The fundamental theorems

In the previous section we saw that the action of $PGL(2,{\Bbb R})$ on the projective line is transitive. In fact, we can say much more than that:

Theorem 2.1 (First fundamental theorem)   If $x,y,z$ and $x',y',z'$ are two triples of distinct points on the projective line, there exits a unique projective transformation that sends $x$ to $x'$, $y$ to $y'$ and $z$ to $z'$.

The proof hinges on the following exercise:

Exercise 2.1 (10)   Let $L_{1},L_{2}$, and $L_{3}$ be three distinct lines throught the origin in ${\Bbb R}^2$. Show that it is possible to find a basis $\{v_{1},v_{2}\}$ of ${\Bbb R}^2$ such that $v_{1}$ lies on $L_{1}$, $v_{2}$ lies on $L_{2}$ and $v_{3} := v_{1} + v_{2}$ lies on $L_{3}$. Moreover, show that such a basis is uniquely defined up to multiples.

Exercise 2.2 (05)   Prove the first fundamental theorem.

Exercise 2.3 (10)   In this exercise, we sketch a computational proof of the first fundamental theorem.

1. Show that there exists a unique transformation of the form
   $$y \longmapsto \frac{c + dy}{a + by}$$
   that takes a triple of numbers $y_{0}, y_{1}, y_{2}$, representing the slopes of three distinct lines, to $\infty, 0, 1$. Hint: if $y_{0}$ is taken to infinity, then the transformation must be of the form $y \mapsto (c + dy)/b(y - y_{0})$.
2. Write the formula for the projective transformation that takes $\infty, 0, 1$ to $y_{0}, y_{1}, y_{2}$ (i.e., the inverse of the above transformation).
3. Write the formula for the transformation that takes $y_{0}, y_{1}, y_{2}$ to $y'_{0},y'_{1},y'_{2}$.

Exercise 2.4   Explain how the following figure constitutes a third proof of the first fundamental theorem (uniqueness excepted).

Having seen that any triple of distinct points can be mapped onto any other triple of distict points by a unique projective transformation, we cannot expect the same to hold for quadruples of points.

Definition 2.1   Let $\ell_{0},\dots,\ell_{3}$ be a quadruple of points on the projective line with $\ell_{0},\ell_{1}$, and $\ell_{2}$ different from each other, and let $T$ be the unique projective transformation taking $\ell_{0},\ell_{1}$, and $\ell_{2}$ respectively onto the lines of slope $\infty, 0$, and $1$. The cross-ratio $[\ell_{0},\ell_{1}, \ell_{2},\ell_{3}]$ is defined as the point $T(\ell_{3}) \in {\Bbb R}P^1$.

Proposition 2.1   Show that if $S$ is a projective transformation and $\ell_{0},\dots,\ell_{3}$ is a quadruple of points on the projective line with $\ell_{0},\ell_{1}$, and $\ell_{2}$ different from each other, then the cross-ratio $[\ell_{0},\ell_{1}, \ell_{2},\ell_{3}]$ equals $[S(\ell_{0}), S(\ell_{1}), S(\ell_{2}),S(\ell_{3})]$.

Exercise 2.5 (10)   Prove the above result.

Exercise 2.6 (05)   By indentifying points in the projective line with the slopes of the corresponding lines, verify that the cross-ratio of four numbers $y_{0},\dots,y_{3}$ can be written as:

$$[y_{0},y_{1},y_{2},y_{3}] = \frac{y_{2} - y_{0}}{y_{2} - y_{1}} \frac{y_{3} - y_{1}}{y_{3} - y_{0}}$$

Exercise 2.7 (05)   Use the preceding exercise to give a purely computational proof of the invariance of the cross-ratio.

Exercise 2.8 (05)   A projective transformation sends 0 to $1$, $1$ to $2$, and $2$ to 0. Where does it send $3$?

We are now ready to state an important characterization of projective transformations of the real projective line.

Theorem 2.2 (Second fundamental theorem)   A map from the projective line to itself is a projective transformation if and only if it preserves cross-ratios.

The proof is given in the following exercises.

Exercise 2.9 (05)   Show that a map from the projective line to itself that preserves cross-ratios and that fixes the lines with slope $\infty, 0$, and $1$ is the identity map.

Exercise 2.10 (05)   Prove theorem 2.2.

Exercise 2.11 (05)   The aim of this exercise is to study the effect of permuting the arguments in the computation of the cross-ratio.

1. Show that $[a,b,c,d] = [b,a,c,d]^{-1} = [a,b,d,c]^{-1}$ and $[a,b,c,d] + [a,c,b,d] = 1$.
2. Use part 1 to show that if $[a,b,c,d] = k$, then the only possible values that can be obtained by permuting $a,b,c$, and $d$ are: $k, k^{-1}, 1 - k, 1 - k^{-1}, (1 - k)^{-1}$, and $k/(k - 1)$.