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 $I \subseteq \mathbb$ be an open interval, and $\gamma : I \to \mathbb^2$ be a parametrisation of a smooth plane curve.

The symmetry set of $\gamma (I) \subset \mathbb^2$ 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 $m$ in $\mathbb^n$ (clearly we need $m < n$). 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 $U \subseteq \mathbb^m$ be an open simply connected domain and $(u_1\ldots,u_m) := \underline \in U$. Let $\underline : U \to \R^n$ be a parametrisation of a smooth piece of manifold.
We may define a $n$ parameter family of functions on the curve, namely
:$F : \mathbb^n \times U \to \mathbb \ , \quad \mbox \quad F(\underline,\underline) = (\underline - \underline) \cdot (\underline - \underline) \ .$
This family is called the family of distance squared functions. This is because for a fixed $\underline_0 \in \mathbb^n$ the value of $F(\underline_0,\underline)$ is the square of the distance from $\underline_0$ to $\underline$ at $\underline(u_1\ldots,u_m).$

The symmetry set is then the bifurcation set of the family of distance squared functions. I.e. it is the set of $\underline \in \R^n$ such that $F(\underline,-)$ has a repeated singularity for some $\underline \in U.$

By a repeated singularity, we mean that the jacobian matrix is singular. Since we have a family of functions, this is equivalent to $\mathcal F = \underline$.

The symmetry set is then the set of $\underline \in \mathbb^n$ such that there exist $(\underline_1, \underline_2) \in U \times U$ with $\underline_1 \neq \underline_2$, and
:$\mathcal F(\underline,\underline_1) = \mathcal F(\underline,\underline_2) = \underline{0}$
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).

Differential geometry