The Frenet formulas
Back to Curves and Surfaces homepage
Recall from the lecture on the Frenet frame that if \(p : I \rightarrow \mathbb{R}^3\) is parametrised by arclength and \(p(s)\) is not a point of inflection (equivalently, \(p''(s) \neq 0\) which is equivalent to \(\kappa(s) \neq 0\)) then we can define the Frenet frame of the curve at the point \(p(s)\) as the orthonormal triple of vectors \(T(s), N(s), B(s)\) given by the formulas
\[
\begin{aligned}
T(s) & = p'(s) \quad \text{(Unit tangent vector)} \\
N(s) & = \frac{T'(s)}{|T'(s)|} \quad \text{(Principal normal vector)} \\
B(s) & = T(s) \times N(s) \quad \text{(Binormal vector)}
\end{aligned}
\]
The following picture shows the Frenet frame for the helix as the point \(P\) moves along the curve.
In this lecture we will prove the Frenet formulas which relate the derivatives \(T'(s), N'(s), B'(s)\) to the Frenet frame \(T(s), N(s), B(s)\). This relationship can be written entirely in terms of the curvature and torsion, which will allow us to prove in the next lecture that the curvature and torsion completely determine the curve up to Euclidean motions.
First recall the definition of the curvature
\[
\begin{aligned}
N(s) & = \frac{T'(s)}{|T'(s)|} \\
\Rightarrow \quad T'(s) & = \kappa(s) N(s)
\end{aligned}
\]
and the definition of the torsion
\[
B'(s) = \tau(s) N(s) .
\]
Therefore we have easily computed \(T'(s)\) and \(B'(s)\) in terms of the curvature, the torsion and the vectors in the original Frenet Frame. The final derivative we need to compute is \(N'(s)\), which we can do by noting that since \((T(s), N(s), B(s))\) form an orthonormal frame, then
\[
\begin{aligned}
\left| N(s) \right| = 1 \quad \Rightarrow \quad & N'(s) \cdot N(s) = 0 \\
& N'(s) = \alpha T(s) + \beta B(s)
\end{aligned}
\]
for some real numbers \(\alpha\) and \(\beta\) which we can compute by
\[
N'(s) \cdot T(s) = \left( \alpha T(s) + \beta B(s) \right) \cdot T(s) = \alpha , \quad N'(s) \cdot B(s) = \left( \alpha T(s) + \beta B(s) \right) \cdot B(s) = \beta .
\]
Now we also know that
\[
\begin{aligned}
N(s) \cdot T(s) & = 0 \\
\Rightarrow \quad \frac{d}{ds} \left( N(s) \cdot T(s) \right) & = 0 \\
\Leftrightarrow \quad N'(s) \cdot T(s) + N(s) \cdot T'(s) & = 0 \\
\Rightarrow \quad \alpha = N'(s) \cdot T(s) & = -N(s) \cdot T'(s) .
\end{aligned}
\]
The definition of curvature tells us that \(T'(s) = \kappa(s) N(s)\), and therefore
\[
\alpha = -N(s) \cdot T'(s) = - \kappa(s) .
\]
Now we can use the same idea to compute \(\beta = N'(s) \cdot B(s)\). By differentiating the equation \(N(s) \cdot B(s) = 0\), we see that
\[
\begin{aligned}
N'(s) \cdot B(s) + N(s) \cdot B'(s) & = 0 \\
\Leftrightarrow \quad \beta = N'(s) \cdot B(s) & = - N(s) \cdot B'(s) .
\end{aligned}
\]
The definition of torsion tells us that \(B'(s) = \tau(s) N(s)\), therefore
\[
\beta = -N(s) \cdot B'(s) = -\tau(s) .
\]
Therefore we have computed the derivative of the normal vector
\[
N'(s) = -\kappa(s) T(s) - \tau(s) B(s) .
\]
The Frenet formulas
These three formulas are together called the Frenet formulas and can be written in matrix form as follows.
\[
\left( \begin{matrix} T'(s) \\ N'(s) \\ B'(s) \end{matrix} \right) = \left( \begin{matrix} 0 & \kappa(s) & 0 \\ -\kappa(s) & 0 & -\tau(s) \\ 0 & \tau(s) & 0 \end{matrix} \right) \left( \begin{matrix} T(s) \\ N(s) \\ B(s) \end{matrix} \right)
\]
Next time
We will use these formulas to prove the Fundamental Theorem of Space Curves, which says that if there are no inflection points, then the curvature and torsion determine the curve up to a rigid motion (e.g. rotation, translation, reflection).
