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.
Let
I\subseteqR
\gamma:I\toR2
The symmetry set of
\gamma(I)\subsetR2
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.
For a smooth manifold of dimension
m
Rn
m<n
Let
U\subseteqRm
(u1\ldots,um):=\underline{u}\inU
\underline{X}:U\to\Rn
n
F:Rn x U\toR , where F(\underline{x},\underline{u})=(\underline{x}-\underline{X}) ⋅ (\underline{x}-\underline{X}) .
\underline{x}0\inRn
F(\underline{x}0,\underline{u})
\underline{x}0
\underline{X}
\underline{X}(u1\ldots,um).
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\Rn
F(\underline{x},-)
\underline{u}\inU.
By a repeated singularity, we mean that the jacobian matrix is singular. Since we have a family of functions, this is equivalent to
l{r}F=\underline{0}
The symmetry set is then the set of
\underline{x}\inRn
(\underline{u}1,\underline{u}2)\inU x U
\underline{u}1 ≠ \underline{u}2
l{r}F(\underline{x},\underline{u}1)=l{r}F(\underline{x},\underline{u}2)=\underline{0}