Convex Hull Algorithms in Computational Geometry

Convex hull algorithms in computational geometry are used to determine the smallest convex polygon that encloses a set of points. Efficient and user-friendly convex hull algorithms are widely applied in computer graphics, path planning, and related fields.

Common convex hull algorithms include:

Graham Scan: This algorithm first identifies the lowest point as a pivot, sorts all points by polar angle, then incrementally processes them while eliminating points that violate convexity. With O(n log n) time complexity, it's suitable for most practical scenarios. The key implementation steps involve sorting using cross-product calculations and maintaining a stack for convex hull points.

Andrew's Algorithm (Monotone Chain): By performing two separate scans (upper and lower), this method constructs the convex hull's top and bottom portions independently. It offers stable O(n log n) performance and straightforward implementation using point coordinate sorting and cross-product checks.

Jarvis March (Gift Wrapping): Starting from the leftmost point, this algorithm iteratively finds the next convex hull vertex by comparing angular relationships. While its O(nh) time complexity (where h is the number of hull vertices) makes it less efficient for large datasets, it performs well for smaller point sets. The implementation typically uses vector cross-products to determine winding directions.

For programming implementations, many libraries (such as Python's `scipy.spatial.ConvexHull` or C++'s CGAL) provide built-in convex hull computation capabilities. Users can simply pass point sets to these functions without implementing underlying algorithms manually. The `ConvexHull` class in SciPy, for example, uses Qhull internally and returns hull vertices, facets, and volume calculations. Choosing the appropriate algorithm requires balancing dataset size with performance requirements, where Graham Scan or Andrew's Algorithm generally suit large datasets, while Jarvis March works better for smaller sets or when hull size is expected to be small.