Riemann sphere said to be the same as complex projective line
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
      
    
    \mathbf {C} ^{2}
  : two non-null vectors 
  
    
      
        (
        w
        ,
        z
        )
      
    
    {\displaystyle (w,z)}
   and 
  
    
      (
      u
      ,
      v
      )
    
    (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 
  
    
      
        [
        w
        ,
        z
        ]
      
    
    {\displaystyle [w,z]}
   using projective coordinates. Given any point 
  
    
      
        [
        w
        ,
        z
        ]
      
    
    {\displaystyle [w,z]}
   in the complex projective line, one of 
  
    w
    w
   and 
  
    z
    z
   must be non-zero, say 
  
    
      
        w
        ≠
        0
      
    
    {\displaystyle w\neq 0}
  . Then by the notion of equivalence, 
  
    
      
        [
        w
        ,
        z
        ]
        =
        
          [
          
            1
            ,
            z
            
              /
            
            w
          
          ]
        
      
    
    {\displaystyle [w,z]=\left[1,z/w\right]}
  , 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.Applications
In complex analysis, a meromorphic function on the complex plane (or on any Riemann surface, for that matter) is a ratio 
  
    
      
        f
        
          /
        
        g
      
    
    {\displaystyle f/g}
   of two holomorphic functions 
  
    f
    f
   and 
  
    g
    g
  . As a map to the complex numbers, it is undefined wherever 
  
    g
    g
   is zero. However, it induces a holomorphic map 
  
    
      
        (
        f
        ,
        g
        )
      
    
    {\displaystyle (f,g)}
   to the complex projective line that is well-defined even where 
  
    
      
        g
        =
        0
      
    
    {\displaystyle g=0}
  . This construction is helpful in the study of holomorphic and meromorphic functions. For example, on a compact Riemann surface there are no non-constant holomorphic maps to the complex numbers, but holomorphic maps to the complex projective line are abundant.
The Riemann sphere has many uses in physics. In quantum mechanics, points on the complex projective line are natural values for photon polarization states, spin states of massive particles of spin 
  
    
      
        1
        
          /
        
        2
      
    
    {\displaystyle 1/2}
  , and 2-state particles in general (see also Quantum bit and Bloch sphere). The Riemann sphere has been suggested as a relativistic model for the celestial sphere. In string theory, the worldsheets of strings are Riemann surfaces, and the Riemann sphere, being the simplest Riemann surface, plays a significant role. It is also important in twistor theory.