Calculus Small Changes, Big Ideas

Section 13.4 The Frenet-Serret frame

• Unit tangent:

  \begin{equation*} \bv{t}=\frac{\bv{r}'}{\norm{\bv{r}'}} \end{equation*}

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• Unit normal:

  \begin{equation*} \bv{n}=\frac{\bv{t}'}{\norm{\bv{t}'}} \end{equation*}

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• Curvature:

  \begin{equation*} \kappa=\norm{\dv{\bv{t}}{s}}=\frac{\norm{\bv{t}'}}{\norm{\bv{r}'}}=\frac{\bv{r}'\times\bv{r}''}{\norm{\bv{r}'}^{3}} \end{equation*}

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• Tangent component of acceleration:

  \begin{equation*} \bv a_{t}=\bv{a}\cdot\bv{n} \end{equation*}

  Normal component of acceleration:

  \begin{equation*} \bv a_{n}=\bv{a}\times\bv{n} \end{equation*}

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• In 3D, we also get the binormal:

  \begin{equation*} \bv{b}=\bv{t}\times\bv{n} \end{equation*}

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• Torsion:

  \begin{equation*} \dv{\bv{b}}{s}=-\tau\bv{n} \end{equation*}

  \begin{equation*} \tau=-\dv{\bv{b}}{s}\cdot\bv{n}=-\dfrac{\bv{b}'\cdot\bv{n}}{\norm{\bv{r}'}}=\dfrac{(\bv{r}'\times\bv{r}'')\cdot\bv{r}'''}{\norm{\bv{r}'\times\bv{r}''}^2} \end{equation*}

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• The unit tangent, unit normal, and binormal together form the Frenet-Serret frame.

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