On the Geometric Dilation of Finite Point Sets 1st edition by Annette Ebbers Baumann, Ansgar Grune, Rolf Klein ISBN 3540206958 9783540206958

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ISBN 10:    3540206958   
ISBN 13: 978-3540206958   
Author:         Annette Ebbers Baumann, Ansgar Grüne, Rolf Klein 
 

On the Geometric Dilation of Finite Point Sets 1st Table of contents:

1. Introduction

   • 1.1 Overview of Geometric Dilation
   • 1.2 Importance of Dilation in Computational Geometry
   • 1.3 Motivation and Applications of Geometric Dilation
   • 1.4 Structure of the Paper

2. Preliminaries in Geometric Geometry

   • 2.1 Basic Definitions in Computational Geometry
   • 2.2 Geometric Transformations and Scaling
   • 2.3 Point Sets in Euclidean Space
   • 2.4 Convex Hulls and Other Geometric Constructs
   • 2.5 Notations and Terminology

3. Understanding Geometric Dilation

   • 3.1 Formal Definition of Geometric Dilation
   • 3.2 Dilation in Euclidean Space
   • 3.3 The Role of Distance Functions in Dilation
   • 3.4 Properties of Geometric Dilation
   • 3.5 Dilation and Convexity: A Relationship

4. Dilation of Finite Point Sets

   • 4.1 Dilation for Finite Point Sets: Key Concepts
   • 4.2 Dilation of Point Sets in One, Two, and Higher Dimensions
   • 4.3 Geometric Dilation and the Shape of Point Sets
   • 4.4 Impact of Distribution and Configuration of Points on Dilation
   • 4.5 Applications to Geometric Clustering and Approximation

5. Computational Methods for Geometric Dilation

   • 5.1 Algorithms for Calculating Geometric Dilation
   • 5.2 Exact vs. Approximate Methods
   • 5.3 Efficient Computation of Convex Hulls and Related Structures
   • 5.4 Approximating Dilation in High Dimensions
   • 5.5 Complexity Analysis of Dilation Algorithms

6. Properties and Bounds of Geometric Dilation

   • 6.1 Upper and Lower Bounds for Dilation
   • 6.2 Extremal Point Configurations and Their Dilation
   • 6.3 Relationships Between Dilation and Other Geometric Parameters (e.g., Diameter, Spread)
   • 6.4 Metric Properties of Dilation in Various Spaces
   • 6.5 Dilation and Optimization: Finding Minimal Dilation Partitions

7. Applications of Geometric Dilation

   • 7.1 Dilation in Clustering and Data Analysis
   • 7.2 Geometric Dilation in Network Design and Facility Location
   • 7.3 Dilation and its Role in Image Processing and Computer Vision
   • 7.4 Dilation in Geospatial Data and Geographic Information Systems (GIS)
   • 7.5 Real-World Examples and Case Studies

8. Advanced Topics in Geometric Dilation

   • 8.1 Dilation in Non-Euclidean Spaces (e.g., Hyperbolic, Spherical)
   • 8.2 Dilation for Non-Convex Sets and Complex Geometries
   • 8.3 Randomized Algorithms for Dilation
   • 8.4 Multi-Scale Dilation and Its Computational Applications
   • 8.5 Geometric Dilation in Graphs and Networks

9. Open Problems and Future Directions

   • 9.1 Challenges in Computing Geometric Dilation in High Dimensions
   • 9.2 Exploring Non-Convex Dilation and Generalizations
   • 9.3 Theoretical and Algorithmic Open Problems in Dilation
   • 9.4 Future Directions in Geometric Dilation and Applications

10. Conclusion

    • 10.1 Summary of Key Insights on Geometric Dilation
    • 10.2 Contributions to Computational Geometry and Optimization
    • 10.3 Final Thoughts and Future Impact
11.  

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