Reparametrisation

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]