A
singular point on a
curveIn mathematics, a curve consists of the points through which a continuously moving point passes. This notion captures the intuitive idea of a geometrical one-dimensional object, which furthermore is connected in the sense of having no discontinuities or gaps. Simple examples include the sine wave...
is one where it is not
smoothIn mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
, for example, at a
cusp__FORCETOC__In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
.
The precise definition of a singular point depends on the type of curve being studied.
Algebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension 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.- Plane algebraic curves...
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
derivativeIn calculus 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; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
s vanish,
A
parameterizedIn mathematics, parametric equations are a method of defining a function using parameters. A simple kinematical 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)).
A
singular point on a
curveIn mathematics, a curve consists of the points through which a continuously moving point passes. This notion captures the intuitive idea of a geometrical one-dimensional object, which furthermore is connected in the sense of having no discontinuities or gaps. Simple examples include the sine wave...
is one where it is not
smoothIn mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
, for example, at a
cusp__FORCETOC__In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
.
The precise definition of a singular point depends on the type of curve being studied.
Algebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension 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.- Plane algebraic curves...
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
derivativeIn calculus 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; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
s vanish,
A
parameterizedIn mathematics, parametric equations are a method of defining a function using parameters. A simple kinematical 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
Many curves can be defined in either fashion, but the two definitions may not agree. For example the
cusp__FORCETOC__In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
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
nodeA 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 of...
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
implicitIn 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 functionIn mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
, 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 WhitneyHassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
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 varietyIn 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. In the case of an algebraic curve, a plane curve that has a double point, such as the cubic curveexhibits at , cannot simply be parametrized near the...
.
Types of singular points
Some of the possible singularities are:
- An isolated point: x2+y2 = 0, an 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
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 of...
- A cusp
__FORCETOC__In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
: x3−y2 = 0, also called a spinode
- A rhamphoid cusp: x5−y2 = 0.