Symmetry set

Last updated February 26, 2019

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.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture or material composition.

Topological skeleton one-dimensional approximation to a given shape

In shape analysis, skeleton of a shape is a thin version of that shape that is equidistant to its boundaries. The skeleton usually emphasizes geometrical and topological properties of the shape, such as its connectivity, topology, length, direction, and width. Together with the distance of its points to the shape boundary, the skeleton can also serve as a representation of the shape.

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 two distinct points (bitangent circles).

In mathematics, a bitangent to a curve C is a line L that touches C in two distinct points P and Q and that has the same direction as C at these points. That is, L is a tangent line at P and at Q.

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.

In the geometry of planar curves, a vertex is a point of where the first derivative of curvature is zero. This is typically a local maximum or minimum of curvature, and some authors define a vertex to be more specifically a local extreme point of curvature. However, other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant. For space curves, on the other hand, a vertex is a point where the torsion vanishes.

In mathematics a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point on the curve must start to move backward. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.

In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the envelope of the normals to a curve.

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.

As a bifurcation set

Let $U\subseteq \mathbb {R} ^{m}$ be an open simply connected domain and $(u_{1}\ldots ,u_{m}):={\underline {u}}\in U$ . Let ${\underline {X}}:U\to \mathbb {R} ^{n}$ be a parametrisation of a smooth piece of manifold. We may define a $n$ parameter family of functions on the curve, namely

F:\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 ${\underline {x}}\in \mathbb {R} ^{n}$ 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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References

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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