The Frenet-Serret Formulas

The Frenet-Serret Formulas

So far, we have looked at three important types of vectors for curves $C$ defined by a vector-valued function. The first type of vector we looked were Unit Tangent Vectors denoted $\hat{T}$, and subsequently Unit Normal and Unit Binormal Vectors denoted $\hat{N}$ and $\hat{B}$ respectively.

Now let $\vec{r}(s)$ be an arc-length parameterization of $\vec{r}(t)$. Therefore:

  • Unit Tangent Vector: $\hat{T}(s) = \vec{r'}(s)$.
  • Unit Normal Vector: $\hat{N}(s) = \frac{\hat{T'}(s)}{\| \hat{T'}(s) \|}$.
  • Unit Binormal Vector: $\hat{B}(s) = \hat{T}(s) \times \hat{N}(s)$.

This set of vectors has an important name which we will define.

Definition: Let $\vec{r}(s) = (x(s), y(s), z(s))$ be a vector-valued function with arc-length parameterization that traces the smooth curve $C$. Then the Frenet Frame of $C$ at $\vec{r}(s)$ is the set of right-handed mutually perpendicular unit vectors $\{ \hat{T}(s), \hat{N}(s), \hat{B}(s) \}$.

Furthermore, the Frenet-Serret formulas are $\frac{d \hat{T}(s)}{ds} = \kappa(s) \hat{N}(s)$, $\frac{d\hat{N}(s)}{ds} = - \kappa(s) \hat{T}(s) + \tau (s) \hat{B}(s)$, and $\frac{d \hat{B}(s)}{ds} = -\tau(s) \hat{N}(s)$ where $\kappa (s)$ is the curvature at the point $\vec{r}(s)$, and $\tau (s)$ is the torsion at the point $\vec{r}(s)$. These formulas can be represented and memorized as the following matrix:

\begin{align} \frac{d}{ds} \begin{bmatrix} \hat{T}(s) \\ \hat{N}(s) \\ \hat{B}(s) \end{bmatrix} = \begin{bmatrix} 0 & \kappa (s) & 0\\ - \kappa (s) & 0 & \tau (s) \\ 0 & -\tau (s) & 0 \end{bmatrix} \begin{bmatrix} \hat{T}(s) \\\hat{N}(s) \\ \hat{B}(s) \end{bmatrix} \end{align}

We will prove the first of the Frenet-Serret formulas.

Theorem 1: Let $\vec{r}(s) = (x(s), y(s), z(s))$ be a vector-valued function with arc length parameterization that traces out the smooth curve $C$. Then $\frac{d}{ds} \hat{T}(s) = \kappa (s) \hat{N}(s)$.
  • Proof: We note that $\kappa (s) = \frac{ \| \hat{T'}(s) \| }{ \| \vec{r'}(s) \|}$ and that $\hat{N}(s) = \frac{\hat{T'}(s)}{\| \hat{T'}(s) \|}$. Therefore:
\begin{align} \quad \kappa (s) \hat{N}(s) = \left ( \frac{ \| \hat{T'}(s) \| }{ \| \vec{r'}(s) \|} \right ) \left ( \frac{\hat{T'}(s)}{\| \hat{T'}(s) \|} \right ) \\ \quad \kappa (s) \hat{N}(s) = \frac{\hat{T'}(s)}{\vec{r'}(s)} \end{align}
  • Now notice that one of the properties of an arc length parameterization of a vector-valued function is that the curve $C$ is traced out at unit speed, that is, $\| \vec{r'}(s) \| = 1$. This can be seen since:
\begin{align} \int_a^t \| \vec{r'}(u) \| \: du = s(t) \\ \| \vec{r'}(t) \| = \frac{ds}{dt} \\ \| \vec{r'}(s) \| = \frac{ds}{ds} \\ \| \vec{r'}(s) \| = 1 \end{align}
  • Thus $\hat{T'}(s) = \frac{d}{ds} \hat{T}(s) = \kappa (s) \hat{N}(s)$. $\blacksquare$
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