Parametrization#

Definition: Surface Parametrization

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

A parametrization of \(\mathcal{S}\) is a continuous function \(\varphi: \mathcal{D} \subseteq \mathbb{R}^2 \to \mathbb{R}^n\) 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 \(\phi: \mathcal{D}_{\phi} \subseteq \mathbb{R}^2 \to \mathbb{R}^n\) be parametrizations of the same surface \(\mathcal{S} \subset \mathbb{R}^n\).

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

\[ \begin{aligned} \psi(\mathbf{x}) = \phi(h_{\mathcal{D}_{\psi} \to \mathcal{D}_{\phi}}(\mathbf{x})) \qquad \forall \mathbf{x} \in \mathcal{D}_{\psi} \\ \phi(\mathbf{x}) = \psi(h_{\mathcal{D}_{\phi} \to \mathcal{D}_{\psi}}(\mathbf{x})) \qquad \forall \mathbf{x} \in \mathcal{D}_{\phi} \end{aligned} \]

Note

This is the most general definition for reparametrization. However, it is quite common to require that both \(h_{I_{\psi} \to I_{\phi}}\) and \(h_{I_{\phi} \to I_{\psi}}\) have additional properties such as Continuity (Real Functions), continuous differentiability or smoothness. In this case, when we say that a reparametrization has some property, we mean that both \(h_{I_{\psi} \to I_{\phi}}\) and \(h_{I_{\phi} \to I_{\psi}}\) 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 (Real Functions), continuous differentiability or smoothness. In this case, we say that \(\psi\) and \(\phi\) are "equivalent up to a PROPERTY reparametrization" such as "equivalent up to a continuous reparametrization" or "equivalent up to a smooth reparametrization".