Introduction

In computational geometry, determining the boundary points of a shape often involves identifying the points that form the minimal enclosing outline or hull of that shape. This process is crucial in various applications like computer graphics, image processing, and geospatial mapping. In this article, we'll explore an efficient algorithm to determine the boundary points of a shape using JavaScript.

Convex Hull Problem

A popular problem in determining boundary points is the convex hull problem. The convex hull of a set of points is the smallest convex polygon that contains all the points. This can be visualized as the shape formed by a rubber band snapped around the outermost points of a dataset.

Graham's Scan Algorithm

Graham's Scan is an efficient algorithm for finding the convex hull of a finite set of points in the plane. It constructs the hull in `O(n log n)` time complexity due to the preliminary sort step and linear pass through the sorted list of points.

Algorithm Explanation

1. Find the Point with the Lowest Y-Coordinate
   Start with the point having the lowest y-coordinate. If there are multiple points with the same y-coordinate, choose the one with the lowest x-coordinate. This point is guaranteed to be part of the convex hull.
2. Sort the Points
   Sort the remaining points based on the polar angle made with the initial point. In case of ties, sort by distance to the initial point.
3. Construct the Convex Hull
   Using a stack, iterate over the sorted points and ensure that each triplet of points (top two on the stack and current point) makes a counter-clockwise turn. If a clockwise turn is detected, pop the stack until a counter-clockwise turn is encountered or the stack is empty.

JavaScript Implementation

Here's the implementation of Graham's Scan in JavaScript:

• Collinear Points: They do not affect the convexity of the shape; they will be handily managed by the same algorithm.
• Edge Cases: Pay attention to special cases such as all points lying on the same line or duplicate points.
• Precision: When dealing with floating-point coordinates, precision issues may arise, requiring careful handling of numerical operations.