Symmetry set

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.

be an open interval, and

γ :

be a parametrisation of a smooth plane curve.

The symmetry set of

γ (

is defined to be the closure of the set of centres of circles tangent to the curve at at least two distinct points (bitangent circles).

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.

For a smooth manifold of dimension

The symmetry set of the manifold is the closure of the centres of hyperspheres tangent to the manifold in at least two distinct places.

be an open simply connected domain and

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

is the square of the distance from

The symmetry set is then the bifurcation set of the family of distance squared functions.

I.e. it is the set of

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

, and together with the limiting points of this set.