----
> [!definition]+ Definition. ([[arc length of a path]])
> The **arc length** of a differentiable, [[regular curve|regular]] [[parameterized curve|Euclidean path]] $\boldsymbol{\alpha}: I \to \mathbb{R}^{n}$ from the point $t_{0}$ is defined $s(t)=\int _{t_{0}}^{t} \|\boldsymbol{\alpha}'(\tau)\|_{2} \, d\tau. $
>
^e7f19d
> [!basicexample]+
> Consider the helix $\boldsymbol{\alpha}(t)=(\sin t, \cos t, t), t \in (-\infty, \infty)$ of pitch $2\pi$. The [[velocity vector of a parameterized curve|tangent vector]] to the trace of $\boldsymbol{\alpha}$ is $\boldsymbol{\alpha}'(t)=(\cos t, -\sin t, 1)$ and consequently we have parameterization speed $\|\boldsymbol{\alpha}'(t)\|_{2}=\sqrt{ 2 }$. Thus the [[arc length of a path|arc length]] function starting at $t_{0}=0$ is $\int_{0}^{t} \sqrt{ 2 } \, dt= t\sqrt{ 2 }$.
>
> We can obtain a [[parameterization by arc length|(re)parameterization]] of $\boldsymbol{\alpha}$ by [[arc length of a path|arc length]] as follows: define $\begin{align}
\Phi : & (-\infty, \infty) \to (-\infty, \infty) \\
s \mapsto & \Phi(s) := \frac{t}{\sqrt{ 2 }} ;\end{align}$
then we can verify that $\tilde{\alpha}:= \boldsymbol{\alpha} \circ \Phi$ is a [[parameterization by arc length]] thus:
>$\begin{align}
\|\tilde{\alpha}'(t)\|_{2}= & \| D\boldsymbol{\alpha} |_{\Phi(t)} D\Phi(t) \|_{2} \\
= & \|(\cos \Phi(t), \sin \phi(t), 1) \frac{1}{\sqrt{ 2 }}\|_{2} \\
= & 1.
\end{align}$
^13e6ff
----
#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```