📐Computational Geometry Unit 2 – Convex hull algorithms

What's a Convex Hull Anyway?

• Convex hull is the smallest convex polygon that encloses all points in a given set
• Imagine stretching a rubber band around a set of points on a plane, the shape it forms is the convex hull
• All points in the set are either on the boundary or inside the convex hull
• The convex hull is unique for a given set of points
• Convex hulls have applications in various fields (computational geometry, computer graphics, robotics)
  • Used for collision detection in video games and simulations
  • Helps in shape analysis and pattern recognition in computer vision
• Convex hulls can be computed in 2D, 3D, or higher dimensions
• The number of points on the convex hull is always less than or equal to the total number of points in the set

Key Concepts and Definitions

• Convexity: A set is convex if for any two points in the set, the line segment connecting them is also entirely within the set
• Convex combination: A point that can be expressed as a weighted average of other points, with non-negative weights that sum to 1
• Extreme points: The vertices of the convex hull, which cannot be expressed as convex combinations of other points in the set
• Supporting line: A line that touches the convex hull at an extreme point without intersecting the interior of the hull
• Facet: A lower-dimensional feature of the convex hull (edges in 2D, faces in 3D)
• Halfspace: A region of space defined by a hyperplane, dividing the space into two parts
  • Points on one side of the hyperplane belong to the halfspace, while points on the other side do not
• Duality: The concept of representing points as lines and lines as points in a dual space, which is useful in some convex hull algorithms

Popular Convex Hull Algorithms

• Gift Wrapping (Jarvis March): Starts with an extreme point and "wraps" around the set, finding the next extreme point at each step
  • Time complexity: $O(nh)$, where $n$ is the number of points and $h$ is the number of points on the hull
• Graham Scan: Sorts points by angle and performs a stack-based scan to construct the hull
  • Time complexity: $O(n \log n)$ due to the sorting step
• Quickhull: Divide-and-conquer approach that recursively partitions the point set and builds the hull
  • Average time complexity: $O(n \log n)$, worst-case: $O(n^2)$
• Monotone Chain: Sorts points lexicographically and builds upper and lower hulls separately
  • Time complexity: $O(n \log n)$
• Incremental: Adds points one by one, updating the hull at each step
  • Time complexity: $O(n^2)$ in the worst case, but can be improved with randomization
• Divide-and-Conquer: Recursively divides the point set, computes hulls for subsets, and merges them
  • Time complexity: $O(n \log n)$

Time Complexity Analysis

• The time complexity of convex hull algorithms depends on the number of input points ($n$) and the output size ($h$)
• Optimal output-sensitive algorithms have a time complexity of $O(n \log h)$
  • Examples: Chan's algorithm, Kirkpatrick-Seidel algorithm
• Worst-case optimal algorithms have a time complexity of $O(n \log n)$
  • Examples: Graham Scan, Monotone Chain, Divide-and-Conquer
• Some algorithms, like Gift Wrapping, have a time complexity that depends on both $n$ and $h$
• Randomized algorithms, like randomized incremental, can achieve better expected time complexity
• Lower bound for computing convex hulls is $\Omega(n \log n)$ in the worst case, based on the sorting lower bound

Implementation Strategies

• Choose the appropriate algorithm based on the problem requirements and input characteristics
• Utilize existing libraries or frameworks (Computational Geometry Algorithms Library (CGAL), Boost Geometry)
• Implement robust geometric primitives (point comparison, orientation tests) to handle degenerate cases and numerical precision issues
  • Use epsilon comparisons or exact arithmetic libraries (GMP, MPFR) when necessary
• Optimize for cache efficiency by using contiguous memory and locality-friendly data structures
• Parallelize computations when possible, especially for large datasets
• Use spatial data structures (kd-trees, range trees) for efficient point queries and updates
• Implement proper memory management and cleanup to avoid leaks and optimize resource usage

Real-World Applications

• Computer graphics and visualization: Constructing bounding volumes for efficient rendering and collision detection
• Robotics and motion planning: Defining safe navigation regions and obstacle avoidance
• Geographic information systems (GIS): Analyzing spatial data and generating maps
• Machine learning and data analysis: Identifying clusters and outliers in high-dimensional datasets
• Computational biology: Studying protein folding and molecular shape analysis
• Computer vision and image processing: Object recognition and tracking
• Facility location and resource allocation: Optimizing placement of service centers or resources
• Meteorology and climate modeling: Analyzing weather patterns and atmospheric phenomena

Common Pitfalls and How to Avoid Them

• Numerical instability and precision issues: Use robust geometric primitives and consider exact arithmetic when necessary
• Degeneracies and special cases: Handle collinear points, duplicate points, and other edge cases properly
• Incorrect algorithm selection: Choose the appropriate algorithm based on the problem requirements and input characteristics
• Inefficient implementations: Optimize for cache efficiency, use appropriate data structures, and consider parallelization when possible
• Not considering the output size: Some algorithms, like Gift Wrapping, have a time complexity that depends on both input size and output size
• Forgetting to free allocated memory: Implement proper memory management and cleanup to avoid leaks
• Overlooking the cost of geometric primitives: Be aware of the time complexity of underlying operations (point comparison, orientation tests) and their impact on overall performance

Advanced Topics and Extensions

• Approximate convex hulls: Computing a simplified or approximate hull for large datasets or real-time applications
  • Algorithms: $\epsilon$-hull, hull peeling, coresets
• Dynamic convex hulls: Maintaining the convex hull under point insertions, deletions, or updates
  • Algorithms: Overmars-van Leeuwen, Chan's dynamic algorithm
• Convex hull in higher dimensions: Extending algorithms to work in 3D or higher-dimensional spaces
  • Algorithms: Quickhull, gift wrapping, beneath-beyond
• Weighted convex hulls: Associating weights with points and computing the weighted convex hull
• Convex layers and depth: Computing nested convex hulls or the depth of points within the hull
• Convex hull under constraints: Computing the convex hull subject to additional constraints (line segments, polygonal obstacles)
• Duality-based algorithms: Leveraging the duality transform to solve convex hull problems in dual space
  • Algorithms: Half-plane intersection, point-line duality