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Singular point of a curve

 

 
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Singular point of a curve
 
 
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A singular point on a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 is one where it is not smooth
Smooth function

In mathematical analysis, a differentiability class is a classification of function according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives....
, for example, at a cusp.

The precise definition of a singular point depends on the type of curve being studied.

Algebraic curve
Algebraic curve

In algebraic geometry, an algebraic curve is an algebraic variety of dimension of an algebraic variety one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections....
s 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 derivative
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
s vanish,

A parameterized
Parametric equation

In mathematics, parametric equations are a method of defining a curve. A simple kinematics example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
 curve in R2 is defined as the image of a function g:R?R2, g(t) = (g1(t),g2(t)).






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A singular point on a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 is one where it is not smooth
Smooth function

In mathematical analysis, a differentiability class is a classification of function according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives....
, for example, at a cusp.

The precise definition of a singular point depends on the type of curve being studied.

Algebraic curve
Algebraic curve

In algebraic geometry, an algebraic curve is an algebraic variety of dimension of an algebraic variety one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections....
s 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 derivative
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
s vanish,

A parameterized
Parametric equation

In mathematics, parametric equations are a method of defining a curve. A simple kinematics example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
 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


Cusp
Many curves can be defined in either fashion, but the two definitions may not agree. For example the cusp
Cusp

Cusp may refer to:*Cusp , a singular point of a curve*Cusp form in modular form theory*Cuspidal representation, a generalization of cusp forms in the theory of automorphic representations...
 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
Crunode

A crunode is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines.For a plane curve, defined as the locus of points f = 0, where f is a smooth function of variables x and y ranging over the real numbers, a crunode of the curve is a singularity theory of the function f, w...
 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
Implicit function

In mathematics, an implicit function is a function in which the dependent variable has not been given "explicitly" in terms of the independent variable....
 curves which are defined as the zero set f-1(0) of a smooth function
Smooth function

In mathematical analysis, a differentiability class is a classification of function according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives....
, 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
Hassler Whitney

Hassler Whitney was an United States mathematician. He was one of the founders of singularity theory....
  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
Singular point of an algebraic variety

In mathematics, a singular point of an algebraic variety V is a point P that is 'special' , in the geometric sense that V is not locally flat there....
.

Types of singular points

Some of the possible singularities are:
  • An isolated point: x2+y2 = 0, an acnode
    Acnode

    An acnode is an isolated point not on a curve, but whose coordinates satisfy the equation of the curve. The term "isolated point" or "hermit point" is an equivalent term....
  • Two lines crossing: x2-y2 = 0, a crunode
    Crunode

    A crunode is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines.For a plane curve, defined as the locus of points f = 0, where f is a smooth function of variables x and y ranging over the real numbers, a crunode of the curve is a singularity theory of the function f, w...
  • A cusp
    Cusp (singularity)

    In singularity theory a cusp is a Singular point of a curve. Spinode is an alternative name, but this is less commonly used today.For a curve defined as the zero set of a function of two variables , the cusps on the curve will have the following properties:...
    : x3-y2 = 0, also called a spinode
  • A rhamphoid cusp: x5-y2 = 0.


See also

  • Singularity theory
    Singularity theory

    In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width....
  • Morse theory
    Morse theory

    In differential topology, the techniques of Morse theory give a very direct way of analyzing the topological space of a manifold by studying differentiable functions on that manifold....