The.Algorithm.Design.Manual.Springer-Verlag.1998
The.Algorithm.Design.Manual.Springer-Verlag.1998 The.Algorithm.Design.Manual.Springer-Verlag.1998
Maintaining Line Arrangements are discussed in some detail in Section . The basic procedure is to sort the intersection points by xcoordinate and then walk from left to right while keeping track of all we have seen. Implementations: Arrange is a package written in C by Michael Goldwasser for maintaining arrangements of polygons in either the plane or on the sphere. Polygons may be degenerate and hence represent arrangements of lines. A randomized incremental construction algorithm is used and efficient point location on the arrangement supported. Polygons may be inserted but not deleted from the arrangement, and arrangements of several thousand vertices and edges can be constructed in a few seconds. Arrange is available from http://theory.stanford.edu/people/wass/wass.html. LEDA (see Section ) provides a function that constructs an embedded planar graph from a set of line segments, essentially constructing their arrangement. Notes: Edelsbrunner [Ede87] provides a comprehensive treatment of the combinatorial theory of arrangements, plus algorithms on arrangements with applications. It is an essential reference for anyone seriously interested in the subject. Good expositions on constructing arrangements include [O'R94]. Arrangements generalize naturally beyond two dimensions. Instead of lines, the space decomposition is defined by planes (or beyond 3-dimensions, hyperplanes). In general dimensions, the zone theorem states that any arrangement of n d-dimensional hyperplanes has total complexity , and any single hyperplane intersects cells of complexity . This provides the justification for the incremental construction algorithm for arrangements. Walking around the boundary of each cell to find the next cell that the hyperplane intersects takes time proportional to the number of cells created by inserting the hyperplane. The history of the zone theorem has become somewhat muddled, because the original proofs were later found to be in error in higher dimensions. See [ESS93] for a discussion and a correct proof. The theory of Davenport-Schintzl sequences is intimately tied into the study of arrangements. It is presented in [SA95]. The naive algorithm for sweeping an arrangement of lines sorts the intersection points by x-coordinate and hence requires time. The topological sweep [EG89, EG91] eliminates the need to sort, and it traverses the arrangement in quadratic time. This algorithm is readily implementable and can be applied to speed up many sweepline algorithms. Related Problems: Intersection detection (see page ), point location (see page ). file:///E|/BOOK/BOOK5/NODE198.HTM (3 of 4) [19/1/2003 1:32:00]
Maintaining Line Arrangements Next: Minkowski Sum Up: Computational Geometry Previous: Motion Planning Algorithms Mon Jun 2 23:33:50 EDT 1997 file:///E|/BOOK/BOOK5/NODE198.HTM (4 of 4) [19/1/2003 1:32:00]
- Page 561 and 562: Convex Hull ● How many dimensions
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- Page 571 and 572: Voronoi Diagrams McDonald's, the ti
- Page 573 and 574: Nearest Neighbor Search Next: Range
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- Page 579 and 580: Range Search subdivisions in C++. I
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- Page 583 and 584: Point Location More recently, there
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- Page 589 and 590: Bin Packing Next: Medial-Axis Trans
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- Page 617 and 618: Set and String Problems Next: Set C
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Maintaining Line Arrangements<br />
Next: Minkowski Sum Up: Computational Geometry Previous: Motion Planning<br />
<strong>Algorithm</strong>s<br />
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