# Torus

## Definition

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.

## 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.

## Parametric Equation

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°` (1a) `β = angle around the x/y-plane, 0° ≤ β < 360°` (1b)

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

 `c(α) = (R cosα, R sinα, 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 = cosβ c1 + sinβ z1` (3a) `c1 = (cosα, sinα, 0)T` (3b) `z1 = (0, 0, 1)T` (3b)

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 cosβ) cosα ` (5a) `y(α, β) = (R + r cosβ) sinα ` (5b) `z(α, β) = r sinβ ` (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
``````