Tacnode

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In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp)^[1] is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to the curve at that point are tangent. This means that two branches of the curve have ordinary tangency at the double point.^[1]

The canonical example is

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

A tacnode of an arbitrary curve may then be defined from this example, as a point of self-tangency locally diffeomorphic 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

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