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{{Short description|Point on a curve not given by a smooth embedding of a parameter}}
In [[Geometry|geometry]], a '''singular point''' on a [[Curve|curve]] is one where the curve is not given by a smooth embedding of a [[Parametrization (geometry)|parameter]]. The precise definition of a singular point depends on the type of curve being studied.

==Algebraic curves in the plane==
Algebraic curves in the plane may be defined as the set of points {{math|(''x'', ''y'')}} satisfying an equation of the form <math>f(x,y) = 0,</math> where {{mvar|f }} is a [[Polynomial|polynomial]] function {{tmath|f: \R^2 \to \R.}} If {{mvar|f }} is expanded as
<math display="block">f = a_0 + b_0 x + b_1 y + c_0 x^2 + 2c_1 xy + c_2 y^2 + \cdots</math>
If the origin {{math|(0, 0)}} is on the curve then {{math|1=''a''{{sub|0}} = 0}}. If {{math|''b''{{sub|1}} ≠ 0}} then the [[Implicit function theorem|implicit function theorem]] guarantees there is a smooth function {{mvar|h}} so that the curve has the form {{math|1=''y'' = ''h''(''x'')}} near the origin. Similarly, if {{math|''b''{{sub|0}} ≠ 0}} then there is a smooth function {{mvar|k}} so that the curve has the form {{math|1=''x'' = ''k''(''y'')}} near the origin. In either case, there is a smooth map from {{tmath|\R}} to the plane which defines the curve in the neighborhood of the origin. Note that at the origin
<math display="block">b_0 = \frac{\partial f}{\partial x}, \; b_1 = \frac{\partial f}{\partial y},</math>
so the curve is non-singular or ''regular'' at the origin if at least one of the [[Partial derivative|partial derivative]]s of {{mvar|f }} is non-zero. The singular points are those points on the curve where both partial derivatives vanish,
<math display="block">f(x,y) = \frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} = 0.</math>

===Regular points===
Assume the curve passes through the origin and write <math>y = mx.</math> Then {{mvar|f }} can be written
<math display="block">f= (b_0 + m b_1) x + (c_0 + 2m c_1 + c_2 m^2)x^2 + \cdots.</math>
If <math>b_0 + mb_1</math> is not 0 then {{math|1=''f'' = 0}} has a solution of multiplicity 1 at {{math|1=''x'' = 0}} and the origin is a point of single contact with line <math>y = mx.</math> If <math>b_0 + mb_1 = 0</math> then {{math|1=''f'' = 0}} has a solution of multiplicity 2 or higher and the line <math>y = mx,</math> or <math>b_0x + b_1y = 0,</math> is tangent to the curve. In this case, if <math>c_0 + 2mc_1 + c_2m^2</math> is not 0 then the curve has a point of double contact with <math>y = mx.</math> If the coefficient of {{math|''x''{{sup|2}}}}, <math>c_0 + 2mc_1 + c_2m^2,</math> is 0 but the coefficient of {{math|''x''{{sup|3}}}} is not then the origin is a [[Inflection point|point of inflection]] of the curve. If the coefficients of {{math|''x''{{sup|2}}}} and {{math|''x''{{sup|3}}}} are both 0 then the origin is called ''point of undulation'' of the curve. This analysis can be applied to any point on the curve by translating the coordinate axes so that the origin is at the given point.<ref>Hilton Chapter II §1</ref>

===Double points===
[[Image:Limacons.svg|thumb|500px|none|Three [[Limaçon|limaçon]]s illustrating the types of double point. When converted to Cartesian coordinates as <math>(x^2 + y^2 - x)^2 = (1.5)^2 (x^2 + y^2),</math> the left curve acquires an acnode at the origin, which is an isolated point in the plane. The central curve, the [[Cardioid|cardioid]], has a cusp at the origin. The right curve has a crunode at the origin and the curve crosses itself to form a loop.]]

If {{math|''b''{{sub|0}}}} and {{math|''b''{{sub|1}}}} are both {{math|0}} in the above expansion, but at least one of {{math|''c''{{sub|0}}}}, {{math|''c''{{sub|1}}}}, {{math|''c''{{sub|2}}}} is not 0 then the origin is called a '''''double point''''' of the curve. Again putting <math>y = mx,</math> {{mvar|f }} can be written
<math display="block">f = (c_0 + 2m c_1 + c_2 m^2)x^2 + (d_0 + 3md_1 + 3 m^2 d_2 + d_3 m^3) x^3 + \cdots.</math>
Double points can be classified according to the solutions of <math>c_0 + 2mc_1 + m^2c_2 = 0.</math>

====Crunodes====
{{main article|Crunode}}
If <math>c_0 + 2mc_1 + m^2c_2 = 0</math> has two real solutions for {{mvar|m}}, that is if <math>c_0c_2 - c_1^2 < 0,</math> then the origin is called a ''[[Crunode|crunode]]''. The curve in this case crosses itself at the origin and has two distinct tangents corresponding to the two solutions of <math>c_0 + 2mc_1 + m^2c_2 = 0.</math> The function {{mvar|f }} has a [[Saddle point|saddle point]] at the origin in this case.

====Acnodes====
{{main article|Acnode}}
If <math>c_0 + 2mc_1 + m^2c_2 = 0</math> has no real solutions for {{mvar|m}}, that is if <math>c_0c_2 - c_1^2 > 0,</math> then the origin is called an ''[[Acnode|acnode]]''. In the real plane the origin is an [[Isolated point|isolated point]] on the curve; however when considered as a complex curve the origin is not isolated and has two imaginary tangents corresponding to the two complex solutions of <math>c_0 + 2mc_1 + m^2c_2 = 0.</math> The function {{mvar|f }} has a [[Maxima and minima|local extremum]] at the origin in this case.

====Cusps====
{{Main article|Cusp (singularity)}}
If <math>c_0 + 2mc_1 + m^2c_2 = 0</math> has a single solution of multiplicity 2 for {{mvar|m}}, that is if <math>c_0c_2 -  c_1^2 = 0,</math> then the origin is called a [[Cusp (singularity)|''cusp'']]. The curve in this case changes direction at the origin creating a sharp point. The curve has a single tangent at the origin which may be considered as two coincident tangents.

====Further classification====
The term ''node'' is used to indicate either a crunode or an acnode, in other words a double point which is not a cusp. The number of nodes and the number of cusps on a curve are two of the invariants used in the [[Plücker formula]]s.

If one of the solutions of <math>c_0 + 2mc_1 + m^2c_2 = 0</math> is also a solution of <math>d_0 + 3md_1 + 3m^2d_2 + m^3d_3 = 0,</math> then the corresponding branch of the curve has a point of inflection at the origin. In this case the origin is called a ''flecnode''. If both tangents have this property, so <math>c_0 + 2mc_1 + m^2c_2</math> is a factor of <math>d_0 + 3md_1 + 3m^2d_2 + m^3d_3,</math> then the origin is called a ''biflecnode''.<ref>Hilton Chapter II §2</ref>

===Multiple points===
[[Image:3 Petal rose.svg|thumb|200px|right|A curve with a triple point at the origin: {{math|1=''x''(''t'') = sin(2''t'') + cos(''t'')}}, {{math|1=''y''(''t'') = sin(''t'') + cos(2''t'')}}]]
In general, if all the terms of degree less than {{mvar|k}} are 0, and at least one term of degree {{mvar|k}} is not 0 in {{mvar|f}}, then curve is said to have a ''multiple point'' of order {{mvar|k}} or a ''k-ple point''. The curve will have, in general, {{mvar|k}} tangents at the origin though some of these tangents may be imaginary.<ref>Hilton Chapter II §3</ref>

==Parametric curves==
A [[Parametric equation|parameterized]] curve in {{tmath|\R^2}} is defined as the image of a function {{tmath|g: \R \to \R^2,}} <math>g(t) = (g_1(t),g_2(t)).</math> The singular points are those points where
<math display="block">\frac{dg_1}{dt} = \frac{dg_2}{dt} = 0.</math>

[[Image:cusp.svg|thumb|200px|A cusp in the [[Semicubical parabola|semicubical parabola]] <math>y^2=x^3</math>]]
Many curves can be defined in either fashion, but the two definitions may not agree. For example, the [[Cusp (singularity)|cusp]] can be defined on an [[Algebraic curve|algebraic curve]], <math>x^3 - y^2 = 0,</math> or on a parametrised curve, <math>g(t) = (t^2, t^3).</math> Both definitions give a singular point at the origin. However, a [[Crunode|node]] such as that of <math>y^2 - x^3 - x^2 = 0</math> at the origin is a singularity of the curve considered as an algebraic curve, but if we parameterize it as <math>g(t) = (t^2 - 1, t(t^2 - 1)),</math> then {{tmath|g'(t)}} never vanishes, and hence the node is ''not'' a singularity of the parameterized curve as defined above.

Care needs to be taken when choosing a parameterization. For instance the straight line {{math|1=''y'' = 0}} can be parameterised by <math>g(t) = (t^3, 0),</math> which has a singularity at the origin. When parametrised by <math>g(t) = (t, 0),</math> it is nonsingular. Hence, it is technically more correct to discuss ''singular points of a smooth mapping'' here rather than a singular point of a curve.

The above definitions can be extended to cover ''[[Implicit function|implicit]] curves'' which are defined as the zero set {{tmath|f^{-1}(0)}} of a smooth function, and it is not necessary just to consider algebraic varieties. The definitions can be extended to cover curves in higher dimensions.

A theorem of [[Biography:Hassler Whitney|Hassler Whitney]]<ref>Th. Bröcker, ''Differentiable Germs and Catastrophes'', London Mathematical Society. Lecture Notes 17. Cambridge, (1975)</ref><ref>Bruce and Giblin, ''Curves and singularities'', (1984, 1992) {{isbn|0-521-41985-9}}, {{isbn|0-521-42999-4}} (paperback)</ref> states

{{math theorem| Any closed set in {{tmath|\R^n}} occurs as the solution set of {{tmath|f^{-1}(0)}} for some '''smooth''' function <math>f: \R^n \to \R.</math>}}

Any parameterized curve can also be defined as an implicit curve, and the classification of singular points of curves can be studied as a classification of [[Singular point of an algebraic variety|singular points of an algebraic variety]].

==Types of singular points==
Some of the possible singularities are:
*An isolated point: <math>x^2 + y^2 = 0, </math> an [[Acnode|acnode]]
*Two lines crossing: <math>x^2 - y^2 = 0,</math> a [[Crunode|crunode]]
*A [[Cusp (singularity)|cusp]]: <math>x^3 - y^2 = 0,</math> also called a ''spinode''
*A [[Tacnode|tacnode]]: <math>x^4 - y^2 = 0</math>
*A rhamphoid cusp: <math>x^5 - y^2 = 0.</math>

==See also==
*[[Singular point of an algebraic variety]]
*[[Singularity theory]]
*[[Morse theory]]

==References==
<references/>
*{{cite book |title=Plane Algebraic Curves|first=Harold|last=Hilton|publisher=Oxford|year=1920
|chapter=Chapter II: Singular Points|url=https://archive.org/stream/cu31924001544216#page/n37/mode/1up}}
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