Computational Geometry Questions Long
Computing the intersection of two polygons with curved and straight boundaries and holes in Computational Geometry can be achieved using the following algorithm:
1. Preprocessing:
a. Convert the curved boundaries of the polygons into a series of straight line segments using an approximation technique such as polygonal approximation or Bézier curve approximation.
b. Identify and mark the holes within each polygon. Holes can be represented as separate polygons with opposite winding order compared to the outer boundary.
2. Decomposition:
a. Decompose each polygon into a set of simple polygons by splitting along the holes. This can be done using the polygon decomposition algorithm, such as the ear-clipping algorithm or the trapezoidal decomposition algorithm.
3. Intersection computation:
a. For each pair of simple polygons (one from each original polygon), compute their intersection using the algorithm for intersecting two simple polygons. This can be achieved by applying the sweep line algorithm or the line sweep algorithm.
b. Repeat step 3a for all pairs of simple polygons.
4. Merge intersections:
a. Merge the computed intersections of the simple polygons to obtain the final intersection of the original polygons. This can be done by applying the Boolean operations (union, intersection, difference) on the computed intersections.
5. Handle holes:
a. Identify the holes within the computed intersection by checking the winding order of the polygons. Holes will have a different winding order compared to the outer boundary.
b. Remove the holes from the computed intersection by applying the Boolean difference operation between the intersection and the holes.
6. Output:
a. The final result is the intersection of the two polygons with curved and straight boundaries, excluding any holes.
It is important to note that the complexity of this algorithm depends on the number of vertices and the complexity of the curved boundaries. The approximation technique used in step 1 can affect the accuracy of the result, so choosing an appropriate technique is crucial. Additionally, the algorithm assumes that the polygons are planar and non-self-intersecting.