Homogeneous Coordinates

by William Shoaff with lots of help

Euclidean space can be extended into projective space by the introduction of a homogeneous coordinate. Given a finite point (number) (x) in one dimension:

$\begin{picture}(144,20) \put(72,10){\vector(1,0){72}} \put(72,10){\vector(-1,0){... ...*{2}} \put(72,0){$O$ } \put(108,10){\circle*{2}} \put(108,0){$x$ } \end{picture}$

(wx, w), $w\neq 0$ are the homogeneous coordinates for (x). Notice that we can think of the homogeneous coordinates as a punctured line is a two dimensional projective space:

$\begin{picture}(144,144) \put(72,70){\vector(1,0){72}} \put(72,70){\vector(0,1){... ...2){$O$ } \put(74,71){\line(2,1){72}} \put(70,69){\line(-2,-1){72}} \end{picture}$

For example if x=2 we get the line $(2w,w),\,-\infty < w < 0,\,0 < w < \infty$ .

If $(a,\,b)$ are homogeneous coordinates, they represent the point (number)

$\begin{displaymath}\frac{a}{b} = x.\end{displaymath}$

Homogeneous coordinates of the form (x, 0) describe the point at infinity in projective space. This might make sense if you consider what happens to x as $b\rightarrow 0$ while a remains fixed in the expression

$\begin{displaymath}\frac{a}{b}.\end{displaymath}$

In two dimensions a point (x, y) has homogeneous coordinates $(wx,\,wy,\,w),\,w\neq0$ . This is again a punctured line in three dimensions. Homogeneous coordinates of the form $(a,\,b,\,c)$ represent the two dimensional point $(x,\,y)$ where

$\begin{displaymath}x = \frac{a}{c},\quad y = \frac{b}{c}, \quad c\neq0.\end{displaymath}$

Coordinates $(a,\,b,\,0)$ for which

$\begin{displaymath}\frac{a}{b} = \lambda\end{displaymath}$

describe the point at infinity along the line with slope $\lambda$ .

In three dimensions a point $(x,\, y,\,z)$ has homogeneous coordinates $(wx,\,wy,\,wz,\,w),\,w\neq0$ . Once again, this is a punctured line, but now it is in four dimensions. Homogeneous coordinates of the form $(a,\,b,\,c,\,d)$ represent the three dimensional point $(x,\, y,\,z)$ where

$\begin{displaymath}x = \frac{a}{d},\quad y = \frac{b}{d}, \quad z = frac{c}{d}, \quad d\neq0.\end{displaymath}$

Coordinates $(a,\,b,\,c,\,0)$ describe the point at infinity.

What's in a name?

The term homogeneous comes from writing expressions and equations in a common (homogeneous) format, for example, we can write the polynomial

p(x) = 3x²+5x-4

in homogeneous form

p(x) = 3x²+5x¹-4x⁰,

or we can write the linear equation

3x+5y-4=0

in homogeneous form

$\begin{displaymath}3wx+5wy-4w=0,\,w\neq 0.\end{displaymath}$

Where are homogeneous coordinates used?

We'll use homogeneous coordinates to write translations using matrix notation, that is, the translation

$\begin{displaymath}(x,\,y) \rightarrow (x+t_x,\,y+t_y)\end{displaymath}$

can be written as

$\begin{displaymath}\left(\begin{array}{ccc}x & y & 1 \end{array}\right) \left[\... ... \left(\begin{array}{ccc}x +t_x & y+t_y & 1 \end{array}\right).\end{displaymath}$

We'll also use homogeneous coordinates to represent prespective transforms and they are used to represent ``pre-multiplied'' colors

$\begin{displaymath}(\alpha r, \,\alpha g, \, \alpha b,\, \alpha).\end{displaymath}$

No References!

William D. Shoaff
2000-09-11