# Geometrical regularity of the projection/normalization of a curve

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 $$0$$.