Torus - Geometry

Geometry

Horn torus Spindle torus Bottom-halves and cross-sections of the three classes

A torus can be defined parametrically by:

where

θ,φ are angles which make a full circle, starting at 0 and ending at 2π, so that their values start and end at the same point,

R is the distance from the center of the tube to the center of the torus,

r is the radius of the tube.

R and r are also known as the "major radius" and "minor radius", respectively. The ratio of the two is known as the "aspect ratio". A doughnut has an aspect ratio of about 2 to 3.

An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is

or the solution of, where

Algebraically eliminating the square root gives a quartic equation,

The three different classes of standard tori correspond to the three possible relative sizes of r and R. When R > r, the surface will be the familiar ring torus. The case R = r corresponds to the horn torus, which in effect is a torus with no "hole". The case R < r describes the self-intersecting spindle torus. When R = 0, the torus degenerates to the sphere.

The surface area and interior volume of this torus are easily computed using Pappus's centroid theorem giving

These formulas are the same as for a cylinder of length 2πR and radius r, created by cutting the tube and unrolling it by straightening out 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.

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 and, angles measured from the center point. As a torus has, effectively, two center points, the centerpoints of the angles are moved; measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of 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 these terms are more commonly used to discuss magnetic confinement fusion devices.