Mathematics
Copyright © 2018 Jiri Kriz, www.nosco.ch

Torus

Definition

Take a hollow cylinder (a tube), bend it to a ring, connect the two open ends and you get a torus:

Torus

Another construction is to revolve a circle in three dimensional space about an axis coplanar with the circle (Wikipedia). The torus is in a standard, canonical position if the circle is perpendicular to the x/y-plane and is rotated about the z-axis.

Implicit Equation

The implicit equation of the canonical torus with inner radius r and revolving radius R is:

( R - √ x2 + y2 ) 2 + z2 = r2
(1)

This formula is derived in the figures below.

Torus

Torus

Parametric Equation

Torus is a 2-dimensional surface and hence can be parametrized by 2 independent variables which are obviously the 2 angles:

Torus

The vector c from the origin O to the inner center C of the torus is:

c(α) = (R cos α, R sin α, 0)T
(2)

The vector d from the inner center C of the torus to the point A on the torus surface can be written as the sum of its orthogonal components:

d = cos β c1 + sin β z1
(3a)
c1 = (cos α, sin α, 0)T
(3b)
z1 = (0, 0, 1)T
(3c)

The vector a = (x, y, z)T from the origin to an arbitrary point A(x, y, z) on the torus surface is:

a = c + d
(4)

By substituting (2) and (3) into (4) we get the parametric equations of the torus:

x(α, β) = (R + r cos β) cos α
(5a)
y(α, β) = (R + r cos β) sin α
(5b)
x(α, β) = z(α, β) = r sin β
(5c)

The parameters α, β are usually denoted by u, v, respectively.

Octave Program

An Octave program (see also SourceForge) for drawing a torus is below (similar to Matlab):

function f = torus(r, R, numGridPoints)
    gridPoints = linspace(0, 2*pi, numGridPoints);
    [u, v] = meshgrid(gridPoints, gridPoints);
    x = (R + r * cos(v)) * cos(u);
    y = (R + r * cos(v)) * sin(u);
    z = r * sin(v);
    surf(x, y, z);
end