4. Concluding Remarks

A new linear algorithm for intersecting convex polygons. Comput. An optimal algorithm for computing the minimum vertex distance between two crossing convex polygons. Computing 32:357-364 13. Toussaint GT, Avis D (1982) On a convex hull algorithm for polygons and its application to triangulation problems. A new O(n log n) algorithm for computing the intersection of convex polygons. Pattern Recognition. 20:419-424-8-backtracking and at the same time can answer the question of whether CH(P ∪ Q) = P or Q. If there are k bridges on CH(P ∪ Q) then STEPDOWN is called k times in step 2. Each call requires time linear in the number of vertices processed and the total number of these vertices is the sum total of the vertices on all the outer chains of P and Q. Thus step 2 runs in O(m+n) time. Finally, if we leave pointers from the intersection points to the inner and outer chains in both directions, as we find them in step 2, then the merge step of the inner chains in step 3 can be done in linear time by a mere traversal of the two lists for P and Q. As a final remark we mention that the " ear-cutting " triangulation algorithm for sail polygons presented in section two can be applied to the problem of triangulating a set of n points on the plane in O(n log n) time via divide-and-conquer. Here, if the points have been presorted, at each step we must merge two triangulations T 1 and T 2 which are linearly separable triangulated convex polygons (Fig. 5.) The merge step consists of triangulating the hourglass polygon " in between " T 1 and T 2. This region lies outside T 1 and T 2 but inside CH(T 1 ∪ T 2). An hourglass polygon is a polygon consisting of two edges called the top (bridge p i , p j) and the bottom (bridge p k , p l) such that p i and p l (as well as p k , p j) are joined by concave chains and (p i , p j , p k , p l) forms a convex quadrilateral. Now consider a critical line of support between T 1 and T 2 at p u and p v. This line decomposes the hourglass polygon into two sail polygons and. Finding the bridges and the edge p u p …