Computational Geometry Questions Long
The algorithm for computing the convex hull of a set of line segments in Computational Geometry can be achieved by following these steps:
1. Input: Start with a set of line segments as input.
2. Find the endpoints: Identify all the endpoints of the line segments in the input set. These endpoints will be the initial points of the convex hull.
3. Sort the endpoints: Sort the endpoints in counterclockwise order based on their polar angles with respect to a reference point. This reference point can be any point within the set of line segments or the leftmost point among the endpoints.
4. Initialize the convex hull: Create an empty stack to store the points of the convex hull.
5. Push the first two points: Push the first two points from the sorted endpoints onto the stack.
6. Traverse the remaining points: Iterate through the sorted endpoints starting from the third point. For each point, check if it makes a left turn or a right turn with the top two points on the stack.
a. Left turn: If the current point makes a left turn with the top two points on the stack, push the current point onto the stack.
b. Right turn: If the current point makes a right turn with the top two points on the stack, pop the top point from the stack and repeat this step until a left turn is obtained.
7. Repeat step 6 until all the points have been traversed.
8. Output: The stack now contains the points of the convex hull in counterclockwise order. Return the stack as the result.
This algorithm is known as the Graham's scan algorithm and it computes the convex hull of a set of line segments in O(n log n) time complexity, where n is the number of line segments.