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Singularity Theory and its ApplicationsPowerPoint Presentation

Singularity Theory and its Applications

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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

Quadratic forms in 2 variables can be classified:

General form:

Hyperbola

Parabola

Ellipse

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.

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

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

Essential variables

Inessential variables

Thom’s Classification

Fold

Cusp

Swallowtail

Butterfly

Elliptic umbilic

Hyperbolic umbilic

Parabolic umbilic

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).

Cusp

Fold

Immersion

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.

Eg. 4-bar mechanism

- 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.

Local models of coupler curves

Stable

Codimension 1

Codimension 2

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.

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

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

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 asfor some smooth functions .

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

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

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

What does this look like?

- Multiplicity 3: cuspCan write the surface locally as
- Eliminating t fromgives

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.

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.

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