Focal surface

For a <a href="/facts/Surface_(mathematics)/whF08jvy">surface</a> in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the <a href="/facts/Tangent/sF0jH3EI">tangential</a> <a href="/facts/Sphere/jywYLNM2">spheres</a> whose radii are the <a href="/facts/Multiplicative_inverse/yczdYyCw">reciprocals</a> of one of the <a href="/facts/Principal_curvature/7K9D3NqU">principal curvatures</a> at the point of tangency. Equivalently it is the surface formed by the centers of the circles which <a href="/facts/Osculating_curve/pIAF1Kxv">osculate</a> the <a href="/facts/Curvature_line/7K9D3NqU">curvature lines</a>.

As the principal curvatures are the eigenvalues of the second fundamental form, there are two at each point, and these give rise to two points of the focal surface on each <a href="/facts/Surface_normal/jhtLIhcu">normal direction</a> to the surface. Away from <a href="/facts/Umbilical_point/kGaaHg4f">umbilical points</a>, these two points of the focal surface are distinct; at umbilical points the two sheets come together. When the surface has a <a href="/facts/Ridge_(differential_geometry)/jIDh1ndu">ridge</a> the focal surface has a <a href="/facts/Cuspidal_edge/QH3fEGry">cuspidal edge</a>, three such edges pass through an elliptical umbilic and only one through a hyperbolic umbilic. At points where the <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> is zero, one sheet of the focal surface will have a point at infinity corresponding to the zero principal curvature.
If 
 
 
 
 
 
 
 p
 →
 
 
 
 
 
 {\displaystyle {\vec {p}}}
 
 is a point of the given surface, 
 
 
 
 
 
 
 n
 →
 
 
 
 
 
 {\displaystyle {\vec {n}}}
 
 the unit normal and 
 
 
 
 
 k
 
 1
 
 
 ,
 
 k
 
 2
 
 
 
 
 {\displaystyle k_{1},k_{2}}
 
 the <a href="/facts/Principal_curvature/7K9D3NqU">principal curvatures</a> at 
 
 
 
 
 
 
 p
 →
 
 
 
 
 
 {\displaystyle {\vec {p}}}
 
, then

b
                →
              
            
          
          
            1
          
        
        (
        
          
            
              p
              →
            
          
        
        )
        =
        
          
            
              p
              →
            
          
        
        +
        
          
            
              
                n
                →
              
            
            
              k
              
                1
              
            
          
        
        
      
    
    {\displaystyle {\vec {b}}_{1}({\vec {p}})={\vec {p}}+{\frac {\vec {n}}{k_{1}}}\quad }
  
 and 
  
    
      
        
        
          
            
              
                b
                →
              
            
          
          
            2
          
        
        (
        
          
            
              p
              →
            
          
        
        )
        =
        
          
            
              p
              →
            
          
        
        +
        
          
            
              
                n
                →
              
            
            
              k
              
                2
              
            
          
        
         
      
    
    {\displaystyle \quad {\vec {b}}_{2}({\vec {p}})={\vec {p}}+{\frac {\vec {n}}{k_{2}}}\ }

are the corresponding two points of the focal surface.

Focal surface open-in-new

Focal surface