By Mark de Berg, Otfried Cheong, Marc van Kreveld, Mark Overmars
Computational geometry emerged from the sector of algorithms layout and research within the overdue Nineteen Seventies. It has grown right into a famous self-discipline with its personal journals, meetings, and a wide neighborhood of lively researchers. The good fortune of the sphere as a learn self-discipline can at the one hand be defined from the wonderful thing about the issues studied and the ideas received, and, however, by means of the various program domains---computer snap shots, geographic info structures (GIS), robotics, and others---in which geometric algorithms play a primary role.
For many geometric difficulties the early algorithmic recommendations have been both gradual or obscure and enforce. lately a few new algorithmic concepts were built that more advantageous and simplified a number of the earlier techniques. during this textbook now we have attempted to make those smooth algorithmic suggestions available to a wide viewers. The e-book has been written as a textbook for a path in computational geometry, however it is additionally used for self research.
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Additional info for Computational Geometry: Algorithms and Applications (3rd Edition)
The algorithm presented in the current chapter sweeps a horizontal line downwards over the plane. For some problems it is more convenient to sweep the plane in another way. For instance, we can sweep the plane with a rotating line—see Chapter 15 for an example—or with a pseudo-line (a line that need not be straight, but otherwise behaves more or less as a line) . The plane sweep technique can also be used in higher dimensions: here we sweep the space with a hyperplane [213, 311, 324]. Such algorithms are called space sweep algorithms.
The next lemma states that this is indeed the case. 2 Algorithm F IND I NTERSECTIONS computes all intersection points and the segments that contain it correctly. Proof. Recall that the priority of an event is given by its y-coordinate, and that when two events have the same y-coordinate the one with smaller x-coordinate is given higher priority. We shall prove the lemma by induction on the priority of the event points. Let p be an intersection point and assume that all intersection points q with a higher priority have been computed correctly.
Of course its size is always bounded by O(n + I), but it would be better if the working storage were always linear. There is a relatively simple way to achieve this: only store intersection points of pairs of segments that are currently adjacent on the sweep line. The algorithm given above also stores intersection points of segments that have been horizontally adjacent, but aren’t anymore. By storing only intersections among adjacent segments, the number of event points in Q is never more than linear.