Is there a classification of the maps from $\phi:U\to\mathbb{R}^m$ that preserve collinearity where $U\subseteq\mathbb{R}^n$ is open?
For example, consider the map $\phi:U\to\mathbb{R}^2$ defined by $\phi((x,y,z)):=(\frac{x}{z},\frac{y}{z})$ where $U=\{p\in\mathbb{R}^3:|p-(0,0,2)|<1\}$
Notice that $\phi$ preserves collinearity, that is, if three points $x,y,z$ are in a line, then $\phi(x),\phi(y),\phi(z)$ are also in a line. Also notice that $\phi$ is defined on its entire domain $U$, however it cannot be extended to the entirety of $\mathbb{R}^3$.
It is clear that all affine maps are collinearity preserving, however there are also these 'projection maps' like the one I gave above which also preserve collinearity. If you compose collinearity preserving maps, you will get another such map. I would like to know if general what form collinearty preserving maps can take. In particular I would like to know if there are any maps that cannot be expressed as a composition of affine maps and projection maps.
I suspect there is some theorem somewhere which address this in full. The name of the theorem would suffice as an answer. Thanks for your time.