Computational Geometry

  1. Convex Hulls
    1. Convexity Fundamentals
      1. Convex Sets
        1. Convex Combinations
          1. Convex Hull Definition
            1. Properties of Convex Hulls
              1. Supporting Lines and Hyperplanes
                1. Extreme Points
                2. Two-Dimensional Convex Hulls
                  1. Problem Formulation
                    1. Lower Bound Analysis
                      1. Sorting Reduction
                        1. Information-Theoretic Lower Bound
                        2. Graham Scan Algorithm
                          1. Preprocessing Step
                            1. Scanning Phase
                              1. Stack Operations
                                1. Time Complexity Analysis
                                  1. Implementation Details
                                  2. Jarvis March Algorithm
                                    1. Gift Wrapping Concept
                                      1. Algorithm Steps
                                        1. Output Sensitivity
                                          1. Time Complexity Analysis
                                          2. Monotone Chain Algorithm
                                            1. Lower Hull Construction
                                              1. Upper Hull Construction
                                                1. Merging Phase
                                                  1. Comparison with Graham Scan
                                                  2. Divide and Conquer Algorithm
                                                    1. Recursive Division
                                                      1. Merging Convex Hulls
                                                        1. Bridge Finding
                                                          1. Time Complexity Analysis
                                                          2. Chan's Algorithm
                                                            1. Output-Sensitive Approach
                                                              1. Grouping Strategy
                                                                1. Mini-Hull Construction
                                                                  1. Overall Algorithm
                                                                    1. Time Complexity Analysis
                                                                  2. Three-Dimensional Convex Hulls
                                                                    1. Problem Complexity
                                                                      1. Divide and Conquer Approach
                                                                        1. Recursive Division
                                                                          1. Merging 3D Hulls
                                                                            1. Handling Degeneracies
                                                                            2. Randomized Incremental Construction
                                                                              1. Point Insertion Strategy
                                                                                1. Conflict Graphs
                                                                                  1. Visibility Computation
                                                                                    1. Expected Time Analysis
                                                                                    2. Gift Wrapping in 3D
                                                                                      1. Face-by-Face Construction
                                                                                        1. Edge Finding
                                                                                      2. Higher-Dimensional Convex Hulls
                                                                                        1. Curse of Dimensionality
                                                                                          1. Complexity in d-Dimensions
                                                                                            1. Output Sensitivity
                                                                                              1. Practical Algorithms
                                                                                                1. Applications and Limitations

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                                                                                              3. Fundamental Geometric Predicates

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                                                                                              5. Line Segment Intersection