The global medial structure of regions in R 3
The global medial structure of regions in R 3
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DOI：
10.2140/gt.2006.10.2385
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发表时间：
2006-12
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影响因子：
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通讯作者：
J. Damon
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文献类型：
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作者：
J. Damon
对于 R 3 中具有通用平滑边界 B 的紧凑区域，我们考虑其几何属性，其几何属性位于其拓扑和几何之间，可以用术语“几何复杂性”来概括。的“几何复杂性”由其 Blum 中轴 M 捕获，它是 Whitney 分层集，其每个点的局部结构由特定的标准局部类型给出。我们通过给出 Blum 中轴 M 的结构定理对几何复杂性进行分类。为此，我们首先给出一种使用局部类型将 M 分解为“不可约分量”的算法，然后将每个中间分量表示为通过将带有边界的曲面附加到四价图而获得的形式。这两个阶段由两级扩展图结构来描述。顶层描述了不可约中间组件相互连接的简化形式，每个不可约组件的第二级扩展图结构指定了如何构造该组件。我们进一步使用与扩展图结构相关的数据来表达拓扑不变量，例如使用本地标准类型和扩展图结构定义的 M 的奇异不变量的同源性和基本群。使用分类，我们根据扩展图结构和相关数据来表征可收缩区域。
For compact regionsin R 3 with generic smooth boundary B , we consider geo- metric properties ofwhich lie midway between their topology and geometry and can be summarized by the term "geometric complexity". The "geometric complexity" ofis 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 topo- logical invariants ofsuch 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.