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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Dr Cathy Hobbs
Quadratic forms in 2 variables can be classified:
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
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
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.
Robotic motions are smooth maps from n-parameter space to 2 or 3 dimensional space.
All can be realised by a four-bar mechanism.
© Henry Moore
© Barbara Hepworth
We have a point of