Parametric surface

<p>A parametric surface is a <a href="/facts/Surface_(mathematics)/whF08jvy">surface</a> in the <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> 
  
    
      
        
          
            R
          
          
            3
          
        
      
    
    {\displaystyle \mathbb {R} ^{3}}
  
 which is defined by a <a href="/facts/Parametric_equation/5Uvrlgqf">parametric equation</a> with two parameters 
  
    
      
        
          r
        
        :
        
          
            R
          
          
            2
          
        
        →
        
          
            R
          
          
            3
          
        
      
    
    {\displaystyle \mathbf {r} :\mathbb {R} ^{2}\to \mathbb {R} ^{3}}
  
. Parametric representation is a very general way to specify a surface, as well as <a href="/facts/Implicit_surface/gKpWgFNH">implicit representation</a>. Surfaces that occur in two of the main theorems of <a href="/facts/Vector_calculus/mXKIJLJE">vector calculus</a>, <a href="/facts/Stokes%27_theorem/z0pAoj2x">Stokes' theorem</a> and the <a href="/facts/Divergence_theorem/pUGKF1N2">divergence theorem</a>, are frequently given in a parametric form. The curvature and <a href="/facts/Arc_length/QgTrmLxY">arc length</a> of <a href="/facts/Curve/xBQ0pTBW">curves</a> on the surface, <a href="/facts/Surface_area/WNowzFCw">surface area</a>, differential geometric invariants such as the <a href="/facts/First_fundamental_form/uGBRcbSg">first</a> and <a href="/facts/Second_fundamental_form/3Ihz5tEc">second</a> fundamental forms, <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian</a>, <a href="/facts/Mean_curvature/lgdJ9b0k">mean</a>, and <a href="/facts/Principal_curvature/7K9D3NqU">principal</a> curvatures can all be computed from a given parametrization.
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Parametric surface open-in-new

Parametric surface