Singularity theory and its applications
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Singularity Theory and its Applications. Dr Cathy Hobbs 30/01/09. Introduction: What is Singularity Theory?. Differential geometry. Singularity Theory. Topology. Singularity Theory. The study of critical points on manifolds (or of mappings) – points where the “derivative” is zero.

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Introduction what is singularity theory l.jpg
Introduction: What is Singularity Theory?



Singularity Theory


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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

Quadratic forms in 2 variables can be classified:

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.

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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

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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

Essential variables

Inessential variables

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Thom’s Classification





Elliptic umbilic

Hyperbolic umbilic

Parabolic umbilic

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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).




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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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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.

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Local models of coupler curves


Codimension 1

Codimension 2

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

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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

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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 asfor some smooth functions .

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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

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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

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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

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What does this look like?

  • Multiplicity 3: cuspCan write the surface locally as

  • Eliminating t fromgives

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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.

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

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  • 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.