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 past due Seventies. It has grown right into a well-known self-discipline with its personal journals, meetings, and a wide group of energetic researchers. The luck of the ?eld as a study self-discipline can at the one hand be defined from the wonderful thing about the issues studied and the strategies acquired, and, however, via the various program domains—computer snap shots, geographic details structures (GIS), robotics, and others—in which geometric algorithms play a primary position. for lots of geometric difficulties the early algorithmic suggestions have been both gradual or dif?cult to appreciate and enforce. lately a couple of new algorithmic thoughts were built that more advantageous and simpli?ed a number of the earlier methods. during this textbook we now have attempted to make those sleek algorithmic options available to a wide viewers. The booklet has been written as a textbook for a path in computational geometry, however it can be used for self-study.
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Additional info for Computational Geometry: Algorithms and Applications
Such an algorithm is called an output-sensitive algorithm: the running time of the algorithm is sensitive to the size of the output. 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.
The following lemma states an even stronger result: the running time is O((n + I) log n), where I is the number of intersections. This is stronger, because for one intersection point the output can consist of a large number of segments, namely in the case where many segments intersect in a common point. 3 The running time of Algorithm F IND I NTERSECTIONS for a set S of n line segments in the plane is O(n log n + I log n), where I is the number of intersection points of segments in S. 28 Proof.
Because we know that the incident face lies to the left, we can compute the angle these two half-edges make inside the incident face. If this angle is smaller than 180◦ then the cycle is an outer boundary, and otherwise it is the boundary of a hole. This property holds for the leftmost vertex of a cycle, but not necessarily for other vertices of that cycle. To decide which boundary cycles bound the same face we construct a graph G. For every boundary cycle—inner and outer—there is a node in G. There is also one node for the imaginary outer boundary of the unbounded face.
Computational Geometry: Algorithms and Applications by Prof. Dr. Mark de Berg, Dr. Otfried Cheong, Dr. Marc van Kreveld, Prof. Dr. Mark Overmars (auth.)