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Imagine you are walking on the campus of a university and suddenly you realize you have to make an urgent phone call. There are many public phones on campus and of course you want to go to the nearest one. But wh ich one is the nearest? It would be helpful to have a map on which you could look up the nearest public phone, wherever on campus you are. The map should show a subdivision of the campus into regions, and for each region indicate the nearest public phone. What would these regions look like? And how could we compute them?Even though this is not such a terribly important issue, it describes the basics of a fundamental geometric concept, which plays a role in many applications. The subdivision of the campus is a so-called Voronoi diagram, and it will be studied in Chapter 7 in this book. It can be used to model trading areas of different cities, to guide robots, and even to describe and simulate the growth of crystals. Computing a geometric structure like a Voronoi diagram requires geometric algorithms. Such algorithms form the topic of this book.A second example. Assurne you located the dosest public phone. With a campus map in hand you will probably have little problem in getting to the phone along a reasonably short path, without hitting walls and other objects. But programming a robot to perform the same task is a lot more difficult. Again, the heart of the problem is geometric: given a collection of geometric obstacles, we have to find a short connection between two points, avoiding collisions with the obstac1es. Solving this so-called motion planning problem is of crucial importance in robotics. Chapters 13 and 15 deal with geometric algorithms required for motion planning. A third example. Assume you don't have one map but two: one with a description of the various buildings, induding the public phones, and one indicating the roads on the campus. To plan a motion to the public phone we have to overlay these maps, that is, we have to combine the information in the two maps. Overlaying maps is one of the basic operations of geographic information systems. It involves locating the position of objects from one map in the other, computing the intersection of various features, and so on. Chapter 2 deals with this problem. M. de Berg et al., Computational Geometry © Springer-Verlag Berlin Heidelberg 2000 Chapter 1 COMPUTATIONAL GEOMETRY convex notconvex 2 These are just three examples of geometrie problems requiring earefully designed geometrie algorithms for their solution.In the 1970s the field of emnputational geometry emerged, dealing with such geometrie problems. It ean be defined as the systematie study of algorithms and data struetures for geometrie objeets, with a foeus on exaet algorithms that are asymptotieally fast. Many researehers were attracted by the challenges posed by the geometrie problems.The road from problem formulation to efficient and elegant solutions has often been long, with many diffieult and sub-optimal intermediate results. Today there is a rieh eolleetion of geometrie algorithm...