Isomorphism from $\mathbb{R}^2$ to $\mathbb{D}$ such that lines become circular arcs

Question

I'm currently working on a hobbyist math project that require taking lines on an infinite plane, and projecting them onto a finite (euclidean) surface such that intersections are preserved.

Does there exist an isomorphism from $\mathbb{R}^2$ to $\mathbb{D}$ (the complex unit disc) such that lines become circular arcs?

@quid Not until now. Based on the wikipedia summary, I think this directly answers my question. You can feel free to turn your comment into an answer so that I can accept it. — Ryan, Jan 26 '15 at 05:16
I am glad it is usefl for you. I will post an answer later today. — quid, Jan 26 '15 at 12:29

score 1 · Accepted Answer · answered Jan 27 '15 at 10:57

A common class of maps that takes lines to circular arcs are Möbius transformations. They are most commonly considered on complex numbers, but this translates to $\mathbb{R}^2$.

A Möbius transformation of the (extended) complex plane is a map $z \mapsto \frac{a z + b}{c z +d}$ with $ad - bc \neq 0$.

Isomorphism from $\mathbb{R}^2$ to $\mathbb{D}$ such that lines become circular arcs

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