Computational Geometry

  1. Motion Planning
    1. Robot Motion Planning Problem
      1. Problem Formulation
        1. Types of Robots
          1. Point Robots
            1. Rigid Body Robots
              1. Articulated Robots
              2. Constraints
                1. Geometric Constraints
                  1. Kinematic Constraints
                    1. Dynamic Constraints
                  2. Configuration Space
                    1. Configuration Definition
                      1. Degrees of Freedom
                        1. Configuration Space Construction
                          1. Dimensionality Issues
                          2. Obstacles and C-Obstacles
                            1. Workspace Obstacles
                              1. Configuration Space Obstacles
                                1. C-Obstacle Construction
                                  1. Translation
                                    1. Rotation
                                      1. Combined Transformations
                                    2. Path Planning for Point Robots
                                      1. Visibility Graphs
                                        1. Construction Algorithm
                                          1. Shortest Path Computation
                                            1. Complexity Analysis
                                            2. Cell Decomposition
                                              1. Exact Decomposition
                                                1. Approximate Decomposition
                                                  1. Trapezoidal Decomposition
                                                  2. Potential Field Methods
                                                    1. Attractive Potentials
                                                      1. Repulsive Potentials
                                                        1. Local Minima Problem
                                                      2. Shortest Paths in Polygons
                                                        1. Shortest Path Trees
                                                          1. Funnel Algorithm
                                                            1. Funnel Maintenance
                                                              1. Path Construction
                                                              2. Visibility-Based Methods
                                                              3. Minkowski Sums
                                                                1. Definition and Properties
                                                                  1. Geometric Interpretation
                                                                    1. Computation Methods
                                                                      1. Convex Polygons
                                                                        1. General Polygons
                                                                        2. Applications
                                                                          1. C-Obstacle Construction
                                                                            1. Robot Motion Planning
                                                                          2. Probabilistic Methods
                                                                            1. Probabilistic Roadmaps
                                                                              1. Rapidly-Exploring Random Trees
                                                                                1. Sampling Strategies
                                                                                2. Grid-Based Methods
                                                                                  1. Discretization Approaches
                                                                                    1. A* Search Algorithm
                                                                                      1. Heuristic Functions
                                                                                        1. Optimality Conditions
                                                                                        2. D* Algorithm
                                                                                          1. Hierarchical Methods

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                                                                                        9. Arrangements and Duality

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                                                                                        11. Advanced Topics