📂Geometry

Coordinate Transformation in Curved Surface Theory

Definition ¹

Assuming that $2$th-dimensional Euclidean space $U \subset \mathbb{R}^{2}$ is a set, let’s say $k \in \mathbb{N}$. If for $k \in \mathbb{N}$ a bijective function $f : U \to \mathbb{R}^{3}$ and its inverse function $f^{-1}$ are both $C^{k}$ functions, it is called a Coordinate Transformation.

Explanation

The definition of coordinate transformation reminds us of the homeomorphism and diffeomorphism discussed in general topology. It is specifically about dealing with $3$-dimensional space within the context of geometry, hence the domain and range are particular. Conceptually, it corresponds to reparameterization in curve theory and, intuitively, can be seen as a one-to-one correspondence to ensure differentiability in the codomain, just as in the domain.

The following theorem indicates that regardless of the coordinate transformation, the regularity of a simple surface is maintained.

Theorem

Let’s say $U , V \subset \mathbb{R}^{2}$ is a set. For a simple surface $\mathbf{x} : U \to \mathbb{R}^{3}$ and a coordinate transformation $f : V \to U$, $\mathbf{y} = \mathbf{x} \circ f : V \to \mathbb{R}^{3}$ is a simple surface having the same image as $\mathbf{x}$.