Computational Geometry Questions Long
To compute the intersection of a line with a set of polygons in Computational Geometry, we can use the following algorithm:
1. Input: A line segment defined by two points P1(x1, y1) and P2(x2, y2), and a set of polygons represented as a collection of vertices.
2. Initialize an empty list to store the intersection points.
3. For each polygon in the set:
a. Check if the line segment intersects with the bounding box of the polygon. If not, skip to the next polygon.
b. For each edge of the polygon, represented by two consecutive vertices V1(x1, y1) and V2(x2, y2):
i. Compute the intersection point between the line segment and the edge using the line-line intersection formula.
- Let A = (x1, y1), B = (x2, y2), C = (V1x, V1y), and D = (V2x, V2y).
- Compute the determinants: det1 = (x2 - x1)(V1y - y1) - (y2 - y1)(V1x - x1) and det2 = (x2 - x1)(V2y - y1) - (y2 - y1)(V2x - x1).
- Compute the intersection point coordinates: Px = x1 + (x2 - x1) * (det1 / (det1 - det2)) and Py = y1 + (y2 - y1) * (det1 / (det1 - det2)).
ii. Check if the intersection point lies within the range of the line segment and the edge. If it does, add it to the list of intersection points.
4. Return the list of intersection points.
Note: This algorithm assumes that the polygons are simple, non-self-intersecting polygons. If the polygons can be self-intersecting or have holes, additional steps may be required to handle these cases. Additionally, if the polygons are represented using different data structures (e.g., as a list of edges or as a list of vertices with connectivity information), the algorithm may need to be adapted accordingly.