Developable surface

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a developable surface (or torse: archaic) is a smooth <a href="/facts/Surface_(mathematics)/whF08jvy">surface</a> with zero <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a>.  That is, it is a surface that can be <a href="/facts/Flattening/32pP4JsR">flattened</a> onto a <a href="/facts/Plane_(mathematics)/tAUtRFcn">plane</a> without <a href="/facts/Distortion/z8RJcwNy">distortion</a> (i.e. it can be bent without stretching or compression).  Conversely, it is a surface which can be made by <a href="/facts/Transformation_(mathematics)/Yr2J3o8H">transforming</a> a plane (i.e. "folding", "bending", "rolling", "cutting" and/or "gluing"). Because of these properties, developable surfaces are widely used in the design and fabrication of items to be made from sheet materials, ranging from <a href="/facts/Textiles/PJrVeM5S">textiles</a> to <a href="/facts/Sheet_metal/3yK7SwL4">sheet metal</a> such as <a href="/facts/Ductwork/34ThHzCt">ductwork</a> to <a href="/facts/Shipbuilding/XLy9lj1W">shipbuilding</a>.  
</p><p>In <a href="/facts/Three_dimensions/hwVgAHtV">three dimensions</a> all developable surfaces are <a href="/facts/Ruled_surface/uAGM9b1X">ruled surfaces</a> (but not vice versa). There are developable surfaces in <a href="/facts/Four-dimensional_space/vFQd0M7T">four-dimensional space</a> ⁠
  
    
      
        
          
            R
          
          
            4
          
        
      
    
    {\displaystyle \mathbb {R} ^{4}}
  
⁠ which are not ruled.  The <a href="/facts/Envelope_(mathematics)/jr5DkCGD">envelope</a> of a single parameter family of planes is called a developable surface.
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Developable surface open-in-new

Developable surface