Singularities

Zero sets of Simple Singularities of functions on R^3

Richard Morris
Department of Statistics, University of Leeds
Department of Statistics,
University of Leeds,
Leeds, LS2 9JT
England
rjm@amsta.leeds.ac.uk
webmaster@pfaf.org
http://www.amsta.leeds.ac.uk/~rjm/

MSC2000 Classification:

Keywords:

A function, f, from R³ to R is singular at a point p in R³ if its Jacobian, (df/dx,df/dy,df/dz)^T vanishes at p. Arnold [1] has classified the types of singularities which can can occur and models of the zero sets of some of these are presented in this collection. See [2] for a detailed introduction to singularity theory.

We will just be concerned with the local behaviour of functions defined near the origin and some notation is necessary.

Notation

Let p be a point in Rⁿ. Then f : Rⁿ,p -> R denotes a function defined on some neighbourhood of p. Two functions are regarded as equal if they coincide on some (possibly smaller) neighbourhood of p. A equivalence class of such functions is called a germ at p. The notation f : Rⁿ,p -> R, c, mean in addition f(p)=c.

We now define a equivalence relation for functions.

Definition

Let U_i, i=1,2 be open subsets of Rⁿ, and let p_i be a point of U_i with f_i : U_i -> R. We say that f₁ (at p₁) and f₂ (at p₂) are right-equivalent if there exists open neighbourhoods of p_i, say V_i, a subset of U_i, a diffeomorphism h: V₁ -> V₂ and a constant c in R such that

• h(p₁) = p₁,
• f₁(p) = f₂(h(p)) + c

This equivalence relation allows smooth changes of coordinates in Rⁿ In particular it is sufficient just to consider functions at the origin as f : Rⁿ,p₁ ->R is right-equivalent to a germ f : Rⁿ,0 -> R, the diffeomorphism h : Rⁿ,p₁ -> Rⁿ,0 simply being the translation h(p) = p - p₁.

We can now state the main classification theorem. Let x be a point in Rⁿ and let x₁, ... ,x_n denote its coordinates.

Theorem (V.I. Arnold)

Any simple singularity f : Rⁿ,0 -> R,0 is right equivalent to one of the following types, where e_i = +/- 1 in each case.

1. The A_k series. f : Rⁿ,0 -> R,0 defined by
• f(x) = +/- x₁^k+1 + e₂ x₂² + ... + e_n x_n², where k≥1.
2. The D_k series. f : Rⁿ,0 -> R,0 defined by
• f(x) = +/- x₁^k-1 + x₁ x₂² + e₃ x₃² + ... + e_n x_n², where k≥4.
3. The E₆, E₇ and E₈ singularities. f : Rⁿ,0 -> R,0 defined by
• f(x) = +/- x₁⁴ + x₂³ + e₃ x₃² + ... + e_n x_n²;
• f(x) = + x₁³ x₂ + x₂³ + e₃ x₃² + ... + e_n x_n²;
• f(x) = + x₁⁵ + x₂³ + e₃ x₃² + ... + e_n x_n²;

It should be noted that permuting the coordinates in Rⁿ will give a right-equivalent function, so not all the possible combinations of signs yield distinct singularities. There are further singularities which are not simple. These occur less frequently and have not been included in this collection.

The models

The models in this collection show the zero sets, f^-1(0), for some of these singularities where n=3. Let x, y, z denote the coordinates in R³. Models are provided for the A_k series, (k=1,2,3,4) the D_k series, (k=4,5,6) and E₆, E₇ and E₈. For most cases two zero sets of two non equvalent forms of the singularity, denoted A1pm and A1mm etc., have been provided. The zero sets of some of these singularities (x^k+1+ y²+z², k=1,3) just consists of an isolated point and their models have not been shown. Some of the other zeros sets have not been shown, but these are simple reflections of the models shown. Additionally the zero sets of two non singular functions, A0pm (x+y²-z²) and A0mm (x-y²-z²) have also been included. These two functions are right-equivalent to each other and also to any other non singular function.

The models shown are:

• A0pm: x+y²-z². Non singular saddle shaped.
• A0mm: x-y²-z². Non singular dome shaped.
• A1pp: x²+ y²+z². Isolated point, not shown.
• A1mm: x²- y²-z².
• A2pm: x³+ y²-z².
• A2mm: x³- y²-z².
• A3pp: x⁴+ y²+z². Isolated point, not shown.
• A3pm: x⁴+ y²-z².
• A3mm: x⁴- y²-z².
• A4pm: x⁵+ y²-z².
• A4mm: x⁵- y²-z².
• D4pm: x² y+ y³-z².
• D4mm: x² y- y³-z².
• D5pm: x² y+ y⁴-z².
• D5mm: x² y- y⁴-z².
• D6pm: x² y+ y⁵-z².
• D6mm: x² y- y⁵-z².
• E6pm: x³ + y⁴-z².
• E6mm: x³ - y⁴-z².
• E7mm: x³ - xy³-z².
• E8mm: x³ - y⁵-z².

Calculation of models

These models have been calculate by a Java applet [5] constructed using the JavaView system [6] which connects to a server which calculates the zero set of a given function. The server is adapted from the program in the LSMP package [3,4]. The algorithm has been constructed to try to get accurate representations of the singular points in the surfaces. For the purpose of this collection the models have been cleaned up by hand to give topologically accurate representations of the singular points.

References

1. V.I. Arnold: Singularity Theory, London Mathematical Society (1981).
2. J.W. Bruce and P.J. Giblin: Curves and Singularities, Cambridge University Press (1992).
3. R.J. Morris: A new method for drawing Algebraic Surfaces, Design and application of curves and surface, Oxford University Press (1994), 31--48.
4. R.J. Morris: The Use of Computer Graphics for Solving Problems in Singularity Theory, Visualization and Mathematics, Springer (1997), 53--66.
5. R.J. Morris: Algebraic Surfaces Web-Server.
6. K. Polthier, S. Khadem-Al-Charieh, E. Preuss and U Reitebuch: JavaView.

Technical Notes

Used Software: Liverpool Surface Modeling Package 2.9 and JavaView

These files have been hand edited to ensure that the boundaries are correct and that they are topologically correct around the singular point.