Tacnode

In classical <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a tacnode (also called a point of osculation or double cusp) is a kind of <a href="/facts/Singular_point_of_a_curve/mGFKPaas">singular point</a> of a <a href="/facts/Curve/xBQ0pTBW">curve</a>. It is defined as a point where two (or more) <a href="/facts/Osculating_circle/mDs0CS3x">osculating circles</a> to the curve at that point are <a href="/facts/Tangent/sF0jH3EI">tangent</a>. This means that two branches of the curve have ordinary tangency at the double point.
The canonical example is

y
          
            2
          
        
        −
        
          x
          
            4
          
        
        =
        0.
      
    
    {\displaystyle y^{2}-x^{4}=0.}

A tacnode of an arbitrary curve may then be defined from this example, as a point of self-tangency <a href="/facts/Local_property/sdBEwIy2">locally</a> <a href="/facts/Diffeomorphic/855N5YYj">diffeomorphic</a> to the point at the origin of this curve. Another example of a tacnode is given by the links curve shown in the figure, with equation

(
        
          x
          
            2
          
        
        +
        
          y
          
            2
          
        
        −
        3
        x
        
          )
          
            2
          
        
        −
        4
        
          x
          
            2
          
        
        (
        2
        −
        x
        )
        =
        0.
      
    
    {\displaystyle (x^{2}+y^{2}-3x)^{2}-4x^{2}(2-x)=0.}

Tacnode open-in-new

Tacnode