A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

28 Linear Algebra?

Figure 2.16: Homogeneous coordinates map from R

n

to R

n+1

.

a linear operator! How does it work? First, let us define what

homogeneous coordinates are.

Homogeneous coordinates map from R

n

to R

n+1

. The best

understood example is in 2D, shown in Figure 2.16. The 2D

plane (x, y) is mapped to a parallel plane in 3D, (X, Y, W ); for

convenience, we choose W = 1. Instead of applying the trans-

formations over (x, y, 0), we apply them over (X, Y, W ). Then,

the transformed point is mapped back by

(X, Y, W ) 7→ ((X/W ), (Y /W )) .

The trick is that translating points in the plane (X, Y, 1) never

translates the origin of R

3

, and the operator that