挠率与弗雷内标架

A third axis completes the moving frame

Torsion measures the twist of the osculating plane

The Frenet-Serret equations

At $t = 0$ on the helix $\mathbf{r}(t) = (\cos t, \sin t, t)$, the frame vectors are $\mathbf{T} = \tfrac{1}{\sqrt{2}}(0, 1, 1)$ and $\mathbf{N} = (-1, 0, 0)$. What is $\mathbf{B} = \mathbf{T} \times \mathbf{N}$?

What is the torsion of a circle traversed in the plane?

In the Frenet-Serret equation $\dfrac{d\mathbf{N}}{ds} = -\kappa\,\mathbf{T} + \tau\,\mathbf{B}$, why is there no component along $\mathbf{N}$ itself?

For the helix $\mathbf{r}(t) = (\cos t, \sin t, t)$, the torsion formula gives $\tau = \dfrac{1}{2}$ and the curvature gives $\kappa = \dfrac{1}{2}$. Give the ratio $\tau/\kappa$, as a plain number.

A space curve with $\kappa > 0$ has a Frenet frame of exactly $n$ mutually perpendicular unit vectors. Give $n$, as a plain number.