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
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
More general background
Consider a smooth real-valued function of two variables, say Template:Math where Template:Mvar and Template:Mvar are real numbers. So Template:Mvar is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target. This action splits the whole function space up into equivalence classes, i.e. orbits of the group action.
One such family of equivalence classes is denoted by Template:Tmath where Template:Mvar is a non-negative integer. This notation was introduced by V. I. Arnold. A function Template:Mvar is said to be of type Template:Tmath if it lies in the orbit of i.e. there exists a diffeomorphic change of coordinate in source and target which takes Template:Mvar into one of these forms. These simple forms are said to give normal forms for the type Template:Tmath-singularities.
A curve with equation Template:Math will have a tacnode, say at the origin, if and only if Template:Mvar has a type Template:Tmath-singularity at the origin.
Notice that a node corresponds to a type Template:Tmath-singularity. A tacnode corresponds to a type Template:Tmath-singularity. In fact each type Template:Tmath-singularity, where Template:Math is an integer, corresponds to a curve with self-intersection. As Template:Mvar increases, the order of self-intersection increases: transverse crossing, ordinary tangency, etc.
The type Template:Tmath-singularities are of no interest over the real numbers: they all give an isolated point. Over the complex numbers, type Template:Tmath-singularities and type Template:Tmath-singularities are equivalent: Template:Math gives the required diffeomorphism of the normal forms.