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

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

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

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: Ellipse Parabola Hyperbola General form:

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.

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 Inessential variables Essential variables

Thoms 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 R 2 to R 3 (1955). Immersion Fold Cusp

Applications: Robotics Robotic motions are smooth maps from n-parameter space to 2 or 3 dimensional space. Stewart-Gough platformRobot 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 R 2 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

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 as for 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 solving i.e. x = 0

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

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 (4),