Editing Differentiable curve (section)

== Main theorem of curve theory ==
{{main|Fundamental theorem of curves}}
Given {{math|''n'' − 1}} functions:
<math display="block">\chi_i \in C^{n-i}([a,b],\mathbb{R}^n) , \quad \chi_i(t) > 0 ,\quad  1 \leq i \leq n-1</math>
then there exists a unique (up to transformations using the [[Euclidean group]]) {{math|''C''<sup>''n'' + 1</sup>}}-curve {{math|''γ''}} which is regular of order {{mvar|n}} and has the following properties:
<math display="block">\begin{align}
\|\gamma'(t)\| &= 1  & t \in [a,b] \\
\chi_i(t) &= \frac{ \langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \boldsymbol{\gamma}'(t) \|}
\end{align}</math>
where the set
<math display="block">\mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)</math>
is the Frenet frame for the curve.

By additionally providing a start {{math|''t''<sub>0</sub>}} in {{math|''I''}}, a starting point {{math|''p''<sub>0</sub>}} in <math>\mathbb{R}^n</math> and an initial positive orthonormal Frenet frame {{math|{{mset|''e''<sub>1</sub>, ..., ''e''<sub>''n'' − 1</sub>}}}} with
<math display="block">\begin{align}
\boldsymbol{\gamma}(t_0) &= \mathbf{p}_0 \\
\mathbf{e}_i(t_0) &= \mathbf{e}_i ,\quad  1 \leq i \leq n-1
\end{align}</math>
the Euclidean transformations are eliminated to obtain a unique curve {{math|''γ''}}.