- In 2 dimensions
- In n dimensions
- As a bifurcation set
- References
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 
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 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 
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.
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.
References
- J. W. Bruce, P. J. Giblin and C. G. Gibson, Symmetry Sets. Proc. of the Royal Soc.of Edinburgh 101A (1985), 163-186.
- J. W. Bruce and P. J. Giblin, Curves and Singularities, Cambridge University Press (1993).
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