Computational Geometry Questions Medium
In Computational Geometry, the concept of convex hull refers to the smallest convex polygon that encloses a given set of points in a plane. It can also be defined as the intersection of all convex sets that contain the given points.
To understand the concept better, let's consider a set of points in a plane. The convex hull of these points is the smallest convex polygon that contains all the points. In other words, if we were to stretch an elastic band around the points, the convex hull would be the shape formed by the band when it is pulled tight.
The convex hull has several important properties. Firstly, it is always a convex polygon, meaning that any line segment connecting two points on the polygon lies entirely within the polygon. Secondly, it is unique, meaning that for a given set of points, there is only one convex hull. Lastly, the convex hull has the minimum number of vertices required to enclose all the points.
Computational Geometry algorithms are used to efficiently compute the convex hull of a set of points. One commonly used algorithm is the Graham's scan algorithm, which starts by finding the point with the lowest y-coordinate (or the leftmost point in case of a tie) and then sorts the remaining points based on their polar angles with respect to this point. The algorithm then iteratively adds points to the convex hull by checking if the last two points and the current point form a left turn. If not, it removes the last point and repeats the process until all points are processed.
The convex hull has numerous applications in various fields, including computer graphics, pattern recognition, robotics, and geographic information systems. It can be used to determine the boundary of a set of points, identify the outermost points in a cluster, or solve optimization problems involving point sets.