Fundamental theorem of the local theory of curves

Let $\alpha$ be a regular $C^\infty$ curve in $\mathbb {R}^{3}$ with nonvanishing curvature. Suppose the normal vector $N(t)$ is proportional to the position vector; that is, $N(t) = c(t)\alpha (t)$ for all $t$, where $c$ is a smooth function. Determine all such curves.