Reparametrisation

This page requires knowledge of a parametrisation of a curve

Definition

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

$$\phi:(\tilde{a},\tilde{b})\rightarrow(a,b)$$ which is smooth and a bijection, and $\phi^{-1}$ is also smooth where:

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