Symmetry set

 ( Differential Geometry ) 

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.

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

The symmetry set of $\backslash gamma\; (I)\; \backslash subset\; \backslash mathbb\{R\}^2$ 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.

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

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As a bifurcation set Let $U\; \backslash subseteq\; \backslash mathbb\{R\}^m$ be an open simply connected domain and $(u\_1\backslash ldots,u\_m)\; :=\; \backslash underline\{u\}\; \backslash in\; U$. Let $\backslash underline\{X\}\; :\; U\; \backslash to\; \backslash R^n$ be a parametrisation of a smooth piece of manifold. We may define a $n$ parameter family of functions on the curve, namely

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

This family is called the family of distance squared functions. This is because for a fixed $\backslash underline\{x\}\_0\; \backslash in\; \backslash mathbb\{R\}^n$ the value of $F(\backslash underline\{x\}\_0,\backslash underline\{u\})$ is the square of the distance from $\backslash underline\{x\}\_0$ to $\backslash underline\{X\}$ at $\backslash underline\{X\}(u\_1\backslash ldots,u\_m).$

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

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

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

$\backslash mathcal\{r\}\; F(\backslash underline\{x\},\backslash underline\{u\}\_1)\; =\; \backslash mathcal\{r\}\; F(\backslash underline\{x\},\backslash underline\{u\}\_2)\; =\; \backslash underline\{0\}$

together with the limiting points of this set.

• 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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