5. Geometry of the real projective plane

The real projective plane is useful in geometry, specifically projective geometry. You already know that its points are the ordinary points together with the points at infinity (which may be regarded as corresponding to families of parallel lines in the ordinary plane). Its lines consist of (a) ordinary lines, except that each ordinary line is considered to contain its corresponding point at infinity, and (b) the ``line at infinity'' consisting of all points at infinity.

With this definition, geometry in the real projective plane obeys very simple rules:

Rule 1. Every two lines meet in exactly one point.

Rule 2. Every two points lie on exactly one line.

Rule 1 says that in projective geometry there are no parallel lines. Rule 2 is the same as in ordinary plane geometry.

There is one theorem in projective geometry that shows why projective transformations are important and is useful as background in computer graphics.

The Fundamental Theorem of Real Projective Geometry: Any one-to-one function $T:$   P$_2 \rightarrow$   P$_2$ that takes lines to lines is a projective transformation.

(In other words, $T$ actually comes from some nonsingular $3 \times 3$ matrix $A$.)