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What is the cross-ratio of four lines $L_1,L_2,L_3,L_4$ if they are parallel? What is the cross-ratio if $L_4$ is the line at infinity?

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I know what the cross ratio of points. What is a cross ratio of lines? –  William Jun 27 '12 at 16:05

3 Answers 3

The fact of being parallel is an affine notion. If you projectivize your affine plane, then this notion is indeed restated as intersecting at infinity. However the cross-ratio is a purely projective notion, it doesn't matter if your projective plane comes from (is the projectivization of) an affine plane or not. Therefore the cross-ratio of four lines crossing at infinity has nothing special compared to the cross-ratio of four lines crossing somewhere else.

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You obtain the cross ratio of four lines $\ell_i$ going through the same point $M$ by intersecting these lines with any line $g$ not going through $M$. In this way you obtain $4$ points $A_i$ on $g$, and the cross-ratio of these four points is the cross-ratio of the $\ell_i$.

Your four lines go through the same point $M$ on the line at infinity, so their cross-ratio makes sense. When they are given, e.g., in the form $$\ell_i:\quad y=mx + a_i\qquad(1\leq i\leq4)\ ,$$ you can take the $y$-axis as your $g$. In this way the numbers $a_i$ can be viewed as admissible (i.e., projective) coordinates of the $4$ points $A_i$. Therefore the cross-ratio $C$ of the $\ell_i$ is the cross-ratio $C$ of the $a_i$.

In order to obtain the correct value when $\ell_4$ is the line at infinity, let $a_4\to\infty$ in the expression $C$.

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