Singular point of a curve

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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 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
A cusp

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, x3−y2 = 0, or as a parametrised curve, g(t) = (t2,t3). Both definitions give a singular point at the origin. However, a node such as that of y2−x3−x2 = 0 at the origin is a singularity of the curve considered as an algebraic curve, but if we parameterize it as g(t) = (t2−1,t(t2−1)), 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) = (t3,0) 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 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

  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)

[edit] See also