Third Frenet's formula - geometrical meaning

Question

I am trying to convince myself that the binormal vector $b=t\wedge n$ (perpendicular to both the normal $n$ and tangent $t$ vectors) can only vary along the curve (locally) in the direction of $n$, as prescribed by the formula $$b'(s)=\tau(s)\ n(s).$$ I can derive the fact that $b'(s)\perp t(s)$, but I can't visualise why the osculating plane (and with it $b$) can't tilt in the direction of $t$ instead.

Answer 1

Maybe you don't see this because the Frenet formula describe how the binormal $b$ tilts but it doesn't show how "$b$ rotates around itself". Said differently, the Frenet formula doesn't show how the osculating plane rotates around the binormal when you move along the curve. We have to take the movement of the point on the curve into account.

As an example, consider a point $q(t)$ on the osculating plane of a curve $\gamma$ through $\gamma(t)$. Then there are constants $a$ and $b$ such that $q(t) = \gamma(t) + a T(t) + b N(t)$. Deriving this expression, we get $$ q'(t) = (1+b \kappa(t))T(t) + a \kappa(t)N(t) - b \tau(t) B(t). $$

So if you would make picture (sorry, not included here), you see that the osculating plane tilts (the binormal changes) but that the plane itself also rotates a bit.

I hope this helps your intuition a bit.