Parametrisation

Definition

A parametrisation [ilmath]\gamma[/ilmath] is a function:

[math]\gamma:(a,b)\rightarrow\mathbb{R}^n[/math] with [math]-\infty\le a< b\le +\infty[/math]

Often [ilmath]t[/ilmath] is the parameter, so we talk of [ilmath]\gamma(t_0)[/ilmath] or [ilmath]\gamma(t)[/ilmath]

Differentiation

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Intuitively we see that the gradient at [ilmath]t[/ilmath] of [ilmath]\gamma[/ilmath] is [math]\approx\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t}[/math] taking the limit of [ilmath]\delta t\rightarrow 0[/ilmath] we get [ilmath]\frac{d\gamma}{dt}=\lim_{\delta t\rightarrow 0}(\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t})[/ilmath] as usual.

Other notations for this include [ilmath]\dot{\gamma} [/ilmath]

Arc Length

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

Noting that [math]\|\gamma(t+\delta t)-\gamma(t)\|\approx\|\dot{\gamma}(t)\delta t\|=\|\dot{\gamma}(t)\|\delta t[/math]

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

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

[math]s(t)=\int_{t_0}^t\|\dot{\gamma}(u)\|du[/math]

Rebasing arc length

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

[math]\tilde{s}(t)=\int_{\widetilde{t_0}}^t\|\dot{\gamma}(u)\|du[/math] [math]=\int_{\widetilde{t_0}}^{t_0}\|\dot{\gamma}(u)\|du+\int_{t_0}^t\|\dot{\gamma}(u)\|du[/math] [math]=\int_{\widetilde{t_0}}^{t_0}\|\dot{\gamma}(u)\|du+s(t)[/math]

Differentiating arc length

Easy:

[math]\frac{d}{dt}\Big[s(t)\Big]=\frac{d}{dt}\Big[\int_{t_0}^t\|\dot{\gamma}(u)\|du\Big][/math][math]=\|\dot{\gamma}(t)\|[/math] 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 [math]\|\dot{\gamma}(t)\|[/math]

See also

• Curve
• Reparametrisation
• Smooth