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1 Computational Geometry

    Introduction

    1.1  An Example: Convex Hulls

    1.2  Degeneracies and Robustness

    1.3  Application Domains

    1.4  Notes and Comments

    1.5  Exercises

   Line Segment Intersection

   Thematic Map Overlay

   2.1  Line Segment Intersection

   2.2  The Doubly-Connected Edge List

   2.3  Computing the Overlay of Two

Subdivisions

   2.4  Boolean Operations

   2.5  Notes and Comments

   2.6  Exercises

   Polygon Triangulation

   Guarding an Art Gallery

   3.1  Guarding and Triangulations

   3.2  Partitioning a Polygon into Monotone

Pieces

   3.3  Triangulating a Monotone Polygon

   3.4  Notes and Comments

   3.5  Exercises

   Linear Programming

   Manufacturing with Molds

   4.1  The Geometry of Casting

   4.2  Half-Plane Intersection

   4.3  Incremental Linear Programming

   4.4  Randomized Linear Programming

   4.5  Unbounded Linear Programs

   4.6* Linear Programming in Higher Dimensions

   4.7* Smallest Enclosing Discs

   4.8  Notes and Comments

   4.9  Exercises

   Orthogonal Range Searching

    Querying a Database

    5.1  1-Dimensional Range Searching

    5.2  Kd-Trees

    5.3  Range Trees

    5.4  Higher-Dimensional Range Trees

    5.5  General Sets of Points

    5.6* Fractional Cascading

    5.7  Notes and Comments

    5.8  Exercises

6 Point Location

    Knowing Where You Are

    6.1  Point Location and Trapezoidal

Maps

    6.2  A Randomized Incremental

Algorithm

    6.3  Dealing with Degenerate Cases

    6.4* A Tail Estimate

    6.5  Notes and Comments

    6.6  Exercises

7 Voronoi Diagrams

    The Post Office Problem

    7.1  Definition and Basic Properties

    7.2  Computing the Voronoi Diagram

    7.3  Voronoi Diagrams of Line

Segments

    7.4  Farthest-Point Voronoi Diagrams

    7.5  Notes and Comments

    7.6  Exercises

   Arrangements and Duality

    Supersampling in Ray Tracing

    8.1  Computing the Discrepancy

    8.2  Duality

    8.3  Arrangements of Lines

    8.4  Levels and Discrepancy

    8.5  Notes and Comments

    8.6  Exercises

    Delaunay Triangulations

    Height Interpolation

    9.1  Triangulations of Planar Point

Sets

    9.2  The Delaunay Triangulation

    9.3  Computing the Delaunay

Triangulation

    9.4  The Analysis

    9.5* A Framework for Randomized Algorithms

    9.6  Notes and Comments

    9.7  Exercises

10 More Geometric Data Structures

    Windowing

    10.1 Interval Trees

    10.2 Priority Search Trees

    10.3 Segment Trees

    10.4 Notes and Comments

    10.5 Exercises

11 Convex Hulls

   Mixing Things

    11.1 The Complexity of Convex Hulls in

3-Space

    11.2 Computing Convex Hulls in 3-Space

    11.3" The Analysis

    11.4' Convex Hulls and Half-Space

Intersection

    11.5' Voronoi Diagrams Revisited

    11.6 Notes and Comments

    11.7 Exercises

12 Binary Space Partitions

   The Painter's Algorithm

    12.1 The Definition of BSP Trees

    12.2 BSP Trees and the Painter's Algorithm

    12.3 Constructing a BSP Tree

    12.4' The Size of BSP Trees in 3-Space

    12.5 BSP Trees for Low-Density Scenes

    12.6 Notes and Comments

    12.7 Exercises

13 Robot Motion Planning

    Getting Where You Want to Be

    13.1 Work Space and Configuration Space

    13.2 A Point Robot

    13.3 Minkowski Sums

    13.4 Translational Motion Planning

    13.5' Motion Planning with Rotations

    13.6 Notes and Comments

    13.7 Exercises

14 Quadtrees

    Non-Uniform Mesh Generation

    14.1 Uniform and Non-Uniform Meshes

    14.2 Quadtrees for Point Sets

    14.3 From Quadtrees to Meshes

    14.4 Notes and Comments

    14.5 Exercises

15 Visibility Graphs

    Finding the Shortest Route

    15.1 Shortest Paths for a Point Robot

    15.2 Computing the Visibility Graph

    15.3 Shortest Paths for a Translating Polygonal

Robot

    15.4 Notes and Comments

    15.5 Exercises

16 Simplex Range Searching

    Windowing Revisited

    16.1 Partition Trees

    16.2 Multi-Level Partition Trees

    16.3 Cutting Trees

    16.4 Notes and Comments

    16.5 Exercises

Bibliography

Index

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