Computational Geometry

  1. Proximity Problems
    1. Closest Pair of Points
      1. Problem Definition
        1. Brute-Force Algorithm
          1. All-Pairs Distance
            1. Time Complexity
            2. One-Dimensional Case
              1. Sorting Approach
                1. Linear Time Algorithm
                2. Divide and Conquer Algorithm
                  1. Recursive Division
                    1. Merging Step
                      1. Strip Processing
                        1. Time Complexity Analysis
                          1. Space Complexity
                          2. Randomized Algorithms
                            1. Higher Dimensions
                            2. Voronoi Diagrams
                              1. Definitions and Properties
                                1. Voronoi Cells
                                  1. Voronoi Edges
                                    1. Voronoi Vertices
                                      1. Bisectors
                                      2. Geometric Properties
                                        1. Convexity of Cells
                                          1. Planarity
                                            1. Connectivity
                                            2. Combinatorial Complexity
                                              1. Number of Vertices
                                                1. Number of Edges
                                                  1. Euler's Formula Application
                                                  2. Construction Algorithms
                                                    1. Fortune's Sweep-Line Algorithm
                                                      1. Beach Line Concept
                                                        1. Site Events
                                                          1. Circle Events
                                                            1. Event Processing
                                                              1. Data Structures
                                                              2. Incremental Construction
                                                                1. Point Insertion
                                                                  1. Cell Updates
                                                                    1. Randomized Analysis
                                                                    2. Divide and Conquer
                                                                      1. Recursive Division
                                                                        1. Merging Voronoi Diagrams
                                                                          1. Merge Curve Computation
                                                                        2. Applications
                                                                          1. Nearest Neighbor Queries
                                                                            1. Facility Location
                                                                              1. Path Planning
                                                                                1. Spatial Analysis
                                                                                  1. Meteorology
                                                                                2. Delaunay Triangulations
                                                                                  1. Definition and Properties
                                                                                    1. Delaunay Condition
                                                                                      1. Empty Circle Property
                                                                                        1. Circumcircle Test
                                                                                          1. Angle Maximization
                                                                                          2. Relationship to Voronoi Diagrams
                                                                                            1. Geometric Duality
                                                                                              1. Edge Correspondence
                                                                                                1. Vertex Correspondence
                                                                                                2. Uniqueness Properties
                                                                                                  1. General Position
                                                                                                    1. Degeneracy Handling
                                                                                                    2. Construction Algorithms
                                                                                                      1. Edge Flipping Algorithm
                                                                                                        1. Illegal Edge Detection
                                                                                                          1. Local Optimality
                                                                                                            1. Convergence Proof
                                                                                                            2. Randomized Incremental Construction
                                                                                                              1. Point Insertion Order
                                                                                                                1. Conflict Graphs
                                                                                                                  1. History DAG
                                                                                                                    1. Expected Time Analysis
                                                                                                                    2. Divide and Conquer
                                                                                                                      1. Recursive Construction
                                                                                                                        1. Merging Triangulations
                                                                                                                      2. Applications
                                                                                                                        1. Mesh Generation
                                                                                                                          1. Interpolation
                                                                                                                            1. Terrain Modeling

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                                                                                                                        6. Polygon Triangulation

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                                                                                                                        8. Geometric Data Structures and Searching