Arc length

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

Definition

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Like before we can take small steps $\delta t$ 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 $t_0$ we can define arc length, $s(t)$ as:

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

Rebasing arc length

Suppose we want the arc length to be measured from $\widetilde{t_0}$ 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 $t$ of $\gamma$ is $$\|\dot{\gamma}(t)\|$$

See also

• The arc length of a regular parametrisation is smooth