stylistic:style_transfer
Geometry

A torus can be parametrized as: x (θ, φ) = (R + r cos θ) cos φ y (θ, φ) = (R + r cos θ) sin φ z (θ, φ) = r sin θ {\displaystyle {\begin{aligned}x(\theta,\varphi)&=(R+r\cos \theta)\cos {\varphi }\\y(\theta,\varphi)&=(R+r\cos \theta)\sin {\varphi }\\z(\theta,\varphi)&=r\sin \theta \\\end{aligned}}}

using angular coordinates θ, φ ∈ [ 0, 2 π), {\displaystyle \theta,\varphi \in [0,2\pi),} representing rotation around the tube and rotation around the torus' axis of revolution, respectively, where the major radius R {\displaystyle R} is the distance from the center of the tube to the center of the torus and the minor radius r {\displaystyle r} is the radius of the tube.

The ratio R / r {\displaystyle 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 equation in Cartesian coordinates for a torus radially symmetric about the z {\displaystyle z} - axis is (x 2 + y 2 − R) 2 + z 2 = r 2. {\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr)}^{2}}+z^{2}=r^{2}.}

Algebraically eliminating the square root gives a quartic equation, (x 2 + y 2 + z 2 + R 2 − r 2) 2 = 4 R 2 (x 2 + y 2). {\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right).}

The three classes of standard tori correspond to the three possible aspect ratios between R and r:

When R ≥ r, the interior (x 2 + y 2 − R) 2 + z 2 < r 2 {\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr)}^{2}}+z^{2}<r^{2}} of this torus is diffeomorphic (and, hence, homeomorphic) to a product of a 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: A = (2 π r) (2 π R) = 4 π 2 R r, V = (π r 2) (2 π R) = 2 π 2 R r 2. {\displaystyle {\begin{aligned}A&=\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr,\\V&=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}.\end{aligned}}}

These formulas are the same as for a cylinder of length 2π R and radius 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 p of an outermost point on the surface of the torus to the center, and the distance q of an innermost point to the center (so that R = p + q / 2 and r = p − q / 2), yields A = 4 π 2 (p + q 2) (p − q 2) = π 2 (p + q) (p − q), V = 2 π 2 (p + q 2) (p − q 2) 2 = 1 4 π 2 (p + q) (p − q) 2. {\displaystyle {\begin{aligned}A&=4\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)=\pi ^{2}(p+q)(p-q),\\V&=2\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)^{2}={\tfrac {1}{4}}\pi ^{2}(p+q)(p-q)^{2}.\end{aligned}}}

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, R, the distance from the center of the coordinate system, and θ and φ, angles measured from the center point.

As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of 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".

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