Whewell equation

<h2 id="properties">Properties</h2>
<p>If a point 
  
    
      
        
          
            
              r
              →
            
          
        
        =
        (
        x
        ,
        y
        )
      
    
    {\displaystyle {\vec {r}}=(x,y)}
  
 on the curve is given <a href="/facts/Parametric_curve/5Uvrlgqf">parametrically</a> in terms of the arc length, 
  
    
      
        s
        ↦
        
          
            
              r
              →
            
          
        
        ,
      
    
    {\displaystyle s\mapsto {\vec {r}},}
  
 then the tangential angle φ is determined by
</p><p>
  
    
      
        
          
            
              d
              
                
                  
                    r
                    →
                  
                
              
            
            
              d
              s
            
          
        
        =
        
          
            (
            
              
                
                  
                    
                      
                        d
                        x
                      
                      
                        d
                        s
                      
                    
                  
                
              
              
                
                  
                    
                      
                        d
                        y
                      
                      
                        d
                        s
                      
                    
                  
                
              
            
            )
          
        
        =
        
          
            (
            
              
                
                  cos
                  ⁡
                  φ
                
              
              
                
                  sin
                  ⁡
                  φ
                
              
            
            )
          
        
        
        
          since
        
        
        
          |
          
            
              
                d
                
                  
                    
                      r
                      →
                    
                  
                
              
              
                d
                s
              
            
          
          |
        
        =
        1
        ,
      
    
    {\displaystyle {\frac {d{\vec {r}}}{ds}}={\begin{pmatrix}{\frac {dx}{ds}}\\{\frac {dy}{ds}}\end{pmatrix}}={\begin{pmatrix}\cos \varphi \\\sin \varphi \end{pmatrix}}\quad {\text{since}}\quad \left|{\frac {d{\vec {r}}}{ds}}\right|=1,}

</p><p>which implies

d
              y
            
            
              d
              x
            
          
        
        =
        tan
        ⁡
        φ
        .
      
    
    {\displaystyle {\frac {dy}{dx}}=\tan \varphi .}

</p><p>Parametric equations for the curve can be obtained by integrating:

x
              
              
                
                =
                ∫
                cos
                ⁡
                φ
                
                d
                s
                ,
              
            
            
              
                y
              
              
                
                =
                ∫
                sin
                ⁡
                φ
                
                d
                s
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}x&=\int \cos \varphi \,ds,\\y&=\int \sin \varphi \,ds.\end{aligned}}}

</p><p>Since the <a href="/facts/Curvature/RFCAEsj8">curvature</a> is defined by

κ
        =
        
          
            
              d
              φ
            
            
              d
              s
            
          
        
        ,
      
    
    {\displaystyle \kappa ={\frac {d\varphi }{ds}},}

</p><p>the <a href="/facts/Ces%25C3%25A0ro_equation/TVDhxpV2">Cesàro equation</a> is easily obtained by differentiating the Whewell equation.
</p>
<h2 id="examples">Examples</h2>
<table><tbody><tr><th>Curve</th><th>Equation</th></tr><tr><td><a href="/facts/Line_(mathematics)/gJqsr4Gj">Line</a></td><td>                    φ        =        c              {\displaystyle \varphi =c}  </td></tr><tr><td><a href="/facts/Circle/9fy5ggKn">Circle</a></td><td>                    s        =        a        φ              {\displaystyle s=a\varphi }  </td></tr><tr><td><a href="/facts/Logarithmic_Spiral/4rwkxL7B">Logarithmic Spiral</a></td><td>                    s        =                                            a                              e                                  φ                  tan                  ⁡                  α                                                                    sin              ⁡              α                                            {\displaystyle s={\frac {ae^{\varphi \tan \alpha }}{\sin \alpha }}}  </td></tr><tr><td><a href="/facts/Catenary/fWVU770E">Catenary</a></td><td>                    s        =        a        tan        ⁡        φ              {\displaystyle s=a\tan \varphi }  </td></tr><tr><td><a href="/facts/Tautochrone/zrfIzYj1">Tautochrone</a></td><td>                    s        =        a        sin        ⁡        φ              {\displaystyle s=a\sin \varphi }  </td></tr></tbody></table>

<ul><li>Whewell, W. Of the Intrinsic Equation of a Curve, and its Application. Cambridge Philosophical Transactions, Vol. VIII, pp. 659-671, 1849. <a href="https://books.google.com/books?id=2vsIAAAAIAAJ&pg=PA659">Google Books</a></li>
<li>Todhunter, Isaac. William Whewell, D.D., An Account of His Writings, with Selections from His Literary and Scientific Correspondence. Vol. I. Macmillan and Co., 1876, London. Section 56: p. 317.</li>
<li>J. Dennis Lawrence (1972). <a href="https://archive.org/details/catalogofspecial00lawr/page/1"><i>A catalog of special plane curves</i></a>. Dover Publications. pp. <a href="https://archive.org/details/catalogofspecial00lawr/page/1">1–5</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-60288-5.</li>
<li>Yates, R. C.: <i>A Handbook on Curves and Their Properties</i>, J. W. Edwards (1952), "Intrinsic Equations" p124-5</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/WhewellEquation.html">"Whewell Equation"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

Whewell equation open-in-new

Whewell equation