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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.
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The symmetry set in 2 dimensions
Let
The symmetry set of
The symmetry set will have endpoints corresponding to vertices of the curve. Such points will lie at cusp of the evolute. At such points the curve will have 4-point contact with the circle.
The symmetry set in n dimensions
For a smooth manifold of dimension
The symmetry set as a bifurcation set
Let
This family is called the family of distance squared functions. This is because for a fixed
The symmetry set is then the bifurcation set of the family of distance squared functions. I.e. it is the set of
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
together with the limiting points of this set.