Frénet-Serret Formulas

Frénet-Serret Formulas

I found these formulas in the Vector Calculus book by Baxandall and Liebeck. They are in chapter 2.8, page 73. These formulas are pretty important in the area of differential geometry. See also this Wikipedia page on the same topic. In the text below the following standard notation is used: $T$ is the unit tangent vector, $N$ is the principal normal vector, $B$ is the binormal vector (these three are in fact vector-valued functions); $\kappa$ is the curvature, $\tau$ is the torsion (these two are in fact scalar-valued functions). The associated collection ($T$, $N$, $B$, $\kappa$, $\tau$) is called the Frénet–Serret apparatus. Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar.

Case #1. Suppose that the function $f(s): E \subseteq \mathbb{R} \rightarrow \mathbb{R^3}$ is a three times differentiable path-length parametrization of a curve in $\mathbb{R^3}$ and also suppose that the curvature $\kappa_f(s) \neq 0$ for all $s \in E$. Then 

$T'_f(s) = \kappa_f(s) N_f(s)$ 

$N'_f(s) = -\kappa_f(s) T_f(s) + \tau_f(s) B_f(s)$ 

$B'_f(s) = -\tau_f(s) N_f(s)$ 

In short one can also write 

$T' = \kappa N$ 

$N' = -\kappa T + \tau B$ 

$B' = -\tau N$ 

The three vectors $T, N, B$ are unit vectors and they form (in this order) a right-hand orthonormal basis of $\mathbb{R^3}$. This means that $B = T \times N$ for all $s \in E$. 

Case #2. Suppose that the function $f(s): E \subseteq \mathbb{R} \rightarrow \mathbb{R^3}$ is a three times differentiable smooth parametrization of a curve in $\mathbb{R^3}$ and also suppose that the curvature $\kappa_f(s) \neq 0$ for all $s \in E$. Then 

$T'_f(s) = v(s) \kappa_f(s) N_f(s)$ 

$N'_f(s) = v(s) [-\kappa_f(s) T_f(s) + \tau_f(s) B_f(s)]$ 

$B'_f(s) = -v(s) \tau_f(s) N_f(s)$ 

where $v(s) = || f'(s) || $ for all $s \in E$

In short one can also write 

$T' = v \kappa N$ 

$N' = v (-\kappa T + \tau B)$ 

$B' = -v \tau N$  

In this case again the three vectors $T, N, B$ are unit vectors and they form (in this order) a right-hand orthonormal basis of $\mathbb{R^3}$. This means that $B = T \times N$ for all $s \in E$. 

Note: In the first case $s$ is a path-length parameter i.e $|| f'(s) || = 1$ for all $s \in E$. The second case is more general since in that case we do not require $s$ to be a path-length parameter, i.e. in that case we may have $|| f'(s) || \ne 1$ for some $s \in E$ or for all $s \in E$.