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A 'singular point' on a curve is one where it is not smooth, for example, at a cusp.
The precise definition of a singular point depends on the type of curve being studied.
Algebraic curves in 'R'² are defined as the zero set ''f''⁻¹(0) for a polynomial function ''f'':'R'²→'R'. The singular points are those points on the curve where both partial derivatives vanish,
:$f(x,y)=\{partial\; foverpartial\; x\}=\{partial\; foverpartial\; y\}=0$.
A parameterized curve in ''R''² is defined as the image of a function ''g'':'R'→'R'², ''g''(''t'') = (''g''₁(''t''),''g''₂(''t'')). The singular points are those points where
: $\{dg\_1over\; dt\}=\{dg\_2over\; dt\}=0.$

Many curves can be defined in either fashion, but the two definitions may not agree. For example the cusp can be defined as an algebraic curve, ''x''³−''y''² = 0, or as a parametrised curve, ''g''(''t'') = (''t''²,''t''³). Both definitions give a singular point at the origin. However, a node such as that of ''y''²−''x''³−''x''² = 0 at the origin is a singularity of the curve considered as an algebraic curve, but if we parameterize it as ''g''(''t'') = (''t''²−1,''t''(''t''²−1)), then ''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 ''y'' = 0 can be parameterised by ''g''(''t'') = (''t''³,0) which has a singularity at the origin. When parametrised by ''g''(''t'') = (''t'',0) it is nonsingular. Hence, it is technically more correct to discuss singular points of a smooth mapping rather than a singular point of a curve.
The above definitions can be extended to cover ''implicit curves'' which are defined as the zero set ''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 Hassler Whitney ^{[1] [2]} states
:'Theorem'. Any closed set in 'R'ⁿ occurs as the solution set of ''f''⁻¹(0) for some 'smooth' function f:'R'ⁿ→'R'.
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.

Contents

Types of singular points

References

See also

Types of singular points

Some of the possible singularities are:

★ An isolated point: ''x''²+''y''² = 0, an acnode

★ Two lines crossing: ''x''²−''y''² = 0, a crunode

★ A cusp: ''x''³−''y''² = 0, also called a ''spinode''.

★ A rhamphoid cusp: ''x''⁵−''y''² = 0, also called a tacnode.

References

1. Brooker and Larden, ''Differential Germs and Catastrophes'', London Mathematical Society. Lecture Notes 17. Cambridge, (1975)
2. Bruce and Giblin, ''Curves and singularities'', (1984, 1992) ISBN 0-521-41985-9, ISBN 0-521-42999-4 (paperback)

See also

★ Singularity theory

★ Morse theory