Computational Geometry Questions Long
The Graham scan algorithm is a popular and efficient method for computing the convex hull of a planar point set in computational geometry. It follows a simple and intuitive approach, which can be described as follows:
1. Choose the point with the lowest y-coordinate (in case of a tie, choose the one with the lowest x-coordinate) as the starting point P0.
2. Sort the remaining points in counterclockwise order with respect to the angle they make with the horizontal line passing through P0. This can be done using the polar angle of each point with respect to P0.
3. Initialize an empty stack and push P0 onto it.
4. Iterate through the sorted points, starting from the first point P1. For each point Pi, do the following:
a. While the stack size is greater than or equal to 2 and the last two points on the stack and Pi make a right turn (i.e., the cross product of the vectors formed by the last two points and Pi is negative), pop the top element from the stack.
b. Push Pi onto the stack.
5. Once all the points have been processed, the stack will contain the vertices of the convex hull in counterclockwise order. Pop the elements from the stack and store them in a list or array to obtain the convex hull.
The Graham scan algorithm works by maintaining a convex hull in the form of a stack. It starts with the point with the lowest y-coordinate as the starting point and then iterates through the remaining points in counterclockwise order. At each step, it checks if the current point makes a right turn with respect to the last two points on the stack. If it does, it means that the current point is not part of the convex hull and should be discarded. This process continues until all the points have been processed, resulting in the convex hull stored in the stack.
The time complexity of the Graham scan algorithm is O(n log n), where n is the number of input points. This is because the algorithm involves sorting the points based on their polar angles, which takes O(n log n) time. The remaining steps, such as pushing and popping elements from the stack, take O(n) time. Overall, the Graham scan algorithm provides an efficient solution for computing the convex hull of a planar point set in computational geometry.