By Mark de Berg, Otfried Cheong, Marc van Kreveld, Mark Overmars
Computational geometry emerged from the sector 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 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 suggestions received, and, nonetheless, via the numerous software domains---computer portraits, geographic details structures (GIS), robotics, and others---in which geometric algorithms play a primary role.
For many geometric difficulties the early algorithmic ideas have been both sluggish or obscure and enforce. in recent times a couple of new algorithmic suggestions were built that superior and simplified a number of the earlier techniques. during this textbook now we have attempted to make those sleek algorithmic ideas obtainable to a wide viewers. The ebook has been written as a textbook for a direction in computational geometry, however it can be used for self examine.
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Additional info for Computational Geometry: Algorithms and Applications (3rd Edition)
Actually, we certainly will, because in our application the segments from one set form a number of chains, and we count coinciding endpoints as intersections. These other intersections can be ﬁltered out afterwards by simply checking for each reported intersection whether the two segments involved belong to the same set. So our problem speciﬁcation is as follows: given a set S of n closed segments in the plane, report all intersection points among the segments in S. This doesn’t seem like a challenging problem: we can simply take each pair of segments, compute whether they intersect, and, if so, report their intersection point.
Si sj p So our approach is correct, at least when we forget about the nasty cases mentioned earlier. Now we can proceed with the development of the plane sweep algorithm. Let’s brieﬂy recap the overall approach. We imagine moving a horizontal sweep line downwards over the plane. The sweep line halts at certain event points; in our case these are the endpoints of the segments, which we know beforehand, and the intersection points, which are computed on the ﬂy. While the sweep line moves we maintain the ordered sequence of segments intersected by it.
We locate these four half-edges from S2 by testing where e and e should be in the cyclic order of the edges around vertex v. There are four pairs of half-edges that become linked by a Next() pointer from the one and a Prev() pointer from the other. Consider the half-edge for e that has v as its destination. It must be linked to the ﬁrst half-edge, seen clockwise from e , with v as its origin. The half-edge for e with v as its origin must be linked to the ﬁrst counterclockwise half-edge with v as its destination.