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In geometry, the symmetry set is a method for representing the local symmetries of a curve, and can be used as a method for representing the shape of objects by finding the topological skeleton. The medial axis, a subset of the symmetry set is a set of curves which roughly run along the middle of an object.
The symmetry set in 2 dimensions
Let be an open interval, and be a parametrisation of a smooth plane curve.
The symmetry set of is defined to be the closure of the set of centres of circles tangent to the curve at at least distinct two points (bitangent circles).
The symmetry set in n dimensions
For a smooth manifold of dimension in (clearly we need ). The symmetry set of the manifold is the closure of the centres of hyperspheres tangent to the manifold in at least two distinct places.
The symmetry set as a bifurcation set
Let be an open simply connected domain and . Let be a parametrisation of a smooth piece of manifold. We may define a parameter family of functions on the curve, namely
This family is called the family of distance squared functions. This is because for a fixed the value of is the square of the distance from to at
The symmetry set is then the bifurcation set of the family of distance squared functions. I.e. it is the set of such that has a repeated singularity for some
By a repeated singularity, we mean that the jacobian matrix is singular. Since we have a family of functions, this is equivalent to .
The symmetry set is then the set of such that there exist with , and
together with the limiting points of this set.
1. J. W. Bruce, P.J.Giblin and C. G. Gibson, Symmetry Sets. Proc. of the Royal Soc.of Edinburgh 101A (1985), 163-186.
2. J. W. Bruce and P.J.Giblin, Curves and Singularities, Cambridge University Press (1993).