Excuse the naive question. I am trying to build a geometric intuition which I always lacked.
I can see that if I have a line in $\mathbb{R}^3,$ then I have a plane where the line is entirely contained.
Actually, I have a sheaf of infinitely many planes containing a given line, because I can rotate in 3D space any given plane where the axis of rotation is the given line, obtain all the other planes containing it.
On the other hand, a plane not parallel and not containing entirely the line but intersecting it will meet the line at exactly one point.
So far so good?
Ok, then what about a generic curve?
For example, I can take $S^1$ in $\mathbb{R}^3.$
Then a generic plane not parallel to this $S^1$ will meet the circle in exactly two points.
On the other hand, there exists for sure a plane where this $S^1$ entirely lies on. Is this plane unique in this case? I think yes.
Is it always unique for any curve different than a line? I think yes also.
Does it have to do with the fact that the line has zero curvature?