Let $C$ be a regular curve embedded in $\mathbb{R}^3$ (i.e. a real 1-dimensional manifold embedded in $\mathbb{R}^3$). Let $S$ be the union of its affine tangent lines: $$S=\bigcup\limits_{p\in C}(p+T_pC).$$
What can be said about $S-C$? Under what conditions on $C$ is $S-C$ a regular surface embedded in $\mathbb{R}^3$?
