Relation between affine parameters

[Identity]

In Euclidean three-space with coordinates $(x,y,z)$ and line element

$$ds^2 = dx^2+dy^2-dz^2$$

It is easy to show using the geodesic equation that:

$x = lu+l'$, $y=mu+m'$, $z=nu+n'$

where $u$ is an affine parameter.

However, is it possible to find a relation between $l,m,n$?

 

[pervect]

Shouldn't l,m, and n be analogous to a four-velocity in Minowskian space, and hence have a constant norm?