# Point_loc - Point location by Triangle refinement method...

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Point location by Triangle refinement method Triangle refinement method PSLG G Directed acyclic search graph T Triangulated PSLG G q 2 q 3 q 1 Triangulate PSLG G Construct sequence of triangulations and directed acyclic search graph T (pp. 56-58, Preparata Shamos) Queries Preprocessing Search T (p. 58)

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Geometric Search, Point location, Triangle refinement method Triangulating G , part 1 We assume that the PSLG given in the point location problem is a triangulation; if not, it is transformed into one in O ( NlogN ) time using O(N) storage. Chazelle published an O ( N ) triangulation algorithm in 1991 if you want to use that. We futher assume that the triangulation has a triangular boundary. If not, one can be added in O (1) time by adding three vertice and triangulating the “inbetween” region. Triangulation Triangulation
Geometric Search, Point location, Triangle refinement method Hereinafter, we assume: 1. G is a triangulation 2. G has a triangular boundary 3. G has exactly 3 N - 6 edges ( O ( N )) Though this triangulated G is not the PSLG we started with, we will call it G , and say it has N vertices .

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equence of triangulations e have triangulation G with N vertices. Construct a sequence of iangulations S 1 , S 2 , . .., S h ( N ) , where S 1 = G and S i is obtained from i -1 as follows: ) Remove a maximal independent set of non-boundary vertices of S i -1 and their incident edges. ( A set of vertices of a graph (in our case a planar graph)is said to be independent if no two pair of vertices are adjacent in the graph. A maximal set is one such that adding one more vertex will make at least one pair of vertices to become adjacent. How this set is chosen will determine the performance of the algorithm.) ) Re-triangulate the polygons arising from the removal of vertices and edges. h ( N ) , the final triangulation in the sequence, has no internal vertices; t is just one triangle. In the example below, the numbers denote ces; the set of circled vertices denote independent set. 4 15 18 19 16 11 12 8 13 17 14 4 8 10 9 6 7 5 1 3 2 0 S 3 S 1 S 2 S 4
Geometric Search, Point location, Triangle refinement method Notation The notation R j denotes a triangle. A triangle R j may appear in more than one triangulation in the sequence, but is said to belong to triangulation S i if R j was created in step (2) while constructing S i .

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## This note was uploaded on 07/14/2011 for the course COT 5520 taught by Professor Mukherjee during the Summer '11 term at University of Central Florida.

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Point_loc - Point location by Triangle refinement method...

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