The next question is from Do-Carmo's baby book, page 30 question 3 in section 1-6.
The question goes as follows: Show that the curvature $k(t)\neq 0$ of a regular parametrized curve $\alpha : I\rightarrow \mathbb{R}^3$ is the curvature at $t$ of of the plane curve $\pi \circ \alpha$, where $\pi$ is the normal projection of $\alpha$ over the osculating plane at $t$.
Now I guess I don't understand what is $\pi$ here, I mean I have $\alpha(s)= (x(s),y(s),z(s))$ assume it's in arclength parametrization, so basically I want to show that:
(\pi \circ \alpha (s))'' = k \cdot n_{\pi \circ \alpha (s)}\;,
but I am not sure here what $\pi$ equals to, I mean if its a projection on the osculating plane which is the plane of $xy$ then shouldn't it be the binormal to $\alpha$ (in which case it's $(0,0,z(s))$)?
Any hints?
Thanks.