Arclength#

Definition: Arclength

Let \(\gamma: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}^n\) be a vector-valued function and let \(I \subseteq \mathcal{D}\) be an interval with endpoints \(a\) and \(b\) (\(a \le b\)).

The arclength traced by \(\gamma\) over \(I\) is the integral of the Euclidean norm of \(\gamma\)'s derivative on \(I\) (if it exists):

\[\int_a^b ||\dot{\gamma}(t)|| \,\mathrm{d}t\]

Example: Circumference

We can use this to calculate the circumference of a circle in \(\mathbb{R}\) which is centered at the origin and has radius \(r\). One parametrization of this circle is the function

\[k: [0; 2\pi] \to \mathbb{R}^2\]

defined as

\[k(t) = \begin{bmatrix}r \cos (t) \\ r \sin (t)\end{bmatrix}\]

for all \(t \in [0;2\pi]\). The circumference is given as follows:

\[\begin{aligned}\int_0^{2\pi} \vert\vert k'(t) \vert\vert \, \mathrm{d}t & = \int_0^{2\pi} \left\vert\left\vert \begin{bmatrix}-r \sin (t) \\ r \cos (t)\end{bmatrix}\right\vert\right\vert\,\mathrm{d}t \\ & = \int_0^{2\pi}\sqrt{r^2 \sin^2 t + r^2 \cos^2 (t)}\,\mathrm{d}t \\ & = \int_0^{2\pi} r\,\mathrm{d}t = 2\pi r \end{aligned}\]

Example: Graph Length

Let \(f: [a,b] \subset \mathbb{R} \to \mathbb{R}\) be a real function.

If \(f\) is differentiable on \([a,b]\), the function

\[k: [a, b] \to \mathbb{R}^2\]

defined as

\[k(t) = \begin{bmatrix}t \\ f(t)\end{bmatrix}\]

is a parametrization of \(f\)'s graph. Therefore,

\[\int_a^b ||k'(t)||\,\mathrm{d}t = \int_a^b \sqrt{1 + f'(t)^2}\,\mathrm{d}t\]

is the length of this graph.

Theorem: Continuously Differentiable Equivalence \(\implies\) Equal Arclength

Let \(\gamma: [a, b] \subset \mathbb{R}\) and \(\varphi: [c, d] \subset \mathbb{R}\) be parametric curves.

If \(\gamma\) and \(\phi\) are continuously differentiable and are equivalent up to a continuously differentiable reparametrization, then they trace the same arclength:

\[\int_a^b ||\dot{\gamma}(t)||\,\mathrm{d}t = \int_c^d ||\dot{\varphi}(t)|| \,\mathrm{d}t\]

Proof

TODO

Arclength Parametrization#

Definition: Arclength Function

Let \(k: [a,b] \subset \mathbb{R} \to \mathbb{R}^n\) be a regular parametric curve.

The arclength function of \(k\) is the function

\[s: [a, b] \subset \mathbb{R} \to \mathbb{R}\]

defined as the arclength traced by \(k\) from \(a\) to \(t\):

\[s(t) = \int_a^t ||k'(\tau)||\,\mathrm{d}\tau\]

Definition: Arclength Parametrization

Let \(\mathcal{C} \subseteq \mathbb{R}^n\) and let \(k: [a,b] \subset \mathbb{R} \to \mathbb{R}^n\) be a regular vector-valued function which is a parametrization of \(\mathcal{C}\).

The arclength parametrization of \(\mathcal{C}\) is the parametric curve \(\tilde{k}: [0, L(k)] \to \mathbb{R}^n\) defined as

\[\tilde{k}(\tau) = k (s^{-1}(\tau)),\]

where \(s\) is the arclength function of \(k\).

Theorem: Derivative of Arclength Parametrization

The derivative of an arclength parametrization is always a unit vector with respect to the Euclidean norm.

Proof

TODO