Let $v:\mathbb{R} \rightarrow \mathbb{R}^3/{(0,0,0)} $ be a $C^\infty$ regular arc-length parametrization of a space curve.

W.l.o.g. let us assume $v(0)=(1,0,0)$, $v'(0)=(1,0,0)$. Let $\bar{v}$ be the projection of the curve on the first coordinate plane $x=1$ and $\hat{v}:=v/||v||$ its normalization. This provides non-regular $C^\infty$ parametrizations of two well defined geometrical objects, which in general will not be more than continuous (link to pictures below).

singularities may arise

singularities may not arise

What is a lean way of assessing the geometrical regularity of the projected/normalized curve (greatest regularity of a regular parametrization) from the properties of $v$?

Note: any planar curve can be obtained as the projection of a $C^\infty$ regular space curve, for example using mollifiers. In the non-analytic case it is not enough to look at the Taylor expansion of $v$ in $0$.

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