structural:summarization, linguistic:translation, information:filtering, stylistic:style_transfer
As the complex projective line

The Riemann sphere can also be defined as the complex projective line. The points of the complex projective line can be defined as equivalence classes of non-null vectors in the complex vector space C 2 {\displaystyle \mathbf {C} ^{2}}: two non-null vectors (w, z) {\displaystyle (w,z)} and (u, v) {\displaystyle (u,v)} are equivalent iff (w, z) = (λ u, λ v) {\displaystyle (w,z)=(\lambda u,\lambda v)} for some non-zero coefficient λ ∈ C {\displaystyle \lambda \in \mathbf {C} }.

In this case, the equivalence class is written {\displaystyle } using projective coordinates. Given any point {\displaystyle } in the complex projective line, one of w {\displaystyle w} and z {\displaystyle z} must be non-zero, say w ≠ 0 {\displaystyle w\neq 0}. Then by the notion of equivalence, = {\displaystyle =\left}, which is in a chart for the Riemann sphere manifold.

This treatment of the Riemann sphere connects most readily to projective geometry. For example, any line (or smooth conic) in the complex projective plane is biholomorphic to the complex projective line. It is also convenient for studying the sphere's automorphisms, later in this article.