What you're intuiting might be projective space, which is secretly fundamental to the study of plane curves! https://math.stackexchange.com/questions/85394/why

What you're intuiting might be projective space, which is secretly fundamental to the study of plane curves! https://math.stackexchange.com/questions/85394/why-the-emphasis-on-projective-space-in-algebraic-geometry https://math.stackexchange.com/questions/85394/why-the-empha... It's sometimes inconvenient to have all lines to intersect at a single infinite point, eg. for Bezout's theorem, so mathematicians instead add a point for every +- direction.