Difference between revisions of "Reparametrisation"
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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 | + | {{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]
