若曲线 $c(s)$ 是以弧长为参数, 则 $\kappa(s)=|\ddot{c}(s)|$, $\tau(s)=\frac{\det(\dot{c}(s),\ddot{c}(s),\dddot{c}(s))}{\kappa^2(s)}$.
若曲线 $c(s)$ 是正则曲线, 且以弧长为参数, 证明
\[\kappa(s)=|\ddot{c}(s)|,\quad\tau(s)=\frac{\det(\dot{c}(s),\ddot{c}(s),\dddot{c}(s))}{\kappa^2(s)}.\]
Frenet 标架为
\[\vec{v}(s)=e_1(s)=\dot{c}(s),\quad\vec{n}(s)=e_2(s)=\frac{\ddot{c}(s)}{|\ddot{c}(s)|},\quad\vec{b}(s)=e_3(s)=\frac{\dot{c}(s)\times\ddot{c}(s)}{|\ddot{c}(s)|}.\]
若曲线 $c(t)$ 以一般参数表示, 证明
\[\kappa(t)=\frac{|\dot{c}(t)\times\ddot{c}(t)|}{|\dot{c}(t)|^3},\quad\tau(t)=\frac{\det(\dot{c}(t),\ddot{c}(t),\dddot{c}(t))}{|\dot{c}(t)\times\ddot{c}(t)|^2}.\]
Frenet 标架为
\[
\begin{aligned}
\vec{v}(t)&=e_1(t)=\frac{\dot{c}(t)}{|\dot{c}(t)|},\\
\vec{n}(t)&=e_2(t)=\frac{(\dot{c}(t)\times\ddot{c}(t))\times\dot{c}(t)}{|(\dot{c}(t)\times\ddot{c}(t))\times\dot{c}(t)|},\\
\vec{b}(t)&=e_3(t)=\frac{\dot{c}(t)\times\ddot{c}(t)}{|\dot{c}(t)\times\ddot{c}(t)|}.
\end{aligned}
\]
并证明不论用什么参数表示曲线, 曲线的曲率和挠率是不依赖于参数的. 即有
\[\kappa(s)=\kappa(t),\qquad\tau(s)=\tau(t).\]
不但如此, Frenet 标架也是一致的. 即有
\[e_i(s)=e_i(t),\qquad i=1,2,3.\]