Smooth projective plane

<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, smooth projective planes are special <a href="/facts/Projective_plane/p6ot8oNv">projective planes</a>. The most prominent example of a smooth projective plane is the <a href="/facts/Real_projective_plane/gGVd0ctY">real projective plane</a> 
  
    
      
        
          
            E
          
        
      
    
    {\displaystyle {\mathcal {E}}}
  
. Its geometric operations of joining two distinct points by a line and of intersecting two lines in a point are not only <a href="/facts/Continuous_map/sKbl02pB">continuous</a> but even <i><a href="/facts/Smoothness/mffL4ch2">smooth</a></i> (infinitely differentiable 
  
    
      
        =
        
          C
          
            ∞
          
        
      
    
    {\displaystyle =C^{\infty }}
  
). Similarly, the classical planes over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, the <a href="/facts/Quaternion/T3tnu7mX">quaternions</a>, and the <a href="/facts/Octonion/nC2sIz3p">octonions</a> are smooth planes. However, these are not the only such planes.
</p>

Smooth projective plane open-in-new

Smooth projective plane