Computational Geometry

  1. Geometric Data Structures and Searching
    1. Range Searching
      1. Problem Definition
        1. Orthogonal Range Queries
          1. Circular Range Queries
            1. Halfspace Range Queries
            2. One-Dimensional Range Searching
              1. Sorted Arrays
                1. Binary Search Trees
                  1. Range Query Algorithms
                    1. Fractional Cascading
                    2. Multi-Dimensional Range Searching
                      1. kd-Trees
                        1. Construction Algorithm
                          1. Splitting Rules
                            1. Balancing Strategies
                              1. Range Query Algorithm
                                1. Time Complexity Analysis
                                  1. Space Complexity
                                  2. Range Trees
                                    1. Layered Structure
                                      1. Construction Algorithm
                                        1. Query Algorithm
                                          1. Fractional Cascading
                                            1. Time and Space Trade-offs
                                            2. Segment Trees
                                              1. Construction
                                                1. Query Processing
                                                  1. Updates
                                                2. Advanced Range Searching
                                                  1. Priority Search Trees
                                                    1. Persistent Data Structures
                                                      1. Dynamic Range Searching
                                                    2. Point Location
                                                      1. Problem Definition
                                                        1. Planar Point Location
                                                          1. Query Types
                                                            1. Preprocessing vs Query Time
                                                            2. Slab Decomposition
                                                              1. Vertical Decomposition
                                                                1. Construction Algorithm
                                                                  1. Query Algorithm
                                                                    1. Time and Space Complexity
                                                                    2. Trapezoidal Maps
                                                                      1. Randomized Incremental Construction
                                                                        1. Trapezoidal Decomposition
                                                                          1. Search Structure
                                                                            1. Query Algorithm
                                                                              1. Expected Complexity
                                                                              2. Triangulation-Based Methods
                                                                                1. Point Location in Triangulations
                                                                                  1. Walking Algorithms
                                                                                    1. Jump-and-Walk
                                                                                    2. Persistent Search Trees
                                                                                      1. Construction
                                                                                        1. Query Processing
                                                                                          1. Space Optimization
                                                                                        2. Spatial Partitioning
                                                                                          1. Quadtrees
                                                                                            1. Node Structure
                                                                                              1. Construction Algorithm
                                                                                                1. Point Insertion
                                                                                                  1. Range Queries
                                                                                                  2. Octrees
                                                                                                    1. 3D Extension
                                                                                                      1. Construction
                                                                                                        1. Applications in Graphics
                                                                                                          1. Collision Detection
                                                                                                          2. Binary Space Partitioning Trees
                                                                                                            1. Splitting Hyperplanes
                                                                                                              1. Construction Strategies
                                                                                                                1. Query Processing
                                                                                                                  1. Applications
                                                                                                                    1. Computer Graphics
                                                                                                                      1. Collision Detection
                                                                                                                        1. Ray Tracing
                                                                                                                      2. R-Trees
                                                                                                                        1. Bounding Rectangles
                                                                                                                          1. Construction
                                                                                                                            1. Query Processing
                                                                                                                              1. Variants

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                                                                                                                          7. Proximity Problems

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                                                                                                                          9. Arrangements and Duality