As part of a recent project, I am seeking theoretical (literature) results with regard to the existence of a one-to-one map (where $S\subset \mathbb{R}^3$ is a closed surface) $h:S\rightarrow\mathbb{R}\times [0,2\pi)\times[-\pi,\pi)$ which maps points on $S$ to spherical coordinates and is the composition of an affine transformation $f$ and a map from Cartesian to Spherical coordinates $g$; in other words, $h=g\circ f$.
What are necessary and/or sufficient conditions for the existence of such a map? What helpful and/or related pieces of literature with results on this are available?
Note: I am a novice who has only taken the basic undergraduate sequence of Mathematics courses and I only have a limited background in differential geometry and topology.
As stated, I believe that you really want $h$ to be the restiction of an affine map to $S$, since an affine map in this instance would be of the form $f: \mathbb{R}^3\to \mathbb{R}^3$ so if $g: \mathbb{R}^3\to \mathbb{R}\times [0,2\pi)\times [-\pi,\pi)$ then the composition to $g\circ f: \mathbb{R}^3\to \mathbb{R}\times [0,2\pi)\times [-\pi,\pi)$ which does not have the correct domain. If we take the restriction of $f$ to $S$, i.e. $f\bigg\rvert_{S}$ we get a map with the correct domain.
There also seems to be some confusion with spherical coordinates. As it stands you imply that spherical coordinates take the form of a map $\mathbb{R}^3\to \mathbb{R}\times [0,2\pi)\times [-\pi,\pi)$. If this map takes the standard form as the inverse of the traditional spherical coordinates it cannot be continuous as points in a neighborhood of the half plane $P=\{ (x,y,z)\in \mathbb{R}^3\mid y=0, x\geq 0\}$ will get sent far away from one another (think of this as having to "cut" along $P$ in order to make such a map continuous). For this reason, one typically restricts to $\mathbb{R}^3\backslash P$ when taking about spherical coordinates (this space is $\mathbb{R}^3$ with the half plane $P$ removed.) Also, in order for spherical coordinates and their inverse map to be well defined, we take the range of the spherical coordinates map to be $\mathbb{R}_{>0}\times (0,2\pi)\times (-\pi,\pi)$
So I will now rephrase your question in the way I understand it: Given $S$ a closed surface in $\mathbb{R}^3$, does there exist an affine map $f: \mathbb{R}^3\to \mathbb{R^3}$ such that $f\vert_{S}(S)$ lies in $\mathbb{R}^3\backslash P$ and the composition $g\circ f\vert_{S}$ is injective.
Such a map is always guaranteed to exist through the following reasoning.
Let $S$ be a closed surface embedded in $\mathbb{R}^3$. If $S\cap P=\varnothing$ simply take $f=\mathrm{Id}_{\mathbb{R}^3}$. Since $g$ is injective and $f$ is injective $g\circ f\vert_S$ is injective.
If $S\cap P\neq \varnothing$, consider the function $F: S\to \mathbb{R}$ given by the $x$ coordinate. Since $S$ is compact, $F$ has a maximum, $x_0$. Applying the translation $f:(x,y,z)\mapsto (x-x_0-1, y,z)$ to $S$ gives $f(S)\cap P=\varnothing$ and hence the composition $g\circ f\vert_S$ is injective.