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]

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 we define speed as:

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

See also

• Unit speed parametrisation
• Arc length
• Curve
• Reparametrisation
• Smooth