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Cusp (singularity)
Encyclopedia
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 plane curve
cusps are all diffeomorphic to one of the following forms:
x2 − y2k+1 = 0, where k ≥ 1 is an integer
.
real-valued function
of two variable
s, say f(x, y) where x and y are real number
s. So f is a function from the plane to the line. The space of all such smooth functions is acted
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 Ak±
, where k is a non-negative integer
. This notation was introduced by V. I. Arnold. A function f is said to be of type Ak±
if it lies in the orbit of x2 ± yk+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 x2 ± yk+1 are said to give normal form
s for the type Ak±
-singularities. Notice that the A2n+ are the same as the A2n− since the diffeomorphic change of coordinate (x,y) → (x, −y) in the source takes x2 + y2n+1 to x2 − y2n+1. So we can drop the ± from A2n± notation.
The cusps are then given by the zero-level-sets of the representatives of the A2n equivalence classes, where n ≥ 1 is an integer.
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 A4-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 A4).
To see where these extra divisibility conditions come from, assume that f has a degenerate quadratic part L2 and that L divides the cubic terms. It follows that the third order taylor series of f is given by L2 ± LQ where Q is quadratic in x and y. We can complete the square to show that L2 ± LQ = (L ± ½Q)2 – ¼Q4. We can now make a diffeomorphic change of variable (in this case we simply substitute polynomials with linearly independent linear parts) so that (L ± ½Q)2 − ¼Q4 → x12 + P1 where P1 is quartic (order four) in x1 and y1. The divisibility condition for type A≥4 is that x1 divides P1. If x1 does not divide P1 then we have type exactly A3 (the zero-level-set here is a tacnode
). If x1 divides P1 we complete the square on x12 + P1 and change coordinates so that we have x22 + P2 where P2 is quintic (order five) in x2 and y2. If x2 does not divide P2 then we have exactly type A4, i.e. the zero-level-set will be a rhamphoid cusp.
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 curve
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:...
. 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:
x2 − y2k+1 = 0, where k ≥ 1 is an integer
Integer
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 number
Real 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 acted
Group 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 Ak±
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 integer
Integer
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 Ak±
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 x2 ± yk+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 x2 ± yk+1 are said to give normal form
Canonical form
Generally, in mathematics, a canonical form of an object is a standard way of presenting that object....
s for the type Ak±
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 A2n+ are the same as the A2n− since the diffeomorphic change of coordinate (x,y) → (x, −y) in the source takes x2 + y2n+1 to x2 − y2n+1. So we can drop the ± from A2n± notation.
The cusps are then given by the zero-level-sets of the representatives of the A2n equivalence classes, where n ≥ 1 is an integer.
Examples
- An ordinary cusp is given by x2 − y3 = 0, i.e. the zero-level-set of a type A2-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 A2-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 x2 – y5 = 0, i.e. the zero-level-set of a type A4-singularity. These cusps are non-generic as caustics and wavefronts. The rhamphoid cusp and the ordinary cusp are non-diffeomorphic.
For a type A4-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 A4).
To see where these extra divisibility conditions come from, assume that f has a degenerate quadratic part L2 and that L divides the cubic terms. It follows that the third order taylor series of f is given by L2 ± LQ where Q is quadratic in x and y. We can complete the square to show that L2 ± LQ = (L ± ½Q)2 – ¼Q4. We can now make a diffeomorphic change of variable (in this case we simply substitute polynomials with linearly independent linear parts) so that (L ± ½Q)2 − ¼Q4 → x12 + P1 where P1 is quartic (order four) in x1 and y1. The divisibility condition for type A≥4 is that x1 divides P1. If x1 does not divide P1 then we have type exactly A3 (the zero-level-set here is a tacnode
Tacnode
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 x1 divides P1 we complete the square on x12 + P1 and change coordinates so that we have x22 + P2 where P2 is quintic (order five) in x2 and y2. If x2 does not divide P2 then we have exactly type A4, i.e. the zero-level-set will be a rhamphoid cusp.
External links
- http://www.sciencedaily.com/releases/2009/04/090414160801.htm