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?

Answer 1

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 $ad - bc \neq 0$.