In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, 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

Shape

The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...

 of objects by finding the topological skeleton

Topological skeleton

In shape analysis, skeleton of a shape is a thin version of that shape that is equidistant to its boundaries. The skeleton usually emphasizes geometrical and topological properties of the shape, such as its connectivity, topology, length, direction, and width...

. The medial axis

Medial axis

The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced by Blum as a tool for biological shape recognition....

, 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

Bitangent

In mathematics, a bitangent to a curve C is a line L that touches C in two distinct points P and Q and that has the same direction to C at these points...

 circles).

The symmetry set will have endpoints corresponding to vertices

Vertex (curve)

In the geometry of curves, a vertex is a point of where the first derivative of curvature is zero. This is typically a local maximum or minimum of curvature. Other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant...

 of the curve. Such points will lie at cusp

Cusp (singularity)

In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....

 of the evolute

Evolute

In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. Equivalently, it is the envelope of the normals to a curve....

. At such points the curve will have 4-point contact

Contact (mathematics)

In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. The equivalence classes are generally called jets...

with the circle.

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.