Cross Ratios and Length Ratios:

In $(\lambda, \mu)$ coordinates on the line MN, N is represented by (0, 1) and serves as the ``origin'' and M is represented by (1, 0) and serves as the ``point at infinity''. For an arbitrary point $A=(\lambda,\mu)\not\sim(1, 0)$, we can rescale $(\lambda, \mu)$ to $\mu=1$, and represent A by its ``affine coordinates'' $(\lambda, 1)$, or just $\lambda$ for short. Since we have mapped M to infinity, this is just linear distance along the line from N. Hence, setting $\mu_i=1$ in 3.1, the cross ratio becomes a ratio of length ratios. The ancient Greek mathematicians already used cross ratios in this form.

Exercise 3.2   : Let D be the point at infinity on the projective line, and let A, B, C be three finite points. Show that

$$\{A, B ; C, D\} = \frac{\overline{AC}}{\overline{BC}}$$

Exercise 3.3   If MN is the line between two points M and N in the projective plane, use the $(\lambda, \mu)$ parameterization and the results of section 2.2.1 to show that the cross ratio of four points A,B,C,D on the line is

$$\{A, B ; C, D\} = \frac{(A\times C)\cdot(B\times D)} {(A\times D)\cdot(B\times C)}$$