In classical

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

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

singular point Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...

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 In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve i ...

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 the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

. 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 In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). P ...

diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two man ...

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 Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...

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 f ...

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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s. So is a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

from the plane to the line. The space of all such smooth functions is

acted Agency for Technical Cooperation and Development, commonly known as ACTED, is a French humanitarian non-governmental organisation. It is a non-governmental, non-political and non-profit organisatio. ACTED works in 37 countries responding to eme ...

upon by the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

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

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 the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...

in both the

source Source may refer to: Research * Historical document * Historical source * Source (intelligence) or sub source, typically a confidential provider of non open-source intelligence * Source (journalism), a person, publication, publishing institute o ...

and the

target Target may refer to: Physical items * Shooting target, used in marksmanship training and various shooting sports ** Bullseye (target), the goal one for which one aims in many of these sports ** Aiming point, in field artillery, f ...

. This action splits the whole function space up into equivalence classes, i.e.

orbits In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...

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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

. 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 Database normalization or database normalisation (see spelling differences) is the process of structuring a relational database in accordance with a series of so-called normal forms in order to reduce data redundancy and improve data integri ...

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 fo ...

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

External links