Symmetry set

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.

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 (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.

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.

The symmetry set as a bifurcation set

Let U subseteq mathbb{R}^m be an open simply connected domian and (u_1ldots,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_1ldots,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

mathcal{r} F(underline{x},underline{u}_1) = mathcal{r} F(underline{x},underline{u}_2) = underline{0}
together with the limiting points of this set.

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