The Elliptic Curve Group Law via the Riemann–roch Theorem

And there are similar disjoint decompositions starting with the (x, z)-plane or the (y, z)-plane. When K = R, our intuition is that the real projective line PR is an ordinary line with a point at infinity identifying its opposite directions, and the projective plane is an ordinary plane surrounded by a circle at infinity identifying its opposite directions. The real projective plane differs from the complex projective line — PC is an ordinary plane with a single point at infinity gathering all of its directions together. If F and K are fields with F ⊂ K then we can identify PF with a subset of PK, PF ∼ −→ {K×(x, y, z) : [x : y : z] ∈ PF}.