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Medial/Skeletal Linking Structures for Multi-Region Configurations

About this Title

James Damon, Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3250 and Ellen Gasparovic, Department of Mathematics, Duke University, Box 90320, Durham, North Carolina 27708-0320

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 250, Number 1193
ISBNs: 978-1-4704-2680-4 (print); 978-1-4704-4210-1 (online)
DOI: https://doi.org/10.1090/memo/1193
Published electronically: September 7, 2017
Keywords: Blum medial axis, skeletal structures, spherical axis, Whitney stratified sets, medial and skeletal linking structures, generic linking properties, model configurations, radial flow, linking flow, multi-distance functions, height-distance functions, partial multijet spaces, transversality theorems, measures of closeness, measures of significance, tiered linking graph
MSC: Primary 53A07, 58A35; Secondary 68U05

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Table of Contents

Chapters

• 1. Introduction

1. Medial/Skeletal Linking Structures

• 2. Multi-Region Configurations in ${\mathbb R}^{n+1}$
• 3. Skeletal Linking Structures for Multi-Region Configurations in ${\mathbb R}^{n+1}$
• 4. Blum Medial Linking Structure for a Generic Multi-Region Configuration
• 5. Retracting the Full Blum Medial Structure to a Skeletal Linking Structure

2. Positional Geometry of Linking Structures

• 6. Questions Involving Positional Geometry of a Multi-Region Configuration
• 7. Shape Operators and Radial Flow for a Skeletal Structure
• 8. Linking Flow and Curvature Conditions
• 9. Properties of Regions Defined Using the Linking Flow
• 10. Global Geometry via Medial and Skeletal Linking Integrals
• 11. Positional Geometric Properties of Multi-Region Configurations

3. Generic Properties of Linking Structures via Transversality Theorems

• 12. Multi-Distance and Height-Distance Functions and Partial Multi-Jet Spaces
• 13. Generic Blum Linking Properties via Transversality Theorems
• 14. Generic Properties of Blum Linking Structures
• 15. Concluding Generic Properties of Blum Linking Structures

4. Proofs and Calculations for the Transversality Theorems

• 16. Reductions of the Proofs of the Transversality Theorems
• 17. Families of Perturbations and their Infinitesimal Properties
• 18. Completing the Proofs of the Transversality Theorems
• A. List of Frequently Used Notation

Abstract

We consider a generic configuration of regions, consisting of a collection of distinct compact regions $\{ \Omega _i\}$ in $\mathbb {R}^{n+1}$ which may be either regions with smooth boundaries disjoint from the others or regions which meet on their piecewise smooth boundaries $\mathcal {B}_i$ in a generic way. We introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions’, individual shapes and geometric properties as well as the “positional geometry”, of the collection. The linking structure extends in a minimal way the individual “skeletal structures”, on each of the regions. This allows us to significantly extend the mathematical methods introduced for single regions to the configuration of regions.

We prove for a generic configuration of regions the existence of a special type of Blum linking structure which builds upon the Blum medial axes of the individual regions. As part of this, we introduce the “spherical axis” , which is the analogue of the medial axis but for directions. These results require proving several transversality theorems for certain associated “multi-distance”, and “height-distance”, functions for such configurations. We show that by relaxing the conditions on the Blum linking structures we obtain the more general class of skeletal linking structures which still capture the geometric properties.

The skeletal linking structure is used to analyze the “positional geometry”, of the configuration. This involves using the “linking flow”, to identify neighborhoods of the configuration regions which capture their positional relations. As well as yielding geometric invariants which capture the shapes and geometry of individual regions, these structures are used to define invariants which measure positional properties of the configuration such as: measures of relative closeness of neighboring regions and relative significance of the individual regions for the configuration.

All of these invariants are computed by formulas involving “skeletal linking integrals”, on the internal skeletal structures of the regions. These invariants are then used to construct a “tiered linking graph”, which for given thresholds of closeness and/or significance, identifies subconfigurations and provides a hierarchical ordering in terms of order of significance.