Cusp (singularity)

Encyclopedia

In the mathematical theory of singularities

a

. Cusps are local singularities in that they are not formed by self intersection points of the curve.

The plane curve

cusps are all diffeomorphic to one of the following forms:

.

real-valued function

of two variable

s, say

s. So

upon by the group

of diffeomorphism

s of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target. This action splits the whole function space

up into equivalence classes, i.e. orbits of the group action.

One such family of equivalence classes is denoted by

, where

. This notation was introduced by V. I. Arnold. A function

if it lies in the orbit of

s for the type

-singularities. Notice that the

The cusps are then given by the zero-level-sets of the representatives of the

Ordinary cusps are very important geometrical objects. It can be shown that caustic

in the plane generically comprise smooth points and ordinary cusp points. By generic we mean that an open

and dense

set of all caustics comprise smooth points and ordinary cusp points. Caustics are, informally, points of exception brightness caused by the reflection of light from some object. In the teacup picture light is bouncing off the side of the teacup and interacting in a non-parallel fashion with itself. This results in a caustic. The bottom of the teacup represents a two-dimensional cross section of this caustic.

For a type

To see where these extra divisibility conditions come from, assume that

). If

Singularity theory

-The notion of singularity: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. What is meant by a singularity can be seen by dropping it on the floor...

a

**cusp**is a type of singular point of a curveSingular 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:...

. Cusps are local singularities in that they are not formed by self intersection points of the curve.

The plane curve

Plane curve

In mathematics, a plane curve is a curve in a Euclidean plane . The most frequently studied cases are smooth plane curves , and algebraic plane curves....

cusps are all diffeomorphic to one of the following forms:

*x*^{2}−*y*^{2k+1}= 0, where*k*≥ 1 is an integerInteger

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

.

## More general background

Consider a smoothSmooth function

In 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...

real-valued function

Real-valued function

In mathematics, a real-valued function is a function that associates to every element of the domain a real number in the image....

of two variable

Variable

Variable may refer to:* Variable , a logical set of attributes* Variable , a symbol that represents a quantity in an algebraic expression....

s, say

*f*(*x*,*y*) where*x*and*y*are real numberReal number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s. So

*f*is a function from the plane to the line. The space of all such smooth functions is actedGroup action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

upon by the group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

of diffeomorphism

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...

s of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target. This action splits the whole function space

Function space

In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

up into equivalence classes, i.e. orbits of the group action.

One such family of equivalence classes is denoted by

*A*_{k}^{±}Ak singularity

In mathematics, and in particular singularity theory an Ak, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold....

, where

*k*is a non-negative integerInteger

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

. This notation was introduced by V. I. Arnold. A function

*f*is said to be of type*A*_{k}^{±}Ak singularity

In mathematics, and in particular singularity theory an Ak, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold....

if it lies in the orbit of

*x*^{2}±*y*^{k+1}, i.e. there exists a diffeomorphic change of coordinate in source and target which takes*f*into one of these forms. These simple forms*x*^{2}±*y*^{k+1}are said to give normal formCanonical form

Generally, in mathematics, a canonical form of an object is a standard way of presenting that object....

s for the type

*A*k_{}^{±}Ak singularity

In mathematics, and in particular singularity theory an Ak, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold....

-singularities. Notice that the

*A*_{2n}^{+}are the same as the*A*_{2n}^{−}since the diffeomorphic change of coordinate (*x*,*y*) → (*x*, −*y*) in the source takes*x*^{2}+*y*^{2n+1}to*x*^{2}−*y*^{2n+1}. So we can drop the ± from*A*_{2n}^{±}notation.The cusps are then given by the zero-level-sets of the representatives of the

*A*_{2n}equivalence classes, where*n*≥ 1 is an integer.## Examples

- An
**ordinary cusp**is given by*x*^{2}−*y*^{3}= 0, i.e. the zero-level-set of a type*A*_{2}-singularity. Let*f*(*x*,*y*) be a smooth function of*x*and*y*and assume, for simplicity, that*f*(0,0) = 0. Then a type*A*_{2}-singularity of*f*at (0,0) can be characterised by:

- Having a degenerate quadratic part, i.e. the quadratic terms in the Taylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

of*f*form a perfect square, say*L*(*x*,*y*)^{2}, where*L*(*x*,*y*) is linear in*x*and*y*,*and* -
*L*(*x*,*y*) does not divide the cubic terms in the Taylor series of*f*(*x*,*y*).

Ordinary cusps are very important geometrical objects. It can be shown that caustic

Caustic (mathematics)

In differential geometry and geometric optics, a caustic is the envelope of rays either reflected or refracted by a manifold. It is related to the optical concept of caustics...

in the plane generically comprise smooth points and ordinary cusp points. By generic we mean that an open

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

and dense

Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

set of all caustics comprise smooth points and ordinary cusp points. Caustics are, informally, points of exception brightness caused by the reflection of light from some object. In the teacup picture light is bouncing off the side of the teacup and interacting in a non-parallel fashion with itself. This results in a caustic. The bottom of the teacup represents a two-dimensional cross section of this caustic.

- The ordinary cusp is also important in wavefrontWavefrontIn physics, a wavefront is the locus of points having the same phase. Since infrared, optical, x-ray and gamma-ray frequencies are so high, the temporal component of electromagnetic waves is usually ignored at these wavelengths, and it is only the phase of the spatial oscillation that is described...

s. A wavefront can be shown to generically comprise smooth points and ordinary cusp points. By generic we mean that an openOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

and denseDense setIn topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

set of all wavefronts comprise smooth points and ordinary cusp points.

- A
**rhamphoid cusp**(coming from the Greek meaning beak-like) is given by*x*^{2}–*y*^{5}= 0, i.e. the zero-level-set of a type*A*_{4}-singularity. These cusps are non-generic as caustics and wavefronts. The rhamphoid cusp and the ordinary cusp are non-diffeomorphic.

For a type

*A*_{4}-singularity we need*f*to have a degenerate quadratic part (this gives type*A*_{≥2}), that*L**does*divide the cubic terms (this gives type*A*_{≥3}), another divisibility condition (giving type*A*_{≥4}), and a final non-divisibility condition (giving type exactly*A*_{4}).To see where these extra divisibility conditions come from, assume that

*f*has a degenerate quadratic part*L*^{2}and that*L*divides the cubic terms. It follows that the third order taylor series of*f*is given by*L*^{2}±*LQ*where*Q*is quadratic in*x*and*y*. We can complete the square to show that*L*^{2}±*LQ*= (*L*± ½*Q*)^{2}– ¼*Q*^{4}. We can now make a diffeomorphic change of variable (in this case we simply substitute polynomials with linearly independent linear parts) so that (*L*± ½*Q*)^{2}− ¼*Q*^{4}→*x*_{1}^{2}+*P*_{1}where*P*_{1}is quartic (order four) in*x*_{1}and*y*_{1}. The divisibility condition for type*A*_{≥4}is that*x*_{1}divides*P*_{1}. If*x*_{1}does not divide*P*_{1}then we have type exactly*A*_{3}(the zero-level-set here is a tacnodeTacnode

In geometry, a tacnode is a kind of singular point of a curve. It is defined as a point where two osculating circles to the curve at that point are tangent. This means that two branches of the curve have ordinary tangency at the double point. The canonical example is = 0...

). If

*x*_{1}divides*P*_{1}we complete the square on*x*_{1}_{2}+*P*_{1}and change coordinates so that we have*x*_{2}^{2}+*P*_{2}where*P*_{2}is quintic (order five) in*x*_{2}and*y*_{2}. If*x*_{2}does not divide*P*_{2}then we have exactly type*A*_{4}, i.e. the zero-level-set will be a rhamphoid cusp.## External links

- http://www.sciencedaily.com/releases/2009/04/090414160801.htm