In class the teacher was talking about normal vectors. $r = \langle x,y\rangle$ then the normal vector is
$ N\left(t\right) = \frac{T^{\prime}\left(t\right)}{||T^{\prime}\left(t\right)||} $
where $T\left(t\right) = \frac{r^{\prime}}{||r^{\prime}||}$
Now, one of the naughty students asked the question in the title.
I thought it was a cool question so I am asking here.
ps. I have no idea, still trying to figure out how the unit norm vector needs the 2nd derivative of the tangent vector T.