Singular point of a curve

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A singular point on an 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 R² are defined as the zero set f⁻¹(0) for a polynomial function f:R²→R. The singular points are those points on the curve where both partial derivatives vanish,

$f(x,y)={\partial f\over\partial x}={\partial f\over\partial y}=0$ .

A parameterized curve in R² is defined as the image of a function g:R→R², g(t) = (g₁(t),g₂(t)). The singular points are those points where

${dg_1\over dt}={dg_2\over dt}=0.$

A cusp

Many curves can be defined in either fashion, and generally the two definitions agree. For example the cusp can be defined as an algebraic curve, x³−y² = 0, or as a parametrised curve, g(t) = (t²,t³). Both definitions give a singular point at the origin.

Care needs to be taken when choosing a parameterization. For instance the straight line y = 0 can be parametrised by g(t) = (t³,0) which has a singularity at the origin. When parametrised by g(t) = (t,0) it is non singular. 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 Rⁿ occurs as the solution set of f⁻¹(0) for some smooth function f:Rⁿ→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.

[edit] Types of singular points

Some of the possible singularities are:

An isolated point: x²+y² = 0, an acnode
Two lines crossing: x²-y² = 0, a crunode
A cusp: x³−y² = 0, also called a spinode.
A rhamphoid cusp: x⁵−y² = 0, also called a tacnode.

[edit] References

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