Singular point of a curve

In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a 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  satisfying an equation of the form  where  is a polynomial function  If  is expanded as

If the origin  is on the curve then . If  then the implicit function theorem guarantees there is a smooth function  so that the curve has the form  near the origin. Similarly, if  then there is a smooth function  so that the curve has the form  near the origin. In either case, there is a smooth map from  to the plane which defines the curve in the neighborhood of the origin. Note that at the origin

so the curve is non-singular or regular at the origin if at least one of the partial derivatives of  is non-zero. The singular points are those points on the curve where both partial derivatives vanish,

Regular points
Assume the curve passes through the origin and write  Then  can be written

If  is not 0 then  has a solution of multiplicity 1 at  and the origin is a point of single contact with line  If  then  has a solution of multiplicity 2 or higher and the line  or  is tangent to the curve. In this case, if  is not 0 then the curve has a point of double contact with  If the coefficient of ,  is 0 but the coefficient of  is not then the origin is a point of inflection of the curve. If the coefficients of  and  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.

Double points

If  and  are both  in the above expansion, but at least one of , ,  is not 0 then the origin is called a double point of the curve. Again putting   can be written

Double points can be classified according to the solutions of

Crunodes

If  has two real solutions for , that is if  then the origin is called a crunode. The curve in this case crosses itself at the origin and has two distinct tangents corresponding to the two solutions of  The function  has a saddle point at the origin in this case.

Acnodes

If  has no real solutions for , that is if  then the origin is called an acnode. In the real plane the origin is an 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  The function  has a local extremum at the origin in this case.

Cusps

If  has a single solution of multiplicity 2 for , that is if  then the origin is called a 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 formulas.

If one of the solutions of  is also a solution of  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  is a factor of  then the origin is called a biflecnode.

Multiple points

In general, if all the terms of degree less than  are 0, and at least one term of degree  is not 0 in , then curve is said to have a multiple point of order  or a k-ple point. The curve will have, in general,  tangents at the origin though some of these tangents may be imaginary.

Parametric curves
A parameterized curve in  is defined as the image of a function   The singular points are those points where

Many curves can be defined in either fashion, but the two definitions may not agree. For example, the cusp can be defined on an algebraic curve,  or on a parametrised curve,  Both definitions give a singular point at the origin. However, a node such as that of  at the origin is a singularity of the curve considered as an algebraic curve, but if we parameterize it as  then  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  can be parameterised by  which has a singularity at the origin. When parametrised by  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 curves which are defined as the zero set  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 states

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 points of an algebraic variety.

Types of singular points
Some of the possible singularities are:
An isolated point:  an acnode
Two lines crossing:  a crunode
A cusp:  also called a spinode
A tacnode: 
A rhamphoid cusp:

See also
Singular point of an algebraic variety
Singularity theory
Morse theory

References

Curves
Algebraic curves
Singularity theory