1Introduction to Computational Geometry
92Binary Search Trees
21.1 Definition and Scope
93Interval Trees: Trapezoidal Maps
3Geometric Primitives:
943.1.1 Point-in-Polygon Testing
4Geometric Algorithms:
953.1.2 Ray Casting
5Data Structures:: Computational Complexity:
963.1.3 Binary Search Tree Approaches
61.2 Historical Overview
97Introduction to Point Location Algorithms
71.2.1 Milestones in the Development of Computational Geometry
98Binary Search Trees: Theory and Properties: Basic Binary Search Tree Operations
81.2.2 Influential Figures in the Field
993.2 Convex Hull Algorithms
9Early Developments and Foundations
100The Convex Hull Problem
10Key Concepts and Problems
101Characteristics of Convex Polygons
11Influential Figures in Computational Geometry
102Convex Hull Algorithms
12Algorithmic Breakthroughs and Advancements
1033.2.1 Graham Scan
13Applications and Impact
104Understanding Convex Hull
14Future Directions and Challenges
105Graham Scan Algorithm: Applications of Convex Hull and Graham Scan
151.2.3 Evolution of Key Algorithms and Techniques
1063.2.2 Quickhull
16Early Developments
1073.3 Voronoi Diagrams
17Convex Hull Algorithms
1083.4 Line Intersection Algorithms
18Voronoi Diagrams
1093.5 Polygon Triangulation
19Delaunay Triangulation
110Introduction to Polygon Triangulation:
20Line Segment Intersection
111Triangulation Algorithms:
21Geometric Data Structures
112Complexity Analysis:: Applications of Polygon Triangulation:
22Computational Geometry in Practice
1133.6 Summary
231.3 Importance and Applications
1143.7 Exercise
24Importance of Computational Geometry:
115Geometric Data Structures
25Applications of Computational Geometry:
1164.1 Quadtree and Octree
261.3.1 Role in Geographic Information Systems (GIS)
117Quadtree:
271.3.2 Impact on Computer Graphics and Animation
118Octree:
281.3.3 Applications in Robotics and Automation
119Comparison and Selection:
29Understanding Computational Geometry:: Key Concepts in Computational Geometry:
1204.1.1 Adaptive Spatial Partitioning
301.4 Challenges and Current Trends: Applications of Computational Geometry
121Quadtree: Basics and Structure
311.5 Summary
122Octree: Basics and Structure
321.6 Exercise
123Adaptive Spatial Partitioning
33Foundations of Geometry
124Strategies for Adaptive Partitioning
34Historical Background:
125Applications
35Euclidean Geometry:
126Implementation Considerations
36Non-Euclidean Geometries:
1274.1.2 Range Query Optimization
37Spherical Geometry:
1284.1.3 Applications in Image Processing and GIS
38Hyperbolic Geometry:
129Quadtree
39Foundational Issues and Axiomatic Systems:: Hilbert’s axioms for geometry included:
130Octree: Applications in Image Processing
402.1 Euclidean Geometry
1314.2 Binary Space Partitioning (BSP) Trees
412.1.1 Properties of Euclidean Space
132Introduction to Geometric Data Structures
422.1.2 Euclidean Transformations
133Motivation for Binary Space Partitioning Trees
43Foundations of Geometry:
134The key advantages of BSP trees include:
44Euclidean Transformations:
135Overview of Binary Space Partitioning Trees
45Properties of Euclidean Transformations:
136Properties of BSP Trees
46Types of Euclidean Transformations:
137Applications of BSP Trees
47Applications of Euclidean Transformations:
138Optimization Techniques
48Conclusion:
139Challenges and Future Directions
492.1.3 Euclidean Distance and Metrics
1404.2.1 Binary Space Partitioning for Collision Detection
50Euclidean Geometry Fundamentals
1414.2.2 BSP Tree Construction Algorithms
51Euclidean Distance
1424.3 Segment Trees
52Properties of Euclidean Distance
1434.3.1 Range Query and Update Operations
53Metrics in Euclidean Geometry
144Overview of Segment Trees
542.2 Analytical Geometry
145Applications in Geometric Data Structures:
552.2.1 Cartesian Coordinates
1464.3.2 Interval Trees: Introduction to Geometric Data Structures
56Origins of Cartesian Coordinates
1474.4 Range Trees
57Equations of Lines and Curves
148Overview of Range Trees
58Transformations in the Coordinate Plane: Applications of Cartesian Coordinates
149Applications of Range Trees: Optimization Techniques
592.2.2 Equations of Lines and Planes
1504.5 Kinetic Data Structures
60Relationship Between Lines and Planes:: Applications in Geometry and Beyond:
1514.6 Summary
612.2.3 Parametric Representations of Curves
1524.7 Exercise
622.3 Geometric Primitives
153Spatial Queries and Analysis
63Foundations of Geometry
1545.1 Nearest Neighbour Queries
64Classification of Geometric Primitives
1555.2 Range Searching
65Roles of Geometric Primitives
1565.3 Closest Pair Problems
66Definition of Geometric Objects
1575.4 Spatial Data Mining
67Construction of Geometric Figures
1585.5 Summary
68Visualization of Geometric Concepts
1595.6 Exercise
69Proof of Geometric Theorems
160Applications in Geographic Information Systems (GIS)
70Applications of Geometric Primitives
1616.1 Spatial Data Structures
712.3.1 Points, Lines, and Segments
1626.2 Geospatial Analysis
72Geometric Relationships and Properties: Applications in Geometry and Beyond
1636.3 Network Analysis
732.3.2 Polygons and Polyhedra
1646.4 Cartographic Modeling
742.3.3 Circles, Ellipses, and Hyperbolas
1656.5 Geodatabases
75Circles:
1666.6 Summary
76Ellipses:: Hyperbolas:
1676.7 Exercise
772.4 Linear Algebra in Geometry
168Emerging Trends and Future Directions
78Vector Spaces and Linear Transformations: Coordinate Systems and Representation
1697.1 Quantum Computational Geometry
792.5 Summary
1707.2 Topological Data Analysis
802.6 Exercise
171Overview of Topological Data Analysis
81Basic Algorithms in Computational Geometry
172Current Landscape of TDA Research: Challenges and Limitations
82Significance of Computational Geometry
1737.3 Blockchain and Computational Geometry
83Geometric Primitives and Operations
174Understanding Blockchain Technology
84Convex Hull Computation
175Overview of Computational Geometry: Future Directions and Challenges
85Line Intersection
1767.4 Bioinformatics and Computational Geometry
86Closest Pair of Points
1777.5 Summary
87Voronoi Diagrams: Challenges and Future Directions
1787.6 Exercise
883.1 Point Location Algorithms
179References
89Problem Statement
180Glossary
90Significance of Point Location
181Index
91Brute-Force Approach
