homogeneous Diophantine equation

I am trying to do the example x^2 + y^2 -1 =0 in the wiki article I linked but I am totally lost maybe because I am also sleep derived. They keep spawning in random variables like t or i. Also function `f` what that is supposed to look like. Are there any better resources for this?

https://en.wikipedia.org/wiki/Diophantine_equation#Homogeneous_equations