The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced in 1967 by Harry Blum as a tool for biological

shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...

recognition. In mathematics the closure of the medial axis is known as the cut locus. In 2D, the medial axis of a subset ''S'' which is bounded by planar curve ''C'' is the locus of the centers of circles that are tangent to curve ''C'' in two or more points, where all such circles are contained in ''S''. (It follows that the medial axis itself is contained in ''S''.) The medial axis of a simple polygon is a tree whose leaves are the vertices of the polygon, and whose edges are either straight segments or arcs of parabolas. The medial axis together with the associated radius function of the maximally inscribed discs is called the medial axis transform (MAT). The medial axis transform is a complete shape descriptor (see also shape analysis), meaning that it can be used to reconstruct the

of the original domain. The medial axis is a subset of the symmetry set, which is defined similarly, except that it also includes circles not contained in ''S''. (Hence, the symmetry set of ''S'' generally extends to infinity, similar to the

Voronoi diagram In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed ...

of a point set.) The medial axis generalizes to ''k''-dimensional hypersurfaces by replacing 2D circles with ''k''-dimension hyperspheres. The 2D medial axis is useful for

character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...

and object recognition, while the 3D medial axis has applications in surface reconstruction for physical models, and for dimensional reduction of complex models. In any dimension, the medial axis of a bounded

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...

homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...

to the given set. If ''S'' is given by a unit speed parametrisation

\gamma:\mathbf\to\mathbf^2

, and

\underline(t) =

is the unit tangent vector at each point. Then there will be a bitangent circle with center ''c'' and radius ''r'' if *

(c-\gamma(s))\cdot\underline(s)=(c-\gamma(t))\cdot\underline(t)=0,

, c-\gamma(s), =, c-\gamma(t), =r.\,

For most curves, the symmetry set will form a one-dimensional curve and can contain cusps. The symmetry set has end points corresponding to the vertices of ''S''.

References

External links

The Scale Axis Transform
– a generalization of the medial axis
Straight Skeleton for polygon with holes
– Straight Skeleton builder implemented in java. Geometric shapes

See also

References

Further reading

External links