptype/torus/standard/eqns.html |archive-date=29 April 2012}}</ref>
<math display="block">\begin{align}
x(\theta, \varphi) &= (R + r \sin \theta) \cos{\varphi}\\
y(\theta, \varphi) &= (R + r \sin \theta) \sin{\varphi}\\
z(\theta, \varphi) &= r \cos \theta\\
\end{align}</math>
using angular coordinates {{math|''θ''}}, {{math|''φ'' ∈ [0, 2π)}}, representing rotation around the tube and rotation around the torus's axis of revolution, respectively, where the ''major radius'' {{math|''R''}} is the distance from the center of the tube to the center of the torus and the ''minor radius'' {{math|''r''}} is the radius of the tube.<ref>{{cite web| title=Torus | url=http://doc.spatial.com/index.php/Torus|publisher=Spatial Corp. | access-date=16 November 2014|url-status=live | archive-url=https://web.archive.org/web/20141213210422/http://doc.spatial.com/index.php/Torus |archive-date=13 December 2014}}</ref>

The ratio {{math|''R''/''r''}} is called the ''[[aspect ratio]]'' of the torus. The typical doughnut confectionery has an aspect ratio of about 3 to 2.

An [[implicit function|implicit]] equation in [[Cartesian coordinates]] for a torus radially symmetric about the z-[[coordinate axis|axis]] is
<math display="block">{\textstyle \bigl(\sqrt{x^2 + y^2} - R\bigr)^2} + z^2 = r^2.</math>

Algebraically eliminating the [[square root]] gives a [[quartic equation]],
<math display="block">\left(x^2 + y^2 + z^2 + R^2 - r^2\right)^2 = 4R^2\left(x^2+y^2\right).</math>

The three classes of standard tori correspond to the three possible aspect ratios between {{mvar|R}} and {{mvar|r}}:
* When {{math|''R'' > ''r''}}, the surface will be the familiar ring torus or anchor ring.
* {{math|1=''R'' = ''r''}} corresponds to the horn torus, which in effect is a torus with no "hole".
* {{math|''R'' < ''r''}} describes the self-intersecting spindle torus; its inner shell is a ''[[lemon (geometry)|lemon]]'' and its outer shell is an ''[[apple (geometry)|apple]]''.
* When {{math|1=''R'' = 0}}, the torus degenerates to the sphere radius {{math|''r''}}.
* When {{math|1=''r'' = 0}}, the torus degenerates to the circle radius {{math|''R''}}.

When {{math|''R'' ≥ ''r''}}, the [[interior (topology)|interior]]
<math display="block">{\textstyle \bigl(\sqrt{x^2 + y^2} - R\bigr)^2} + z^2 < r^2</math>
of this torus is [[diffeomorphism|diffeomorphic]] (and, hence, homeomorphic) to a [[Cartesian product|product]] of a [[disk (geometry)|Euclidean open disk]] and a circle. The [[volume]] of this solid torus and the [[surface area]] of its torus are easily computed using [[Pappus's centroid theorem]], giving:<ref>{{MathWorld|Torus|Torus}}</ref>
<math display="block">\begin{align}
A &= \left( 2\pi r \right) \left(2 \pi R \right) = 4 \pi^2 R r, \\[5mu]
V &= \left ( \pi r^2 \right ) \left( 2 \pi R \right) = 2 \pi^2 R r^2.
\end{align}</math>

These formulae are the same as for a cylinder of length {{math|2π''R''}} and radius {{mvar|r}}, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.

Expressing the surface area and the volume by the distance {{mvar|p}} of an outermost point on the surface of the torus to the center, and the distance {{mvar|q}} of an innermost point to the center (so that {{math|1=''R'' = {{sfrac|''p'' + ''q''|2}}}} and {{math|1=''r'' = {{sfrac|''p'' − ''q''|2}}}}), yields
<math display="block">\begin{align}
A &= 4 \pi^2 \left(\frac{p+q}{2}\right) \left(\frac{p-q}{2}\right) = \pi^2 (p+q) (p-q), \\[5mu]
V &= 2 \pi^2 \left(\frac{p+q}{2}\right) \left(\frac{p-q}{2}\right)^2 = \tfrac14 \pi^2 (p+q) (p-q)^2.
\end{align}</math>

As a torus is the product of two circles, a modified version of the [[spherical coordinate system]] is sometimes used.
In traditional spherical coordinates there are three measures, {{mvar|R}}, the distance from the center of the coordinate system, and {{mvar|θ}} and {{mvar|φ}}, angles measured from the center point.

As a torus has, effectively, two center points, the centerpoints of the angles are moved; {{mvar|φ}} measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of {{mvar|θ}} is moved to the center of {{mvar|r}}, and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles".<ref>{{cite web |url=http://dictionary.oed.com/cgi/entry/50183023?single=1&query_type=word&queryword=poloidal&first=1&max_to_show=10 |work=Oxford English Dictionary Online |access-date=10 August 2007 |title=poloidal |publisher=Oxford University Press}}</ref>

In modern use, [[toroidal and poloidal]] are more commonly used to discuss [[magnetic confinement fusion]] devices.