Equivalence of Parametric Surfaces

Definition: Equivalence of Parametric Surfaces

Let $\varphi :{\mathcal{D}}_{\varphi }\subseteq {\mathbb{R}}^2\to {\mathbb{R}}^n$ and $\psi :{\mathcal{D}}_{\psi }\subseteq {\mathbb{R}}^2\to {\mathbb{R}}^n$ be parametric surfaces.

We say that $\varphi$ and $\psi$ are equivalent if they have the same image and there exists a differentiable, bijective real vector field $h:{\mathcal{D}}_{\varphi }\to {\mathcal{D}}_{\psi }$ with a differentiable inverse such that

$$\psi \circ h=\varphi$$

Definition: Reparameterisation

Equivalent parametric surfaces are also known as reparameterisations.

Definition: Smooth Reparameterisation

We say that $\varphi$ and $\psi$ are smooth reparameterisations if they are both smooth and $h$ and $h^{-1}$ are continuously differentiable.

Theorem: Surface Normals of Smooth Parameterisations

If $\varphi :{\mathcal{D}}_{\varphi }\subseteq {\mathbb{R}}^2\to {\mathbb{R}}^n$ and $\psi :{\mathcal{D}}_{\psi }\subseteq {\mathbb{R}}^2\to {\mathbb{R}}^n$ are smooth reparameterisations, then their surface normal vectors ${\mathbf{N}}_{\psi }$ and ${\mathbf{N}}_{\varphi }$ at each $\mathbf{p}\in \varphi ({\mathcal{D}}_{\varphi })$ (or equivalently $\mathbf{p}\in \psi ({\mathcal{D}}_{\psi })$) are related by

$${\mathbf{N}}_{\varphi }=k(\mathbf{p}){\mathbf{N}}_{\psi },$$

where $k(\mathbf{p})\in \mathbb{R}$.

PROOF

TODO