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

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

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If the two sets of segments form connected sub divisions then the situation is even better: in this c:ase the overlay can be computed in O( n + k) time, as has been shown by Finke and Hinrichs [148]. Their result generalizes and improves previous results on map overlay by Nievergelt and Preparata [261], Guibas and Seidel [172], and Mairson and Stolfi [230]. 5 NOTES AND COMMENTS Plane sweep is one of the most important paradigms for designing geometrie algorithms. The first algorithms in computational geometry based on this paradigm are by Shamos and Hoey [315], Lee and Preparata [219], and Bentley and Ottmann [35].

An operation one could ask for is to determine the face containing a given point. This is definitely useful in some applications-indeed, in a later chapter we shall design a data structure for this-but it is a bit too much to ask from abasie representation. The things we can ask for should be more local. For example, it is reasonable to require that we can walk around the boundary of a given face, or that we can access one face from an adjacent one if we are given a common edge. Another operation that could be useful is to visit all the edges around a given vertex.

I g ······················l· - interseetion detected . g 23 Chapter2 UNE SEGMENT INTERSECTION r'p''''''' \; sk After this sketch of the algorithm, it's time to go into more detail. It's also time to look at the degenerate cases that can arise, like three or more segments meeting in a point. We should first specify what we expect from the algorithm in these cases. We could require the algorithm to simply report each intersection point once, but it seems more useful if it reports for each intersection point a list of segments that pass through it or have it as an endpoint.

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