Parametrization

Definition: Surface Parametrization

Let $\mathcal{S}$ be a surface in ${\mathbb{R}}^n$.

A parametrization of $\mathcal{S}$ is a continuous function $\phi :\mathcal{D}\subseteq {\mathbb{R}}^2\to {\mathbb{R}}^n$ on a connected set $\mathcal{D}$ which is injective on $\mathcal{D}$ except possibly at the boundary $\partial \mathcal{D}$ and whose image is $\mathcal{S}$.

Equivalence of Parametrizations

Definition: Reparametrization

Let $\psi :{\mathcal{D}}_{\psi }\subseteq {\mathbb{R}}^2\to {\mathbb{R}}^n$ and $\varphi :{\mathcal{D}}_{\varphi }\subseteq {\mathbb{R}}^2\to {\mathbb{R}}^n$ be parametrizations of the same surface $\mathcal{S}\subset {\mathbb{R}}^n$.

A reparametrization between $\psi$ and $\varphi$ is a bijective function $h_{{\mathcal{D}}_{\psi }\to {\mathcal{D}}_{\varphi }}:{\mathcal{D}}_{\psi }\to {\mathcal{D}}_{\varphi }$ with inverse $h_{{\mathcal{D}}_{\varphi }\to {\mathcal{D}}_{\psi }}:{\mathcal{D}}_{\varphi }\to {\mathcal{D}}_{\psi }$ such that

$$\begin{array}{r}\psi (\mathbf{x})=\varphi (h_{{\mathcal{D}}_{\psi }\to {\mathcal{D}}_{\varphi }}(\mathbf{x}))\forall \mathbf{x}\in {\mathcal{D}}_{\psi }\\ \varphi (\mathbf{x})=\psi (h_{{\mathcal{D}}_{\varphi }\to {\mathcal{D}}_{\psi }}(\mathbf{x}))\forall \mathbf{x}\in {\mathcal{D}}_{\varphi }\end{array}$$

NOTE

This is the most general definition for reparametrization. However, it is quite common to require that both $h_{I_{\psi }\to I_{\varphi }}$ and $h_{I_{\varphi }\to I_{\psi }}$ have additional properties such as Continuity, continuous differentiability or smoothness. In this case, when we say that a reparametrization has some property, we mean that both $h_{I_{\psi }\to I_{\varphi }}$ and $h_{I_{\varphi }\to I_{\psi }}$ have this property.

Definition: Equivalence of Parametrizations

Two parametrizations of a surface $\mathcal{S}$ are equivalent if and only if there exists a reparametrization between them.

Note

This is the most general definition of equivalence for parametrizations. However, sometimes we require that such a reparametrization also has additional properties such as Continuity, continuous differentiability or smoothness. In this case, we say that $\psi$ and $\varphi$ are “equivalent up to a PROPERTY reparametrization” such as “equivalent up to a continuous reparametrization” or “equivalent up to a smooth reparametrization”.