When constructing a parametric cubic spline in three dimensions, I get three splines $x(h)$, $y(h)$ and $z(h)$. When calculating the parameter $h$ I would intuitively use the $l_2$ norm between each successive point.
I'm making some modifications in a legacy application where $h$ is calculated as the $l_1$ norm between each successive point. When changing this to $l_2$ norm (for simpler calculation of tangents and because it feels like the right thing to do) I got very different shapes of the splines.
Can one make a general statement about how this choice affects the shape of the spline, or which one is more correct?