4. The real projective plane

This last observation means that a projective transformation is not just a function on R$^2 \rightarrow$   R$^ 2$. After all, the function notation $f: A \rightarrow B$ is supposed to mean that one is considering the domain of $f$ to be $A$ and that all values of $f$ are in $B$. The following definition is therefore handy:

Definition. The (real) projective plane, denoted P$_2$, is the set of all ordinary points and points at infinity.

To summarize:

Fact 1. The projective plane P$_2$ has two kinds of points: ordinary points and points at infinity.

Fact 2. Each point of the projective plane P$_2$ can be represented by homogeneous coordinates, in many ways.

Fact 3. Every triple $(a,b,c)_h$, except $(0,0,0)$, represents a point of P$_2$. If $c \neq 0$ then the point is the ordinary point $( {\frac a c}, {\frac b c})$; if $c = 0$ then the point is a point at infinity in the direction given by the direction vector $(a,b)$.

Fact 4. If $(a,b,c)_h$ is one name for a point, its other names have the form $(ra, rb, rc)_h$, where $r$ is any nonzero scalar.

Fact 5. A projective transformation of the projective plane is a transformation $T:$   P$_2 \rightarrow$   P$_2$ that is a nonsingular homogeneous linear transformation in homogeneous coordinates. A projective transformation has the form $T(pt$   x$_h) = pt ~$   x$_h A$, where $A$ is a nonsingular $3 \times 3$ matrix.