Here are 100 chapter titles for a Computational Geometry curriculum, progressing from beginner to advanced, tailored for competitive programming:
I. Foundations (20 Chapters)
- Introduction to Computational Geometry: What and Why?
- Points, Vectors, and Coordinate Systems
- Basic Geometric Objects: Lines, Segments, Rays
- Representing Geometric Objects in Code
- Distance Calculations: Point-to-Point, Point-to-Line
- Area of Triangles and Polygons
- Angles and Trigonometric Functions
- Cross Product: Geometric Interpretation and Applications
- Dot Product: Geometric Interpretation and Applications
- Line Equations: Parametric and Implicit Forms
- Intersection of Lines and Line Segments
- Checking if a Point Lies on a Line/Segment
- Orientation of Three Points: Clockwise or Counterclockwise?
- Convex and Concave Polygons
- Simple Polygons: Definition and Properties
- Representing Polygons: Vertex Lists
- Basic Geometric Transformations: Translation, Rotation, Scaling
- Implementing Geometric Transformations in Code
- Introduction to Floating-Point Arithmetic and Precision Issues
- Handling Precision Errors in Geometric Computations
II. Intermediate Techniques (25 Chapters)
- Point in Polygon Test: Ray Casting Algorithm
- Point in Polygon Test: Winding Number Algorithm
- Area of a Polygon: Shoelace Formula
- Centroid of a Polygon
- Convex Hull: Introduction and Definitions
- Graham Scan Algorithm: Convex Hull Construction
- Jarvis' March (Gift Wrapping) Algorithm: Convex Hull
- Divide and Conquer Algorithm for Convex Hull
- Line Segment Intersection: Bentley-Ottmann Algorithm
- Closest Pair of Points: Divide and Conquer Approach
- Diameter of a Polygon
- Width of a Polygon
- Minimum Bounding Box of a Polygon
- Polygon Clipping: Sutherland-Hodgman Algorithm
- Triangulation of Polygons: Ear Clipping Method
- Monotone Polygons: Definition and Properties
- Trapezoidal Decomposition: Polygon Triangulation
- Voronoi Diagrams: Introduction and Properties
- Delaunay Triangulation: Relationship with Voronoi Diagrams
- Line Arrangements: Introduction and Properties
- Duality in Geometric Transformations
- Half-Plane Intersection: Efficient Algorithms
- Arrangement of Lines: Zone Theorem
- Geometric Data Structures: Segment Trees for Intervals
- Range Searching in Geometric Space
III. Advanced Strategies (30 Chapters)
- Convex Hull: Chan's Algorithm (Optimal)
- Linear Programming in 2D: Geometric Interpretation
- Simplex Algorithm: Geometric Intuition
- Point Location in a Planar Subdivision
- Persistent Data Structures for Geometric Problems
- Dynamic Convex Hull Maintenance
- Kinetic Data Structures: Moving Points
- Geometric Intersection Problems: Line-Line, Line-Circle, Circle-Circle
- Minkowski Sum: Geometric Operations
- Geometric Optimization: Finding the Optimal Solution
- Rotating Calipers Technique: Applications
- Smallest Enclosing Circle: Algorithm and Implementation
- Minimum Area Enclosing Rectangle
- Geometric Packing Problems: Introduction
- Geometric Covering Problems: Introduction
- Visibility Problems: Point Visibility in a Polygon
- Art Gallery Problem: Introduction
- Motion Planning: Basic Concepts
- Robot Motion Planning: Geometric Approaches
- Computational Geometry Libraries: CGAL, Boost.Geometry
- Implementing Geometric Algorithms: Code Optimization
- Handling Degeneracies in Geometric Algorithms
- Robust Geometric Computation: Techniques
- Geometric Predicates: Exact Computation
- Interval Trees: Efficient Range Searching
- Kd-Trees: Multi-Dimensional Data Structures
- R-Trees: Spatial Data Structures
- Quadtrees and Octrees: Hierarchical Spatial Data Structures
- Geometric Hashing: Efficient Data Retrieval
- Randomized Algorithms in Computational Geometry
IV. Expert Level & Applications (25 Chapters)
- Computational Geometry in Higher Dimensions
- Arrangements of Hyperplanes
- Convex Polytopes: Properties and Algorithms
- Geometric Duality: Advanced Concepts
- Geometric Transformations: Advanced Topics
- Geometric Data Structures: Advanced Topics
- Geometric Algorithms: Parallel Implementations
- Computational Geometry in Practice: Case Studies
- Applications in Computer Graphics
- Applications in Robotics
- Applications in Geographic Information Systems (GIS)
- Applications in Computer-Aided Design (CAD)
- Applications in Image Processing
- Applications in Machine Learning
- Competitive Programming Strategies: Geometric Problems
- Debugging Geometric Algorithms: Common Pitfalls
- Performance Tuning: Geometric Code Optimization
- Advanced Topics in Voronoi Diagrams and Delaunay Triangulation
- Mesh Generation: Triangulation and Tetrahedralization
- Surface Reconstruction: From Point Clouds
- Computational Geometry Research: Open Problems
- Quantum Computational Geometry: Introduction
- The Future of Computational Geometry: Emerging Trends
- Geometric Deep Learning: Applications
- Advanced Topics in Geometric Optimization
This list provides a comprehensive roadmap for learning computational geometry. Remember that practice is essential. Solve coding challenges and participate in competitive programming contests to apply these techniques and solidify your understanding. Working with visualization tools can also be extremely helpful in understanding geometric concepts.