# Tacnode

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> Source: https://en.wikipedia.org/wiki/Tacnode
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{{short description|Point on a curve at which two or more osculating circles are tangent}}
{{refimprove|date=February 2010}}
thumb|right|214px|A tacnode at the origin of the curve defined by <math>(x^2+y^2-3x)^2 - 4x^2(2-x) = 0.</math>

In classical [algebraic geometry](/source/algebraic_geometry), a '''tacnode''' (also called a '''point of osculation''' or '''double cusp''')<ref name="words">{{citation|title=The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English|series=MAA Spectrum|first=Steven|last=Schwartzman|publisher=[Mathematical Association of America](/source/Mathematical_Association_of_America)|year=1994|isbn=978-0-88385-511-9|page=217|url=https://books.google.com/books?id=SRw4PevE4zUC&pg=PA217}}.</ref> is a kind of [singular point](/source/singular_point_of_a_curve) of a [curve](/source/curve). It is defined as a point where two (or more) [osculating circle](/source/osculating_circle)s to the curve at that point are [tangent](/source/tangent). This means that two branches of the curve have ordinary tangency at the double point.<ref name="words" />

The canonical example is
:<math>y^2-x^4= 0.</math>
A tacnode of an arbitrary curve may then be defined from this example, as a point of self-tangency [locally](/source/Local_property) [diffeomorphic](/source/diffeomorphic) to the point at the origin of this curve. Another example of a tacnode is given by the [links curve](/source/links_curve) shown in the figure, with equation
:<math>(x^2+y^2-3x)^2 - 4x^2(2-x) = 0.</math>

== More general background ==
Consider a [smooth](/source/smooth_function) [real-valued function](/source/real-valued_function) of two [variables](/source/variable_(mathematics)), say {{math|''f'' (''x'', ''y'')}} where {{mvar|x}} and {{mvar|y}} are [real number](/source/real_number)s. So {{mvar|f}} is a [function](/source/function_(mathematics)) from the plane to the line. The space of all such smooth functions is [acted](/source/Group_action_(mathematics)) upon by the [group](/source/group_(mathematics)) of [diffeomorphism](/source/diffeomorphism)s of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of [coordinate](/source/coordinate) in both the [source](/source/Domain_of_a_function) and the [target](/source/range_of_a_function). This action splits the whole [function space](/source/function_space) up into [equivalence class](/source/equivalence_class)es, i.e. [orbits](/source/Group_orbit) of the group action.

One such family of equivalence classes is [denoted by](/source/Ak_singularity) {{tmath|A_k^\pm,}} where {{mvar|k}} is a non-negative [integer](/source/integer). This notation was introduced by [V. I. Arnold](/source/V._I._Arnold). A function {{mvar|f}} is said to be of type {{tmath|A_k^\pm}} if it lies in the orbit of <math>x^2 \pm y^{k+1},</math> i.e. there exists a diffeomorphic change of coordinate in source and target which takes {{mvar|f}} into one of these forms. These simple forms <math>x^2 \pm y^{k+1}</math> are said to give [normal forms](/source/Canonical_form) for the type {{tmath|A_k^\pm}}-singularities.

A curve with equation {{math|1=''f'' = 0}} will have a tacnode, say at the origin, if and only if {{mvar|f}} has a type {{tmath|A_3^-}}-singularity at the origin.

Notice that a [node](/source/Node_(algebraic_geometry)) <math>(x^2-y^2=0)</math> corresponds to a type {{tmath|A_1^-}}-singularity. A tacnode corresponds to a type {{tmath|A_3^-}}-singularity. In fact each type {{tmath|A_{2n+1}^-}}-singularity, where {{math|''n'' ≥ 0}} is an integer, corresponds to a curve with self-intersection. As {{mvar|n}} increases, the order of self-intersection increases: transverse crossing, ordinary tangency, etc.

The type {{tmath|A_{2n+1}^+}}-singularities are of no interest over the real numbers: they all give an isolated point. Over the [complex number](/source/complex_number)s, type {{tmath|A_{2n+1}^+}}-singularities and type {{tmath|A_{2n+1}^-}}-singularities are equivalent: {{math|(''x'', ''y'') → (''x'', ''iy'')}} gives the required diffeomorphism of the normal forms.

== See also ==
* [Acnode](/source/Acnode)
* [Cusp](/source/Cusp_(singularity)) or ''Spinode''
* [Crunode](/source/Crunode)

== References ==
{{reflist}}
== Further reading ==
* {{cite book |url={{Google books|rjDR3Q4_RRgC&q|keywords="Tacnode"|plainurl=yes}}| title=A Treatise on the Higher Plane Curves: Intended as a Sequel to a Treatise on Conic Sections | last1=Salmon | first1=George | year=1873 }}

== External links ==
* {{MathWorld|title=Tacnode|urlname=Tacnode}}
* {{eom|id=32528|first1=M.|last1=Hazewinkel|author-link=Michiel Hazewinkel|title=Tacnode}}
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Adapted from the Wikipedia article [Tacnode](https://en.wikipedia.org/wiki/Tacnode) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Tacnode?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.