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. In 2 dimensions. Let formula_1 be an open interval, and formula_2 be a parametrisation of a smooth plane curve. The symmetry set of formula_3 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. In "n" dimensions. For a smooth manifold of dimension formula_4 in formula_5 (clearly we need formula_6). The symmetry set of the manifold is the closure of the centres of hyperspheres tangent to the manifold in at least two distinct places. As a bifurcation set. Let formula_7 be an open simply connected domain and formula_8. Let formula_9 be a  parametrisation of a smooth piece of manifold. We may define a formula_10 parameter family of functions on the curve, namely This family is called the family of distance squared functions. This is because for a fixed formula_12 the value of formula_13 is the square of the distance from formula_14 to formula_15 at formula_16 The symmetry set is then the bifurcation set of the family of distance squared functions. I.e. it is the set of formula_17 such that formula_18 has a repeated singularity for some formula_19 By a repeated singularity, we mean that the jacobian matrix is singular. Since we have  a family of functions, this is equivalent to formula_20. The symmetry set is then the set of formula_21 such that there exist formula_22 with formula_23, and together with the limiting points of this set.