2. Projective transformations

A projective transformation of the plane is simply a transformation that is a homogeneous linear transformation for homogeneous coordinates. Thus a projective transformation corresponds to a $3 \times 3$ matrix $A$ so that the point whose name in homogeneous coordinates is $(x,y,s)_h$ is mapped to the point whose name in homogeneous coordinates is $(x,y,s)_h A$. In symbols,

$T(pt$   x$_h) = pt ~$   x$_h A$.

Actually, this definition of a projective transformation requires a few clarifications. First, $A$ should be nonsingular. The others can wait until Section 4 below.

Example 2.1 . Let's transform the corners of the rectangle with vertices $(1,1)$, $(-1,1)$, $(-1,0)$, $(1,0)$ using $A = \left[\begin{array}{ccc} 1&0&0\\ 0&1& \frac 12\\ 0&0&1 \end{array}\right]$. First,

$(1,1) = pt (1,1,1)_h \rightarrow pt ~ (1,1,1) \left[\begin{array}{ccc} 1&0&0\\ 0&1& {\frac 1 2}\\ 0&0&1 \end{array}\right]$ $=$ $pt (1,1, {\frac{3} {2}})_h$ $=$ $pt ( {\frac{2} {3}}, {\frac{2} {3}},1)_h$ $=$ $( {\frac{2} {3}}, {\frac{2} {3}})$.

Thus $T(1,1) = ( {\frac{2} {3}}, {\frac{2} {3}})$. The same sort of calculation gives $T(-1,1) = (- {\frac{2} {3}}, {\frac{2} {3}})$, $T(-1,0) = (-1,0)$, $T(1,0) = (1,0)$. This gives a picture somewhat like that of the football field:

book/04dir/example.1.eps