Projective geometry

Projective geometry is a beautiful area of geometry which focuses on properties that are invariant under projective transformations. Unlike Euclidean geometry, there is no notion of length; rather, any configuration of four collinear points or four concurrent lines is characterized by the cross-ratio. A particularly striking result of projective geometry is the duality of points and lines.

The first theorems of a projective character were established by the ancient Greeks, most notably Pappus of Alexandria.

Projective Transformations

Given any two planes $M$ and $N$ and a point $P$ , we can define a projective correspondence $\phi(X): M \rightarrow N$ as follows: for any point $X$ on $M$ , $\phi(X)$ is the intersection of line $PX$ with $N$ .

Note that $\phi(X)$ appears to be undefined when $PX$ is parallel to $N$ . This issue will be addressed in the next section.

The Projective Plane

A key axiom of projective geometry is that any two lines meet in exactly one point, and through any two points there passes exactly one line. The first part of this axiom breaks down in Euclidean geometry with parallel lines.

This problem is remedied by adding a "point at infinity" for each family of parallel lines. Now through any two points at infinity, there must be exactly one line passing through them, so we add a "line at infinity." Finally, if we have a point at infinity and a regular point, the parallel postulate assures that there is exactly one line passing through both points.

The Cross-Ratio

Suppose we are given four points $A, B, C, D$ lying on a line. We define the cross-ratio $(A, B; C, D) = \frac{(AC)(BD)}{(AD)(BC)},$ where directed lengths are used on the right side.

Now suppose we have four lines $a, b, c, d$ passing through a point $O$ , and any fifth line $m$ not passing through $O$ . Let $A$ be the intersection of $m$ with $a$ , let $B$ be the intersection of $m$ with $b$ , and so on. Then we define the cross-ratio of the four lines, $(a, b; c, d)$ , as the cross-ratio of the four points, $(A, B; C, D)$ . To check that this cross-ratio is well-defined, note that by the area formula $K = \frac{1}{2}ab\sin\theta$ we obtain $\begin{align*} (A, B; C, D) &= \frac{(AC)(BD)}{(AD)(BC)} \\ &= \frac{\sin(\angle{AOC})\sin(\angle{BOD})}{\sin(\angle{AOD})\sin(\angle{BOC})}, \end{align*}$ which is clearly independent of the transversal $m$ .

The most important property of the cross-ratio is that it is preserved under projections.

Also, a set of four points or four lines are said to be harmonic if their cross-ratio is $-1$ . The complete quadrilateral illustrates the importance of harmonic points and lines:

Let $EFGI$ be a quadrilateral. Sides $EI$ and $FG$ intersect at $A$ , and sides $EF$ and $IG$ intersect at $B$ . If $EG$ and $IF$ intersect line $AB$ at $C$ and $D$ , respectively, then $(A, B; C, D) = -1.$