Arclength

The arclength of a parametric curve is quite literally the ‘length’ of the path.

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Definition §

Arclength ^definition

Let $C$ be the parametric curve traced out by the 1-parameter parametric equation $\vec{c}:\mathbb{R}\to {\mathbb{R}}^n$ where $c(t)=(f_1(t),f_2(t),\dots ,f_n(t))$ over an interval $t\in [a,b]$. Then the arclength, $s$, of $C$ is given by:

$$\begin{array}{rl}s& ={\int }_a^b\Vert \vec{T}(t)\Vert \text{d}t={\int }_a^b\Vert \frac{\text{d}\vec{c}}{\text{d}t}\Vert \text{d}t\\ & ={\int }_a^b\sqrt{(\frac{\text{d}{f}_1}{\text{d}t}{)}^2+(\frac{\text{d}{f}_2}{\text{d}t}{)}^2+\cdots +(\frac{\text{d}{f}_n}{\text{d}t}{)}^2}\text{d}t\end{array}$$

• $\vec{T}$: The tangent vector to the curve.
• $s:\mathbb{R}\to \mathbb{R}$

We can see how this makes intuitive sense if we use connect its kinematic properties.

• If $\vec{c}$ represents the position of a particle over time.
• Then $\Vert \frac{\text{d}\vec{c}}{\text{d}t}\Vert$ would be the speed of the particle.
• If we take the speed over time, then we have tiny parts of ‘distance’.
• Then, taking the integral of these distances gives the total distance of the particle!

Arclength Parameterisation §

Arc lengths can also be parameterised just like the curve itself. In that sense, we can see how the curve’s length grows as time progresses:

$$s(t)={\int }_a^t\Vert \frac{\text{d}\vec{c}}{\text{d}\tau }\Vert \text{d}\tau$$

We use $\tau$ to represent a dummy variable here.

The advantage of doing this parameterisation is that if we have $s(t)$ being an injective function, so we can just represent $t$ using $s$! For example, if $s=5t$, then $t=\frac{s}{5}$ and we can do:

$$\vec{c}(t)=\vec{c}(\frac{s}{5})$$

Thus we can represent a path as a function of arclength, $s$.

Connection to Line Integrals §

The arclength is used to construct line integrals, hence it is also the most degenerate case of a line integral. If we take a parametric curve $C$ traced by $\vec{c}$, then:

$$s={\int }_{C}1\text{d}s$$

That is, we can get the arclength of a curve using a curve integral.

Generic Univariate Function Arclength §

For any univariate function $y=f(x)$, the arclength is given by:

#todo

$$s=\int \text{d}x$$