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Singularity Theory and its Applications Dr Cathy Hobbs 30/01/09

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Introduction: What is Singularity Theory? Singularity Theory Differential geometry Topology

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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 (1970s). 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: Ellipse Parabola Hyperbola General form:

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Morse Theory of Functions Consider a smooth function. If all partial derivatives are zero for a particular value x 0 we say that y has a critical point at x 0. 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 Inessential variables Essential variables

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Thoms Classification Fold Cusp Swallowtail Butterfly 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 R 2 to R 3 (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 platformRobot 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 R 2 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

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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 as for 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 solving i.e. x = 0

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What does this look like? Multiplicity 3: cusp Can write the surface locally as Eliminating t from gives

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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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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.

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