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 (x, y) satisfying an equation of the form f (x, y) = 0, where f is a polynomial function f : R2 → R. If f is expanded as
Regular points[]
Assume the curve passes through the origin and write y = mx. Then f can be written
Double points[]
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Limacons.svg/500px-Limacons.svg.png)
If b0 and b1 are both 0 in the above expansion, but at least one of c0, c1, c2 is not 0 then the origin is called a double point of the curve. Again putting y = mx, f can be written
Crunodes[]
If c0 + 2mc1 + m2c2 = 0 has two real solutions for m, that is if c0c2 − c12 < 0, 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 c0 + 2mc1 + m2c2 = 0. The function f has a saddle point at the origin in this case.
Acnodes[]
If c0 + 2mc1 + m2c2 = 0 has no real solutions for m, that is if c0c2 − c12 > 0, 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 c0 + 2mc1 + m2c2 = 0. The function f has a local extremum at the origin in this case.
Cusps[]
If c0 + 2mc1 + m2c2 = 0 has a single solution of multiplicity 2 for m, that is if c0c2 − c12 = 0, 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 c0 + 2mc1 + m2c2 = 0 is also a solution of d0 + 3md1 + 3m2d2 + m3d3 = 0 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 c0 + 2mc1 + m2c2 is a factor of d0 + 3md1 + 3m2d2 + m3d3, then the origin is called a biflecnode.[2]
Multiple points[]
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/03/3_Petal_rose.svg/200px-3_Petal_rose.svg.png)
In general, if all the terms of degree less than k are 0, and at least one term of degree k is not 0 in f, then curve is said to have a multiple point of order k or a k-ple point. The curve will have, in general, k tangents at the origin though some of these tangents may be imaginary.[3]
Parametric curves[]
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
![](http://upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Cusp.svg/200px-Cusp.svg.png)
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, x3 − y2 = 0, or on 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 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[4][5] 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.
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 tacnode: x4−y2 = 0
- A rhamphoid cusp: x5 − y2 = 0.
See also[]
References[]
- Hilton, Harold (1920). "Chapter II: Singular Points". Plane Algebraic Curves. Oxford.
- Curves
- Algebraic curves
- Singularity theory