In classical

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a tacnode (also called a point of osculation or double cusp). is a kind of singular point of a

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

. It is defined as a point where two (or more)

osculating circle An osculating circle is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that point. The osculating circle provides a way to unders ...

s to the curve at that point are

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

. This means that two branches of the curve have ordinary tangency at the double point. The canonical example is :

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 :

(x^2+y^2-3x)^2 - 4x^2(2-x) = 0.

More general background

Consider a smooth

real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real ...

of two variables, say where and are

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s. So is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

s of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of

coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...

in both the source and the target. This action splits the whole

function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...

up into

equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...

es, i.e.

orbits In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificia ...

of the group action. One such family of equivalence classes is denoted by where is a non-negative

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

. This notation was introduced by V. I. Arnold. A function is said to be of type if it lies in the orbit of

x^2 \pm y^,

i.e. there exists a diffeomorphic change of coordinate in source and target which takes into one of these forms. These simple forms

x^2 \pm y^

are said to give normal forms for the type -singularities. A curve with equation will have a tacnode, say at the origin, if and only if has a type -singularity at the origin. Notice that a

node In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex). Node may refer to: In mathematics * Vertex (graph theory), a vertex in a mathematical graph *Vertex (geometry), a point where two or more curves, lines ...

(x^2-y^2=0)

corresponds to a type -singularity. A tacnode corresponds to a type -singularity. In fact each type -singularity, where is an integer, corresponds to a curve with self-intersection. As increases, the order of self-intersection increases: transverse crossing, ordinary tangency, etc. The type -singularities are of no interest over the real numbers: they all give an isolated point. Over the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s, type -singularities and type -singularities are equivalent: gives the required diffeomorphism of the normal forms.

References

External links

* * {{Algebraic curves navbox Curves Singularity theory Algebraic curves

More general background

See also

References

Further reading

External links