Convex Hull Algorithm Implementation

Convex hull algorithms represent a classic problem in computational geometry, aiming to find the smallest convex polygon that contains all points in a given two-dimensional point set, with all points lying either inside or on the edges of this polygon. Graham's Scan is one of the commonly used algorithms to solve this problem, whose core concept involves efficiently constructing the convex hull through polar angle sorting and stack operations.

The implementation of Graham's Scan typically involves several key steps. First, select a pivot point, usually the point with the smallest y-coordinate (if multiple points share the same y-coordinate, choose the one with the smallest x-coordinate). Then, sort the remaining points based on their polar angles relative to the pivot point, with secondary sorting by distance from the pivot point for points sharing the same polar angle. Subsequently, traverse these points in order, using a stack structure to maintain candidate points for the convex hull. Whenever a new point is added, check if it violates the convex hull property (i.e., creates a concave point). If a concave point is detected, backtrack by removing points from the top of the stack until convexity is restored.

When implementing Graham's Scan in MATLAB, built-in sorting functions and vector operations can be leveraged to optimize performance. For instance, polar angle sorting can be achieved using the atan2 function, while maintaining the convex hull point set can be accomplished through array-based stack simulation. Improved versions of Graham's algorithm may optimize sorting strategies or backtracking conditions to reduce computational complexity or enhance numerical stability. Key MATLAB functions involved include sortrows for coordinate sorting, cross product calculations for convexity checks, and array indexing for efficient stack operations.

The time complexity of this algorithm is primarily determined by the sorting step, typically O(n log n), where n represents the number of points. MATLAB's matrix operation characteristics make the algorithm implementation more concise and efficient, particularly suitable for processing large-scale two-dimensional point sets. Convex hull algorithms find wide applications in computer vision, path planning, geographic information systems, and other computational geometry domains. The implementation can be further enhanced by incorporating vectorized operations for angle calculations and using MATLAB's built-in convex hull functions for verification purposes.