#### Volume 10, issue 4 (2006)

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The global medial structure of regions in $\mathbb{R}^3$

### James Damon

Geometry & Topology 10 (2006) 2385–2429
 arXiv: 0903.0394
##### Abstract

For compact regions $\Omega$ in ${ℝ}^{3}$ with generic smooth boundary $\mathsc{ℬ}$, we consider geometric properties of $\Omega$ which lie midway between their topology and geometry and can be summarized by the term “geometric complexity”. The “geometric complexity” of $\Omega$ is captured by its Blum medial axis $M$, which is a Whitney stratified set whose local structure at each point is given by specific standard local types.

We classify the geometric complexity by giving a structure theorem for the Blum medial axis $M$. We do so by first giving an algorithm for decomposing $M$ using the local types into “irreducible components” and then representing each medial component as obtained by attaching surfaces with boundaries to $4$–valent graphs. The two stages are described by a two level extended graph structure. The top level describes a simplified form of the attaching of the irreducible medial components to each other, and the second level extended graph structure for each irreducible component specifies how to construct the component.

We further use the data associated to the extended graph structures to express topological invariants of $\Omega$ such as the homology and fundamental group in terms of the singular invariants of $M$ defined using the local standard types and the extended graph structures. Using the classification, we characterize contractible regions in terms of the extended graph structures and the associated data.

##### Keywords
geometric complexity of regions, Blum medial axis, Whitney stratified sets, irreducible medial components, fin curves, extended graphs, Y-network, weighted genus
##### Mathematical Subject Classification 2000
Primary: 57N80, 58A38
Secondary: 68U05, 53A05, 55P55