Difference between revisions of "Reparametrisation"

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(Created page with "This page requires knowledge of a parametrisation of a curve ==Definition== A function {{M|\tilde{\gamma}:(\tilde{a},\tilde{b})\rightarrow\mathb...")
 
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* <math>\tilde{\gamma}(\phi^{-1}(t))=\gamma(t)</math> for all <math>t\in(a,b)</math>
 
* <math>\tilde{\gamma}(\phi^{-1}(t))=\gamma(t)</math> for all <math>t\in(a,b)</math>
  
{{Definition|Differential Geometry|Geometry of Curves and Surface}}
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{{Definition|Differential Geometry|Geometry of Curves and Surfaces}}

Revision as of 20:58, 29 March 2015

This page requires knowledge of a parametrisation of a curve

Definition

A function [ilmath]\tilde{\gamma}:(\tilde{a},\tilde{b})\rightarrow\mathbb{R}^n[/ilmath] is a reparametrisation of the parametrisation [math]\gamma:(a,b)\rightarrow\mathbb{R}^n[/math] if there exists:

[math]\phi:(\tilde{a},\tilde{b})\rightarrow(a,b)[/math] which is smooth and a bijection, and [ilmath]\phi^{-1} [/ilmath] is also smooth where:

  • [math]\tilde{\gamma}(\tilde{t})=\gamma(\phi(\tilde{t}))[/math] for all [math]\tilde{t}\in(\tilde{a},\tilde{b})[/math]
  • [math]\tilde{\gamma}(\phi^{-1}(t))=\gamma(t)[/math] for all [math]t\in(a,b)[/math]