Symmetry set

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A ellipse (red), its evolute (blue), and its symmetry set (green and yellow). the medial axis is just the green portion of the symmetry set. One bi-tangent circle is shown.

The symmetry set is a method for representing the local symmetries of a curve, and can be used as a method for 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.

[edit] The Symmetry Set in 2 dimensions

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

The symmetry set of $\gamma (I) \subset \mathbb{R}^2$ is defined to be the closure of the set of centres of circles tangent to the curve at at least distinct two points (bi-tangent 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.

[edit] The Symmetry set in $n$ dimensions

For a smooth manifold of dimension $m$ in $\mathbb{R}^n$ (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.

[edit] The Symmetry Set as a Bifurcation Set

Let $U \subseteq \mathbb{R}^m$ be an open simply connected domian and $(u_1\ldots,u_m) := \underline{u} \in U$ . Let $\underline{X} : U \to \R^n$ be a parametrisation of a smooth piece of manifold. We may define a $n$ parameter faily of functions on the curve, namely

$F : \mathbb{R}^n \times U \to \mathbb{R} \ , \quad \mbox{where} \quad F(\underline{x},\underline{u}) = (\underline{x} - \underline{X}) \cdot (\underline{x} - \underline{X}) \ .$

This family is called the family of distance squared functions. This is because for a fixed $\underline{x}_0 \in \mathbb{R}^n$ the value of $F(\underline{x}_0,\underline{u})$ is the square of the distance from $\underline{x}_0$ to $\underline{X}$ at $\underline{X}(u_1\ldots,u_m).$

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

By a repeated singularity, we mean that the jacobian matrix is singular. Since we have a family of functions, this is equivalent to $\mathcal{r} F = \underline{0}$ .

The symmetry set is then the set of $\underline{x} \in \mathbb{R}^n$ such that there exist $(\underline{u}_1, \underline{u}_2) \in U \times U$ with $\underline{u}_1 \neq \underline{u}_2$ , and