Editing Frenet–Serret formulas (section)

== Formulas in ''n'' dimensions ==

The Frenet–Serret formulas were generalized to higher-dimensional Euclidean spaces by [[Camille Jordan]] in 1874.

Suppose that {{math|'''r'''(''s'')}} is a smooth curve in <math>\R^n,</math> and that the first {{mvar|n}} derivatives of {{math|'''r'''}} are linearly independent.<ref>Only the first {{math|''n'' &minus; 1}} actually need to be linearly independent, as the final remaining frame vector {{math|'''e'''<sub>''n''</sub>}} can be chosen as the unit vector orthogonal to the span of the others, such that the resulting frame is positively oriented.</ref> The vectors in the Frenet–Serret frame are an [[orthonormal basis]] constructed by applying the [[Gram–Schmidt process]] to the vectors {{math|('''r'''&prime;(''s''), '''r'''&prime;&prime;(''s''), ..., '''r'''<sup>(''n'')</sup>(''s''))}}.

In detail, the unit tangent vector is the first Frenet vector {{math|''e''<sub>1</sub>(''s'')}} and is defined as

<math display=block>\mathbf{e}_1(s) = \frac{\overline{\mathbf{e}_1}(s)} {\| \overline{\mathbf{e}_1}(s) \|}</math>

where

<math display=block>\overline{\mathbf{e}_1}(s) = \mathbf{r}'(s)</math>

The '''normal vector''', sometimes called the '''curvature vector''', indicates the deviance of the curve from being a straight line. It is defined as
<math display=block>\overline{\mathbf{e}_2}(s) = \mathbf{r}''(s) - \langle \mathbf{r}''(s), \mathbf{e}_1(s) \rangle \, \mathbf{e}_1(s)</math>

Its normalized form, the '''unit normal vector''', is the second Frenet vector {{math|'''e'''<sub>2</sub>(''s'')}} and defined as

<math display=block>\mathbf{e}_2(s) = \frac{\overline{\mathbf{e}_2}(s)} {\| \overline{\mathbf{e}_2}(s) \|}
</math>

The tangent and the normal vector at point {{mvar|s}} define the ''[[osculating plane]]'' at point {{math|'''r'''(''s'')}}.

The remaining vectors in the frame (the binormal, trinormal, etc.) are defined similarly by

:<math>\begin{align}
  \mathbf{e}_{j}(s) &= \frac{\overline{\mathbf{e}_{j}}(s)}{\|\overline{\mathbf{e}_{j}}(s) \|}, \\
  \overline{\mathbf{e}_{j}}(s) &= \mathbf{r}^{(j)}(s) - \sum_{i=1}^{j-1} \langle \mathbf{r}^{(j)}(s), \mathbf{e}_i(s) \rangle \, \mathbf{e}_i(s).
\end{align} </math>

The last vector in the frame is defined by the cross-product of the first {{math|''n'' &minus; 1}} vectors:
<math display=block>
  \mathbf{e}_n(s) = \mathbf{e}_1(s) \times \mathbf{e}_2(s) \times \dots \times \mathbf{e}_{n-2}(s) \times \mathbf{e}_{n-1}(s)</math>

The real valued functions used below {{math|''χ<sub>i</sub>''(''s'')}} are called '''generalized curvature''' and are defined as

<math display=block>\chi_i(s) = \frac{\langle \mathbf{e}_i'(s), \mathbf{e}_{i+1}(s) \rangle}{\| \mathbf{r}'(s) \|} </math>

The '''Frenet–Serret formulas''', stated in matrix language, are

<math display=block>
\begin{bmatrix}
  \mathbf{e}_1'(s)\\
       	\vdots \\
 \mathbf{e}_n'(s) \\
\end{bmatrix}

= \| \mathbf{r}'(s) \| \cdot

\begin{bmatrix}
 0          & \chi_1(s) & 0              & 0 \\[4pt]
 -\chi_1(s) & \ddots    & \ddots         & 0 \\[4pt]
 0      	& \ddots    & \ddots         & \chi_{n-1}(s) \\[4pt]
 0          & 0         & -\chi_{n-1}(s) & 0
\end{bmatrix}

\begin{bmatrix}
 \mathbf{e}_1(s) \\
      	\vdots \\
 \mathbf{e}_n(s) \\
\end{bmatrix}
</math>

Notice that as defined here, the generalized curvatures and the frame may differ slightly from the convention found in other sources.
The top curvature {{math|''χ''{{sub|''n''−1}}}} (also called the torsion, in this context) and the last vector in the frame {{math|'''e'''{{sub|''n''}}}},
differ by a sign

<math display=block> \operatorname{or}\left(\mathbf{r}^{(1)},\dots,\mathbf{r}^{(n)}\right) </math>

(the orientation of the basis) from the usual torsion.
The Frenet–Serret formulas are invariant under flipping the sign of both {{math|''χ''{{sub|''n''−1}}}} and {{math|'''e'''{{sub|''n''}}}}, and this change of sign makes the frame positively oriented. As defined above, the frame inherits its orientation from the jet of {{math|'''r'''}}.