There are at least two types of extended real lines: the affinely extended real line, $\mathbb{R}\cup\{-\infty, \infty \}$, and the projectively extended real line, $\mathbb{R}\cup\{\infty\}$.
The latter can be identified with the real projective line $\mathbb{RP}^1$ so it is clearly related to algebraic geometry. However, both the terms "projective" as well as "affine" occur quite frequently (at least it seems to me) in algebraic geometry -- affine/projective varieties, affine/projective changes of coordinate systems, etc. There also seems to be a concept of affine extension; I am not sure.
Does the term "affinely extended real line" also come from algebraic geometry? Or is "projectively extended real line" the only one of the two which is related to algebraic geometry?