# Arc length

Arc length of curves here is defined with respect to parametrisations - it is fundamental for defining unit speed parametrisations

## Definition

Like before we can take small steps [ilmath]\delta t[/ilmath] apart, the length of the line joining such points is $\|\gamma(t+\delta t)-\gamma(t)\|$ (where $\|\cdot\|$ denotes the Euclidean norm)

Noting that $\|\gamma(t+\delta t)-\gamma(t)\|\approx\|\dot{\gamma}(t)\delta t\|=\|\dot{\gamma}(t)\|\delta t$

We can now sum over intervals, taking the limit of $\delta t\rightarrow 0$ we see that an infinitesimal section of arc length is $\|\dot{\gamma}(t)\|dt$.

Choosing a starting point [ilmath]t_0[/ilmath] we can define arc length, [ilmath]s(t)[/ilmath] as:

$s(t)=\int_{t_0}^t\|\dot{\gamma}(u)\|du$

### Rebasing arc length

Suppose we want the arc length to be measured from [ilmath]\widetilde{t_0} [/ilmath] then:

$\tilde{s}(t)=\int_{\widetilde{t_0}}^t\|\dot{\gamma}(u)\|du$ $=\int_{\widetilde{t_0}}^{t_0}\|\dot{\gamma}(u)\|du+\int_{t_0}^t\|\dot{\gamma}(u)\|du$ $=\int_{\widetilde{t_0}}^{t_0}\|\dot{\gamma}(u)\|du+s(t)$

### Differentiating arc length

Easy:

$\frac{d}{dt}\Big[s(t)\Big]=\frac{d}{dt}\Big[\int_{t_0}^t\|\dot{\gamma}(u)\|du\Big]$$=\|\dot{\gamma}(t)\|$ by the Fundamental theorem of Calculus

## Speed

Speed is the rate of change of distance (velocity is the rate of change of position - which are both vector quantities) - from differentiating the arc length (above) we define speed as:

The speed at [ilmath]t[/ilmath] of [ilmath]\gamma[/ilmath] is $\|\dot{\gamma}(t)\|$