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Download Computational Geometry: Algorithms and Applications by Dr. Mark de Berg, Dr. Marc van Kreveld, Prof. Dr. Mark PDF

By Dr. Mark de Berg, Dr. Marc van Kreveld, Prof. Dr. Mark Overmars, Dr. Otfried Schwarzkopf (auth.)

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Extra resources for Computational Geometry: Algorithms and Applications

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This means that we only test any new segment against two segments, namely, the ones immediately left and right of the upper endpoint. Later, when the sweep line has moved downwards to another position, a segment can become adjacent to other segments against which it will be tested. Our new strategy should be reflected in the status of our algorithm: the status now corresponds to the ordered sequence of segments intersecting the sweep line. The new status not only changes at endpoints of segments; it also changes at intersection points, where the order of the intersected segments changes.

We could also call such an algorithm intersection-sensitive, since the number of intersections is what determines the size of the output. How can we avoid testing all pairs of segments for intersection? Here we must make use of the geometry of the situation: segments that are close together are candidates for intersection, unlike segments that are far apart. Below we shall see how we can use this observation to obtain an output-sensitive algorithm for the line segment intersection problem. LetS:= {s1 ,s2 , ...

First of all, we update 'T and Q as in the line segment intersection algorithm. If the event involves only edges from one of the two subdivisions, this is all; the event point is a vertex that can be re-used. If the event involves edges from both subdivisions, we must make local changes to 'D to link the doubly-connected edge lists of the two original subdivisions at the intersection point. This is tedious but not difficult. 5. The edge e must be replaced by two edges denoted e' and e". In the doubly-connected edge list, the two half-edges for e must become four.

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