Tacnode

<h2 id="more-general-background">More general background</h2>
<p>Consider a <a href="/facts/Smooth_function/mffL4ch2">smooth</a> <a href="/facts/Real-valued_function/jetqlJpK">real-valued function</a> of two <a href="/facts/Variable_(mathematics)/VbQlrvKz">variables</a>, say <i>f</i> (<i>x</i>, <i>y</i>) where x and y are <a href="/facts/Real_number/R02gw5Pb">real numbers</a>. So f is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> from the plane to the line. The space of all such smooth functions is <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">acted</a> upon by the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphisms</a> of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of <a href="/facts/Coordinate/Mtd0q4SP">coordinate</a> in both the <a href="/facts/Domain_of_a_function/FQvoeUpX">source</a> and the <a href="/facts/Range_of_a_function/66CeTV6u">target</a>. This action splits the whole <a href="/facts/Function_space/lLUE2R1t">function space</a> up into <a href="/facts/Equivalence_class/1d2sh0TB">equivalence classes</a>, i.e. <a href="/facts/Group_orbit/Vh9VtUUw">orbits</a> of the group action.
</p><p>One such family of equivalence classes is <a href="/facts/Ak_singularity/DsCmXH7u">denoted by</a> ⁠
  
    
      
        
          A
          
            k
          
          
            ±
          
        
        ,
      
    
    {\displaystyle A_{k}^{\pm },}
  
⁠ where k is a non-negative <a href="/facts/Integer/PUwwrawx">integer</a>. This notation was introduced by <a href="/facts/V._I._Arnold/IMBhyfMS">V. I. Arnold</a>. A function f is said to be of type ⁠
  
    
      
        
          A
          
            k
          
          
            ±
          
        
      
    
    {\displaystyle A_{k}^{\pm }}
  
⁠ if it lies in the orbit of 
  
    
      
        
          x
          
            2
          
        
        ±
        
          y
          
            k
            +
            1
          
        
        ,
      
    
    {\displaystyle x^{2}\pm y^{k+1},}
  
 i.e. there exists a diffeomorphic change of coordinate in source and target which takes f into one of these forms. These simple forms 
  
    
      
        
          x
          
            2
          
        
        ±
        
          y
          
            k
            +
            1
          
        
      
    
    {\displaystyle x^{2}\pm y^{k+1}}
  
 are said to give <a href="/facts/Canonical_form/K2jhB44k">normal forms</a> for the type ⁠
  
    
      
        
          A
          
            k
          
          
            ±
          
        
      
    
    {\displaystyle A_{k}^{\pm }}
  
⁠-singularities.
</p><p>A curve with equation <i>f</i> = 0 will have a tacnode, say at the origin, if and only if f has a type ⁠
  
    
      
        
          A
          
            3
          
          
            −
          
        
      
    
    {\displaystyle A_{3}^{-}}
  
⁠-singularity at the origin.
</p><p>Notice that a <a href="/facts/Node_(algebraic_geometry)/8nvkrkE9">node</a> 
  
    
      
        (
        
          x
          
            2
          
        
        −
        
          y
          
            2
          
        
        =
        0
        )
      
    
    {\displaystyle (x^{2}-y^{2}=0)}
  
 corresponds to a type ⁠
  
    
      
        
          A
          
            1
          
          
            −
          
        
      
    
    {\displaystyle A_{1}^{-}}
  
⁠-singularity. A tacnode corresponds to a type ⁠
  
    
      
        
          A
          
            3
          
          
            −
          
        
      
    
    {\displaystyle A_{3}^{-}}
  
⁠-singularity. In fact each type ⁠
  
    
      
        
          A
          
            2
            n
            +
            1
          
          
            −
          
        
      
    
    {\displaystyle A_{2n+1}^{-}}
  
⁠-singularity, where <i>n</i> ≥ 0 is an integer, corresponds to a curve with self-intersection. As n increases, the order of self-intersection increases: transverse crossing, ordinary tangency, etc.
</p><p>The type ⁠
  
    
      
        
          A
          
            2
            n
            +
            1
          
          
            +
          
        
      
    
    {\displaystyle A_{2n+1}^{+}}
  
⁠-singularities are of no interest over the real numbers: they all give an isolated point. Over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, type ⁠
  
    
      
        
          A
          
            2
            n
            +
            1
          
          
            +
          
        
      
    
    {\displaystyle A_{2n+1}^{+}}
  
⁠-singularities and type ⁠
  
    
      
        
          A
          
            2
            n
            +
            1
          
          
            −
          
        
      
    
    {\displaystyle A_{2n+1}^{-}}
  
⁠-singularities are equivalent: (<i>x</i>, <i>y</i>) → (<i>x</i>, <i>iy</i>) gives the required diffeomorphism of the normal forms.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Acnode/DPchyTYP">Acnode</a></li>
<li><a href="/facts/Cusp_(singularity)/QH3fEGry">Cusp</a> or <i>Spinode</i></li>
<li><a href="/facts/Crunode/xysf5ckp">Crunode</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Salmon, George (1873). <a href="https://books.google.com/books?id=rjDR3Q4_RRgC&q&q=%22Tacnode%22"><i>A Treatise on the Higher Plane Curves: Intended as a Sequel to a Treatise on Conic Sections</i></a>.</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/Tacnode.html">"Tacnode"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li>
<li>Hazewinkel, M. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Tacnode">"Tacnode"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Schwartzman, Steven (1994), The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, MAA Spectrum, Mathematical Association of America, p. 217, ISBN 978-0-88385-511-9. <a href="978-0-88385-511-9" target="_blank">978-0-88385-511-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Schwartzman, Steven (1994), The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, MAA Spectrum, Mathematical Association of America, p. 217, ISBN 978-0-88385-511-9. <a href="978-0-88385-511-9" target="_blank">978-0-88385-511-9</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Tacnode open-in-new

Tacnode