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Singularity Theory and its Applications
Dr Cathy Hobbs30/01/09
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Introduction: What is Singularity Theory?
Singularity Theory
Differential
geometryTopology
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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
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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?
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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:
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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
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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
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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.
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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
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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
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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.
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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.
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Example: Whitney classification
Whitney classified stable mappings R2 to R3 (1955).
Immersion Fold Cusp
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Applications: Robotics
Robotic motions are smooth maps from n-parameter space to 2 or 3 dimensional space.
Stewart-Gough platform Robot arm
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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)?
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Eg. 4-bar mechanism
Used in many engineering applications.
Generally planar.
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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
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Stable
Codimension 1
Codimension 2
Local models of coupler curves
All can be realised by a four-bar mechanism.
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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.
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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.
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Examples of singularities on outlines
© Henry Moore
© Barbara Hepworth
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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?
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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.
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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
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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
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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
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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
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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
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What does this look like?
Multiplicity 1: Diffeomorphism
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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
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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
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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
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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.
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Conclusions
Singularity Theory provides some useful tools for the study of local geometry of curves and surfaces.
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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.