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

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

Computational geometry emerged from the ?eld of algorithms layout and research within the overdue Seventies. It has grown right into a famous self-discipline with its personal journals, meetings, and a wide group of energetic researchers. The good fortune of the ?eld as a examine self-discipline can at the one hand be defined from the great thing about the issues studied and the ideas received, and, nonetheless, via the numerous program domains—computer snap shots, geographic details structures (GIS), robotics, and others—in which geometric algorithms play a basic position. for lots of geometric difficulties the early algorithmic suggestions have been both gradual or dif?cult to appreciate and enforce. lately a few new algorithmic strategies were constructed that more suitable and simpli?ed the various past ways. during this textbook now we have attempted to make those sleek algorithmic recommendations obtainable to a wide viewers. The ebook has been written as a textbook for a path in computational geometry, however it can be used for self-study.

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The overlay of S1 and S2 can be constructed in O(n log n + k log n) time, where k is the complexity of the overlay. Proof. 3. Steps 4– 7, where we fill in the face records, takes time linear in the complexity of O(S1 , S2 ). ) Finally, labeling each face in the resulting subdivision with the faces of the original subdivisions that contain it can be done in O(n log n + k log n) time. 4 Boolean Operations The map overlay algorithm is a powerful instrument that can be used for various other applications.

If a vertex is the endpoint of an edge, then we say that the vertex and the edge are incident. Similarly, a face and an edge on its boundary are incident, and a face and a vertex of its boundary are incident. What should we require from a representation of a subdivision? 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 a basic representation.

Suppose that four segments s j , sk , sl , and sm appear in this order on the sweep line when the intersection point of sk and sl is reached. Then sk and sl switch position and we must test sl and s j for intersection below the sweep line, and also sk and sm . The new intersections that we find are, of course, also event points for the algorithm. Note, however, that it is possible that these events have already been detected earlier, namely if a pair becoming adjacent has been adjacent before. si sj sk intersection detected s j sk sl sm 23 Chapter 2 LINE SEGMENT INTERSECTION sk sl sm When the event point is the lower endpoint of a segment, its two neighbors now become adjacent and must be tested for intersection.

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