Take a hollow cylinder (a tube), bend it to a ring, connect the two open ends and you get a 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.
The implicit equation of the canonical torus with inner radius r and revolving radius R is:
(R - √ )2 + z2 = r2
This formula is derived in the figures below.
Torus is a 2-dimensional surface and hence can be parametrized by 2 independent variables which are obviously the 2 angles:
α = angle in the x/y-plane, around the z-axis, 0° ≤ α < 360°
β = angle around the x/y-plane, 0° ≤ β < 360°
The vector c from the origin O to the inner center C of the torus is:
c(α) = (R cosα, R sinα, 0)T
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 = cosβ c1 + sinβ z1
c1 = (cosα, sinα, 0)T
z1 = (0, 0, 1)T
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
By substituting (2) and (3) into (4) we get the parametric equations of the torus:
x(α, β) = (R + r cosβ) cosα
y(α, β) = (R + r cosβ) sinα
z(α, β) = r sinβ
The parameters α, β are usually denoted by u, v, respectively.
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