Motivation:
Consider an arbitrary conic section in $\mathbb{R}^2$ given by
$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$
Now consider the map $$\phi: \mathbb{R}^2 \rightarrow \mathbb{R}^2, \phi_{\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} }(x,y) = \left( \frac{ax + by + c}{dx+ey+f}, \frac{gx+hy+i}{dx+ey+f} \right) $$
This map sends a conic section to a conic section. This is easy enough to verify with algebra by observing that if for some (x,y):
$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$
Then there exists an $x'$ and $y'$ s.t. (if $\phi$ is invertible)$$ A \phi(x')^2 + B\phi(x')\phi(y') + C\phi(y')^2 + D\phi(x') + E\phi(y') + F = 0$$
And therefore
$$ A \left( \frac{ax'+ by'+ c}{dx'+ey'+f} \right)^2 + B \left( \frac{ax'+ by'+ c}{dx'+ey'+f} \right) \left( \frac{gx'+hy'+i}{dx'+ey'+f} \right) + C \left( \frac{gx'+hy'+i}{dx'+ey'+f} \right)^2 + D \left( \frac{ax'+ by'+ c}{dx'+ey'+f} \right) + E \left( \frac{gx'+hy'+i}{dx'+ey'+f} \right) + F = 0 $$
But by multiplying out all the denominators and grouping like powers of $(x')^n(y')^m$ we can clearly see that $(x',y')$ itself lies on some conic section.
So then (it feels clear to me that) we can conclude $\phi$ sends a conic section to a section.
The question:
Now is this set of $\phi$ the ONLY continuous functions $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ that send conics to conics?
Or do there exist some more exotic functions with this property?
It does seem this family of functions is well known as the set of homographies which I found out about via this question.