Relation between affine parameters

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The discussion focuses on the relationship between affine parameters in Euclidean three-space, specifically using the line element ds² = dx² + dy² - dz². It establishes that the coordinates can be expressed as x = lu + l', y = mu + m', and z = nu + n', where u represents an affine parameter. The conversation raises the question of whether l, m, and n should maintain a constant norm, similar to a four-velocity in Minkowskian space, suggesting a deeper connection between these parameters.

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In Euclidean three-space with coordinates [itex](x,y,z)[/itex] and line element

[tex]ds^2 = dx^2+dy^2-dz^2[/tex]

It is easy to show using the geodesic equation that:

[itex]x = lu+l'[/itex], [itex]y=mu+m'[/itex], [itex]z=nu+n'[/itex]

where [itex]u[/itex] is an affine parameter.

However, is it possible to find a relation between [itex]l,m,n[/itex]?
 
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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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