Please give me constructive feedback and references.
Does there exist an isomorphic compactification from the $x,y$ plane to $[0,1]^2?$ And $x,y \ge 1.$ The reason I'm asking is because I want to map linear functions of the form $y=x+b$ to vertical lines of the form $x=1/2+\sigma$ compactified into $[0,1],^2$ study them in this space, and then be able to transform them back. Maybe "isomorphic compactification" is not the right term?
If I could rigorously define and understand such a map, then I could transport these linear functions to a compact space and this would allow me to study them as loops on a complex torus. I could parameterize the class of vertical lines in $[0,1]^2$ as $x=1/2 \pm \sigma$ where $\sigma=it;$ $t\in\Bbb R(0,1).$
I read this wikipedia page and this question on mathoverflow and they were very helpful, but I still don't fully understand how to think about the complex torus, varieties, and algebraic torus.
https://mathoverflow.net/questions/20516/complex-torus-cn-%ce%9b-versus-cn