Laguerre formula

The Laguerre formula (named after Edmond Laguerre) provides the acute angle formula_1 between two proper real lines, as follows:
where:
The expression between vertical bars is a real number.
Laguerre formula can be useful in computer vision, since the absolute conic has an image on the retinal plane which is invariant under camera displacements, and the cross ratio of four collinear points is the same for their images on the retinal plane.
Derivation.
It may be assumed that the lines go through the origin. Any isometry leaves the absolute conic invariant, this allows to take as the first line the "x" axis and the second line lying in the plane "z"=0. The homogeneous coordinates of the above four points are
respectively. Their nonhomogeneous coordinates on the infinity line of the plane "z"=0 are formula_13, formula_14, 0, formula_15. (Exchanging formula_7 and formula_8 changes the cross ratio into its inverse, so the formula for formula_1 gives the same result.) Now from the formula of the cross ratio we have
formula_19