Tacnode

In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp) 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.

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

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