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| {{Unreferenced|date=March 2008}}
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| [[Image:Ellipse symmetry set.svg||right|thumb|240px|An [[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.]]
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| 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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| ==The symmetry set in 2 dimensions==
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| Let <math> I \subseteq \mathbb{R} </math> be an open interval, and <math>\gamma : I \to \mathbb{R}^2</math> be a parametrisation of a smooth plane curve.
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| The symmetry set of <math> \gamma (I) \subset \mathbb{R}^2</math> 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).
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| The symmetry set will have endpoints corresponding to [[vertex (curve)|vertices]] of the curve. Such points will lie at [[cusp (singularity)|cusp]] of the [[evolute]]. At such points the curve will have [[Contact (mathematics)|4-point contact]] with the circle.
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| ==The symmetry set in ''n'' dimensions==
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| For a smooth manifold of dimension <math>m</math> in <math>\mathbb{R}^n</math> (clearly we need <math>m < n</math>). 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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| ==The symmetry set as a bifurcation set==
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| Let <math> U \subseteq \mathbb{R}^m</math> be an open simply connected domain and <math>(u_1\ldots,u_m) := \underline{u} \in U</math>. Let <math>\underline{X} : U \to \R^n</math> be a parametrisation of a smooth piece of manifold.
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| We may define a <math>n</math> parameter family of functions on the curve, namely
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| :<math> 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}) \ . </math> | |
| This family is called the family of distance squared functions. This is because for a fixed <math>\underline{x}_0 \in \mathbb{R}^n</math> the value of <math>F(\underline{x}_0,\underline{u})</math> is the square of the distance from <math>\underline{x}_0</math> to <math>\underline{X}</math> at <math>\underline{X}(u_1\ldots,u_m).</math>
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| The symmetry set is then the bifurcation set of the family of distance squared functions. I.e. it is the set of <math>\underline{x} \in \R^n</math> such that <math>F(\underline{x},-)</math> has a repeated singularity for some <math>\underline{u} \in U.</math>
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| By a repeated singularity, we mean that the jacobian matrix is singular. Since we have a family of functions, this is equivalent to <math>\mathcal{r} F = \underline{0}</math>.
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| The symmetry set is then the set of <math>\underline{x} \in \mathbb{R}^n</math> such that there exist <math>(\underline{u}_1, \underline{u}_2) \in U \times U</math> with <math>\underline{u}_1 \neq \underline{u}_2</math>, and
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| :<math> \mathcal{r} F(\underline{x},\underline{u}_1) = \mathcal{r} F(\underline{x},\underline{u}_2) = \underline{0} </math>
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| together with the limiting points of this set.
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| ==References ==
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| 1. J. W. Bruce, P.J.Giblin and C. G. Gibson, Symmetry Sets. ''Proc. of the Royal Soc.of Edinburgh'' 101A (1985), 163-186.
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| 2. J. W. Bruce and P.J.Giblin, Curves and Singularities, Cambridge University Press (1993).
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| [[Category:Differential geometry]]
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