Relation between affine parameters

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