Difference between revisions of "Parametrisation"

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==Differentiation==
 
==Differentiation==
 
{{Todo|Add picture}}
 
{{Todo|Add picture}}
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Intuitively we see that the gradient at {{M|t}} of {{M|\gamma}} is <math>\approx\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t}</math> taking the limit of {{M|\delta t\rightarrow 0}} we get {{M|1=\frac{d\gamma}{dt}=\lim_{\delta t\rightarrow 0}(\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t})}} as usual.
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Intuitively we see that the gradient at {{M|t}} of {{M|\gamma}} is <m>\approx\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t}</m> taking the limit of {{M|\delta t\rightarrow 0}} we get {{M|1=\frac{d\gamma}{dt}=\lim_{\delta t\rightarrow 0}(\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t})}} as usual.
  
 
Other notations for this include {{M|\dot{\gamma} }}
 
Other notations for this include {{M|\dot{\gamma} }}
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==Arc Length==
 
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{{Todo|Add picture}}
 
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Like before we can take small steps {{M|\delta t}} 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]])
 
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Noting that <math>\|\gamma(t+\delta t)-\gamma(t)\|\approx\|\dot{\gamma}(t)\delta t\|=\|\dot{\gamma}(t)\|\delta t</math>
 
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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>.
 
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Choosing a starting point {{M|t_0}} we can define arc length, {{M|s(t)}} as:
 
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<math>s(t)=\int_{t_0}^t\|\dot{\gamma}(u)\|du</math>
 
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===Rebasing arc length===
 
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Suppose we want the arc length to be measured from {{M|\widetilde{t_0} }} then:
 
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<math>\tilde{s}(t)=\int_{\widetilde{t_0}}^t\|\dot{\gamma}(u)\|du</math>
 
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<math>=\int_{\widetilde{t_0}}^{t_0}\|\dot{\gamma}(u)\|du+\int_{t_0}^t\|\dot{\gamma}(u)\|du</math>
 
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<math>=\int_{\widetilde{t_0}}^{t_0}\|\dot{\gamma}(u)\|du+s(t)</math>
 
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===Differentiating arc length===
 
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Easy:
 
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<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==
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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:
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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 {{M|t}} of {{M|\gamma}} is <math>\|\dot{\gamma}(t)\|</math>
 
The speed at {{M|t}} of {{M|\gamma}} is <math>\|\dot{\gamma}(t)\|</math>
  
 
==See also==
 
==See also==
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* [[Unit speed parametrisation]]
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* [[Arc length]]
 
* [[Curve]]
 
* [[Curve]]
 
* [[Reparametrisation]]
 
* [[Reparametrisation]]

Latest revision as of 11:10, 12 June 2015

Definition

A parametrisation [ilmath]\gamma[/ilmath] is a function[1]:

[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

TODO: Add picture

Intuitively we see that the gradient at [ilmath]t[/ilmath] of [ilmath]\gamma[/ilmath] is [ilmath]\approx\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t}[/ilmath] 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

References

  1. ↑ Elementary Differential Geometry - Pressley - Springer SUMS
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