Computational Geometry Questions Long
To compute the intersection of a line with a polygon in Computational Geometry, we can use the following algorithm:
1. Input: A line defined by two points (P1, P2) and a polygon defined by its vertices.
2. Initialize an empty list called "intersections" to store the intersection points.
3. Iterate through each edge of the polygon:
a. Get the two vertices of the current edge, let's call them A and B.
b. Calculate the direction vector of the line segment AB, let's call it AB_vector.
c. Calculate the normal vector of AB_vector, let's call it AB_normal.
d. Calculate the direction vector of the line segment P1P2, let's call it P1P2_vector.
e. Calculate the dot product between AB_normal and P1P2_vector, let's call it dot_product.
f. If dot_product is zero, it means the line is parallel to the edge, so continue to the next edge.
g. Calculate the vector from P1 to A, let's call it P1A_vector.
h. Calculate the dot product between AB_normal and P1A_vector, let's call it dot_product2.
i. Calculate the parameter t by dividing dot_product2 by dot_product.
j. If t is between 0 and 1, it means the intersection point lies within the line segment AB.
k. Calculate the intersection point by adding t times P1P2_vector to P1.
l. Add the intersection point to the "intersections" list.
4. Return the "intersections" list.
This algorithm works by iterating through each edge of the polygon and checking if the line intersects with it. It calculates the intersection point by finding the parameter t, which represents the position of the intersection point along the line segment P1P2. If t is between 0 and 1, it means the intersection point lies within the line segment AB, and it is added to the "intersections" list. Finally, the algorithm returns the list of intersection points.