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| {{refimprove|date=February 2010}}
| | Marvella is what you can call her but it's not the most feminine name out there. California is our birth location. One of the things she loves most is to do aerobics and now she is trying to make cash with it. My day job is a meter reader.<br><br>My page ... [http://ironptstudio.com/rev/239375 http://ironptstudio.com/] |
| [[Image:Tacnode.gif|thumb|right|214px|A tacnode at the origin of the curve defined by (''x''<sup>2</sup>+''y''<sup>2</sup> −3''x'')<sup>2</sup>−4''x''<sup>2</sup>(2−x)=0]]
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| In [[Glossary of classical algebraic geometry|classical 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]]|year=1994|isbn=9780883855119|page=217|url=http://books.google.com/books?id=SRw4PevE4zUC&pg=PA217}}.</ref> is a kind of [[singular point of a curve]]. It is defined as a point where two (or more) [[osculating circle]]s to the curve at that point are [[tangent]]. This means that two branches of the curve have ordinary tangency at the double point.<ref name="words"/>
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| The canonical example is
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| :<math>(y-x^2)(y+x^2) = 0.</math>
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| A tacnode of an arbitrary curve may then be defined from this example, as a point of self-tangency [[Local property|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
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| :<math> (x^2+y^2-3x)^2 -4x^2(2-x) = 0.</math>
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| ==More general background==
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| Consider a [[smooth function|smooth]] [[real-valued function]] of two [[variable (mathematics)|variables]], say ''f''(''x'', ''y'') where ''x'' and ''y'' are [[real number]]s. So ''f'' is a function from the plane to the line. The space of all such smooth functions is [[group action|acted]] upon by the [[group (mathematics)|group]] of [[diffeomorphism]]s of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of [[coordinate]] in both the [[Domain of a function|source]] and the [[range of a function|target]]. This action splits the whole [[function space]] up into [[equivalence class]]es, i.e. [[Group_orbit#Orbits_and_stabilizers|orbit]]s of the group action.
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| One such family of equivalence classes is denoted by [[Ak singularity|''A<sub>k</sub>''<sup>±</sup>]], where ''k'' is a non-negative [[integer]]. This notation was introduced by [[V. I. Arnold]]. A function ''f'' is said to be of type [[Ak singularity|''A<sub>k</sub>''<sup>±</sup>]] if it lies in the orbit of ''x''<sup>2</sup> ± ''y''<sup>''k''+1</sup>, i.e. there exists a diffeomorphic change of coordinate in source and target which takes ''f'' into one of these forms. These simple forms ''x''<sup>2</sup> ± ''y''<sup>''k''+1</sup> are said to give [[Canonical form|normal form]]s for the type [[Ak singularity|''A<sub>k</sub>''<sup>±</sup>]]-singularities.
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| A curve with equation ''f'' = 0 will have a tacnode, say at the origin, if and only if ''f'' has a type ''A''<sub>3</sub><sup>−</sup>-singularity at the origin.
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| Notice that a [[Node (graph theory)|node]] (''x''<sup>2</sup> − ''y''<sup>2</sup> = 0) corresponds to a type ''A''<sub>1</sub><sup>−</sup>-singularity. A tacnode corresponds to a type ''A''<sub>3</sub><sup>−</sup>-singularity. In fact each type ''A''<sub>2''n''+1</sub><sup>−</sup>-singularity, where ''n'' ≥ 0 is an integer, corresponds to a curve with self intersection. As ''n'' increases the order of self intersection increases: transverse crossing, ordinary tangency, etc.
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| The type ''A''<sub>2''n''+1</sub><sup>+</sup>-singularities are of no interest over the real numbers: they all give an isolated point. Over the complex numbers type ''A''<sub>2''n''+1</sub><sup>+</sup>-singularities and type ''A''<sub>2''n''+1</sub><sup>−</sup>-singularities are equivalent: (''x'',''y'') → (''x'', ''iy'') gives the required diffeomorphism of the normal forms.
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| ==See also==
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| *[[Acnode]]
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| *[[Cusp (singularity)|Cusp]] or ''Spinode''
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| *[[Crunode]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{MathWorld|title=Tacnode|urlname=Tacnode}}
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| {{Algebraic curves navbox}}
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| [[Category:Curves]]
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| [[Category:Singularity theory]]
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| [[Category:Algebraic curves]]
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Marvella is what you can call her but it's not the most feminine name out there. California is our birth location. One of the things she loves most is to do aerobics and now she is trying to make cash with it. My day job is a meter reader.
My page ... http://ironptstudio.com/