Define $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that maps $(x,y)$ to $(\arctan{x}, \arctan{y})$. It maps the $\mathbb{R}^2$ plane to the interior of a square.
Define $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that maps $(x,y)$ to $\displaystyle \left(\frac{x}{\sqrt{x^2+y^2+1}}, \frac{y}{\sqrt{x^2+y^2+1}} \right)$. It maps the $\mathbb{R}^2$ plane to an open (unit) disk.
I realize that $f$ and $g$ are different in the following sense:
- Add the boundaries to their images to make them a square with the boundary and a closed disk. (It's like compactification or "closure".)
- The square with the boundary is a manifold with corners, and the closed disk is a manifold with boundary, and they are not diffeomorphic to each other.
I suspect the difference between $f$ and $g$ as I noted above is a manifestation of something more fundamental. Hence my somewhat vague-wording question: How should I understand this difference between the two functions? What makes them have different "boundary structures"?
[I'd like to add some words on the background of what I am working on in case it helps people understand where I come from.
As a consequence of the different "boundary structures" between the square and the closed disk, $f$ has four "limiting functions" corresponding to the four sides of the square. For example, the upper side of the square corresponds $\lim_{y \rightarrow \infty} f = \tilde{f}$, where $\tilde{f}(x) = \left(\arctan{x}, \frac{\pi}{2}\right)$. In contrast, $g$ has only one "limiting function", the one corresponding to the circle.
Why am I interested in this kind of "limiting functions"? Because I am studying how to approximate certain functions. If we add a scaling factor $N$ to the definition of $f$ and have it now send $(x,y)$ to $(\arctan{x}, \frac{1}{N}\arctan{y})$, where $N \gg 1$, the square will now be squeezed along the y direction and become a thin rectangle. Then I can approximate $f$ with, say, $\tilde{f}$ as defined above, which, in a certain sense, is simpler but has little loss of fidelity. For $g$, similarly I can have it map $(x,y)$ to $\left(\frac{x}{\sqrt{x^2+y^2+1}}, \frac{1}{N}\frac{y}{\sqrt{x^2+y^2+1}} \right)$ and have the disk squeezed to an ellipse shape, but in this case I have only one "limiting" or approximating function.]
Any input (perspectives, references, correcting my terminologies or notations, etc.) on this will be received gratefully.