Let $I,J$ be open intervals. Let $\gamma$ and $w$ be two unit-speed curves $I \to \mathbb{R}^{3}$. In particular, assume that, for every $t \in I$,
- $w(t) \cdot w(t) =1$,
- $w(t) \neq \pm \gamma'(t)$,
- $\gamma''(t) \neq 0$.
I am interested in finding conditions on $\gamma$ so that the (image of the) map $\sigma \colon I \times J \to \mathbb{R}^{3}$,
$$\sigma(t,u) = \gamma(t)+uw(t),$$
which is clearly regular (i.e., immersive) in a neighborhood of $\gamma(I)$, has no planar point. I suspect that choosing $\gamma$ so that its torsion is never zero (or at most only at isolated points) will be sufficient. Can anybody confirm so? Is there a way to prove it without getting into tedious calculations? By planar point I mean of course a point where both principal curvatures vanish.