Roman surface

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Roman surface or Steiner surface is a self-intersecting <a href="/facts/Map_(mathematics)/FABF1l2f">mapping</a> of the <a href="/facts/Real_projective_plane/gGVd0ctY">real projective plane</a> into <a href="/facts/Three-dimensional_space/KAqKWUbS">three-dimensional space</a>, with an unusually high degree of <a href="/facts/Symmetry/hpu7DK8p">symmetry</a>. This mapping is not an <a href="/facts/Immersion_(mathematics)/QMqIgkVY">immersion</a> of the projective plane; however, the figure resulting from removing six <a href="/facts/Singular_point_of_a_curve/mGFKPaas">singular points</a> is one. Its name arises because it was discovered by <a href="/facts/Jakob_Steiner/cXzHkeYt">Jakob Steiner</a> when he was in <a href="/facts/Rome/bh1Ttx3T">Rome</a> in 1844.
The simplest construction is as the image of a <a href="/facts/Sphere/4Ivb1cTj">sphere</a> centered at the origin under the map 
 
 
 
 f
 (
 x
 ,
 y
 ,
 z
 )
 =
 (
 y
 z
 ,
 x
 z
 ,
 x
 y
 )
 .
 
 
 {\displaystyle f(x,y,z)=(yz,xz,xy).}
 
 This gives an implicit <a href="/facts/Formula/BosTcKnv">formula</a> of

x
          
            2
          
        
        
          y
          
            2
          
        
        +
        
          y
          
            2
          
        
        
          z
          
            2
          
        
        +
        
          z
          
            2
          
        
        
          x
          
            2
          
        
        −
        
          r
          
            2
          
        
        x
        y
        z
        =
        0.
        
      
    
    {\displaystyle x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}-r^{2}xyz=0.\,}

Also, taking a parametrization of the sphere in terms of <a href="/facts/Longitude/HsC61ML9">longitude</a> (θ) and <a href="/facts/Latitude/IBYqkjhU">latitude</a> (φ), gives <a href="/facts/Parametric_equation/5Uvrlgqf">parametric equations</a> for the Roman surface as follows:

x
        =
        
          r
          
            2
          
        
        cos
        ⁡
        θ
        cos
        ⁡
        φ
        sin
        ⁡
        φ
      
    
    {\displaystyle x=r^{2}\cos \theta \cos \varphi \sin \varphi }

y
        =
        
          r
          
            2
          
        
        sin
        ⁡
        θ
        cos
        ⁡
        φ
        sin
        ⁡
        φ
      
    
    {\displaystyle y=r^{2}\sin \theta \cos \varphi \sin \varphi }

z
        =
        
          r
          
            2
          
        
        cos
        ⁡
        θ
        sin
        ⁡
        θ
        
          cos
          
            2
          
        
        ⁡
        φ
      
    
    {\displaystyle z=r^{2}\cos \theta \sin \theta \cos ^{2}\varphi }

The origin is a triple point, and each of the xy-, yz-, and xz-planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each coordinate axis which terminate in six pinch points. The entire surface has <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedral</a> <a href="/facts/Symmetry_group/CNqeseMp">symmetry</a>. It is a particular type (called type 1) of Steiner surface, that is, a 3-dimensional <a href="/facts/Linear_projection/sElXGkxD">linear projection</a> of the <a href="/facts/Veronese_surface/mXhQ3ouK">Veronese surface</a>.

Roman surface open-in-new

Roman surface