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# Lec17 - Proximity Greedy triangulation star of spokes Star...

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“Star of spokes” greedy triangulation The simple greedy triangulation had two phases: 1. Generation and presort of possible edges. O ( N 2 log N ) time. 2. Determining if each possible edge belongs in the triangulation. O ( N 3 ) time. The “star of spokes” algorithm reduces the second phase to O ( N 2 log N ) overall by “processing” each of the O ( N 2 ) possible edges in O (log N ) time. Here “processing” means to test a possible edge for intersection with edges already in the triangulation. We will see a way to perform that test in O (log N ) time. That time suggests a search data structure of some kind. Proximity Greedy triangulation, star of spokes

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Star of spokes Suppose the current possible edge is edge p 1 p i . Consider a set of edges connecting p 1 to every other point in S . These edges, called spokes , are not in the triangulation, there are defined for the search structure. The point p 1 and the spokes are together referred to as STAR( p 1 ), the “star of spokes” of p 1 . The spokes are ordered, say counterclockwise, and they subdivide the 2 π angle around p 1 into N - 1 angular intervals, called sectors . Proximity Greedy triangulation, star of spokes p 1 p i
Triangulation edges and sectors Suppose edge p j p k is already part of the triangulation, i.e., a triangulation edge. It spans a set of consecutive sectors relative to p 1 , and the same is true for any p l S - { p j , p k }. Observe that because triangulation edges run from point to point, and points define sectors, edges always entirely span sectors, rather than partially spanning them. Note also that triangulation edges do not intersect each other (by definition), so the triangulation edges present in each sector of STAR( p l ) p l S can be ordered from p l outwards. Proximity Greedy triangulation, star of spokes p 1 p i p k p j

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What to check Recall that edge p 1 p i is the possible triangulation edge that is being tested for intersection with other triangulation edges. Assume that in STAR( p 1 ) point p i is located in sector p k p 1 p j . To determine whether edge p 1 p i can be added to the triangulation, it must be determined if edge p 1 p i intersects any other triangulation edge. It is sufficient to check those triangulation edges that span sector p k p 1 p j , and if there are > 1 such edges, it is sufficient to check the one closest to p 1 in that sector. Conceptually, if p i is closer to p 1 than the closest spanning edge, edge p 1 p i can be added; if not, it cannot. For each sector of STAR( p l ) for every p l S , the closest triangulation edge must be maintained. Preparata calls this “a special case of planar point location”. Proximity Greedy triangulation, star of spokes p 1 p i p k p j
Example situation The figure shows STAR( p 1 ) in some typical situation.

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Lec17 - Proximity Greedy triangulation star of spokes Star...

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