Singularity Theory and its Applications

Dr Cathy Hobbs30/01/09

Introduction: What is Singularity Theory?

Singularity Theory

Differential

geometryTopology

Singularity Theory

The study of critical points on manifolds (or of mappings) – points where the “derivative” is zero.

Developed from ‘Catastrophe Theory’ (1970’s).

Rigorous body of mathematics which enables us to study phenomena which re-occur in many situations

Singularity Theory

provides framework to classify critical points up to certain types of ‘natural’ equivalence

gives precise local models to describe types of behaviour

studies stability – what happens if we change our point of view a little?

Analogous example: Quadratic forms

2 2,F x y ax bxy cy

Quadratic forms in 2 variables can be classified:

Ellipse Parabola Hyperbola

2 2

2 21

x y

a b

2 2

2 21

x y

a b 2y ax

General form:

Morse Theory of Functions

Consider a smooth function .

If all partial derivatives are zero for a particular value x0 we say that y has a critical point at x0.

If the second differential at this point is a nondegenerate quadratic form then we call the point a non-degenerate critical point.

n R R

Morse Lemma

In a neighbourhood of a non-degenerate critical point a function may be reduced to its quadratic part, for a suitable choice of local co-ordinate system whose origin is at the critical point.

i.e. the function can be written as 2 2 2 2 21 2 1... ...k k ny x x x x x

Morse Lemma

Local theory – only valid in a neighbourhood of the point.

Explains ubiquity of quadratic forms.

Non-degenerate critical points are stable – all nearby functions have non-deg critical points of same type.

Splitting Lemma

Let be a smooth function with a degenerate critical point at the origin, whose Hessian matrix of second derivatives has rank r.

Then f is equivalent, around 0, to a function of the form

:R Rnf

2 21 1... ,...,r r nx x f x x

Inessential variablesEssential variables

Thom’s Classification

FoldCuspSwallowtailButterflyElliptic umbilicHyperbolic umbilicParabolic umbilic

31 1 1x a x M

4 21 1 1 2 1x a x a x M

5 3 21 1 1 2 1 3 1x a x a x a x M

6 4 3 21 1 1 2 1 3 1 4 1x a x a x a x a x M

3 2 2 21 1 2 1 1 2 2 1 3 23x x x a x x a x a x N

3 21 2 1 1 2 2 1 3 2x x a x x a x a x N

2 4 2 21 2 2 1 1 2 2 3 1 4 2x x x a x a x a x a x N

Singularities of Mappings

In many applications it is mappings that interest us, rather than functions.

For example, projecting a surface to a

plane is a mapping from 3-d to 2-d.

Singularities of Mappings

Can classify mappings from n-dim space to p-dim space for many (n,p) pairs (eg. n+p < 6).

Appropriate equivalence relations used eg diffeomorphisms.

Can list stable phenomena. Can investigate how unstable

phenomena break up as we perturb parameters.

Example: Whitney classification

Whitney classified stable mappings R2 to R3 (1955).

Immersion Fold Cusp

Applications: Robotics

Robotic motions are smooth maps from n-parameter space to 2 or 3 dimensional space.

Stewart-Gough platform Robot arm

Questions we might tackle:

What kinds of points might we see on the curve/surface traced out by a robotic motion?

Which points are stable, which are unstable (so likely to degenerate under small perturbance of the design)?

Eg. 4-bar mechanism

Used in many engineering applications.

Generally planar.

One parameter generates the motion.

There is a 2-parameter choice of coupler point.

Singularities from R to R2 have been classified.

The 2-dim choice of coupler point gives a codimension restriction to < 3.

Eg. 4-bar mechanism

Stable

Codimension 1

Codimension 2

Local models of coupler curves

All can be realised by a four-bar mechanism.

Other types of mechanism

Two-parameter planar motions – eg 5 bar planar linkage.

One-parameter spatial motions- eg 4 bar spatial linkage.

Two-parameter spatial motions

After this, classification gets complicated.

Applications: Vision

Think of viewing an object as a smooth mapping from a 3-d object to 2-d viewing plane.

Concentrate only on the outline of the object –points on surface where light rays coming from the eye graze it.

Examples of singularities on outlines

© Henry Moore

© Barbara Hepworth

Questions we might tackle:

What do smooth 3-d objects ‘look like’? i.e. what do their outlines look like locally?

What about non-smooth 3-d objects, eg those with corners, edges?

What are the effects of lighting on views, eg shadows, specular highlights?

What happens when motion occurs?

Some maths!

Think of a surface as the inverse image of a regular value of some smooth function.

Any smooth surface can be so described, and we can approximate actual expression with nice, smooth polynomial functions.

Expressing surface algebraically

Consider a smooth surface given by taking the inverse image of the value 0.

Choose co-ordinates so that the orthogonal projection onto the 2-d viewing plane is given by

Then F is given by 1

1 0, , , , ... ,n nn nF t x y a x y t a x y t a x y

, , ,t x y x y

Conditions for outline

Surface M is given by Suppose M goes through the origin,

i.e. Origin yields a point on the outline

exactly when and

1 0F

0 00F

000F

t

0 00F

Conditions for singularities on outline

If but

then t = 0 is a p-fold root of

In a neighbourhood of the origin we are able to rewrite our surface as

for some smooth functions .

1 2

1 2

0 0... 0 00 0

0

p p

p p

F FF

t t

00

p

p

F

t

0F t

11 0... 0w wp p

pt b t b

jb

Simplified local expression

Simplify by applying the Tschirnhaus transformation

Geometrically consists of sliding the surface up/down vertically – no change to outline.

Now local expression is

1

1wpt t b

p

12 0... 0w wp p

pt c t c

How large is p for a general surface?

We have a point of Multiplicity 1 if

Multiplicity 2 if

Multiplicity 3 if

Multiplicity > 3 if

0, 0F

Ft

2

20, 0

F FF

t t

2 3

2 30, 0

F F FF

t t t

2 3

2 30

F F FF

t t t

What does this look like?

Multiplicity 1: Diffeomorphism

What does this look like?

Multiplicity 2: Fold.

Write surface locally as

Outline is given by solvingi.e. x = 0

2 0t x

2 2 0t x t

What does this look like?

Multiplicity 3: cuspCan write the surface locally as

Eliminating t fromgives

3 0t xt y 3 23 0t xt y t x

2 327 4 0y x

Double points Fourth possibility: outline could have a double point.

Stable (and generic) – arises from two separated

parts of the surface projecting to the same neighbourhood.

Can consider such multiple mappings. In this case, it is a mapping .

Only stable cases are overlapping sheets or transverse crossings.

Codimension 3 – will only occur at isolated points along the outline.

3 3 2R R R

Motion

Can allow for motion, either of the object or camera.

Introduces further parameters so projection becomes a mapping from 4 or 5 variables into 2.

This allows the codimension to be higher and so we observe more types of singular behaviour.

Conclusions

Singularity Theory provides some useful tools for the study of local geometry of curves and surfaces.

References

Catastrophe Theory and its applications, Poston & Stewart.

Solid Shape, Koenderink Visual Motion of Curves and

Surfaces, Cipolla & Giblin Seeing – the mathematical

viewpoint, Bruce, Mathematical Intelligencer 1984 6 (4), 18-25.