For compact regions in
with generic smooth
boundary , we consider
geometric properties of
which lie midway between their topology and geometry and can be summarized
by the term “geometric complexity”. The “geometric complexity” of
is captured by its
Blum medial axis ,
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 .
We do so by first giving an algorithm for decomposing
using
the local types into “irreducible components” and then representing each
medial component as obtained by attaching surfaces with boundaries to
–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
such as the homology and fundamental group in terms of the singular invariants of
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
