The goal of this and subsequent chapters is to introduce the algorithmic methods that are used most frequently to solve geometric problems. Generally speaking, computational geometry has recourse to all of the classical algorithmic techniques. Readers examining all the algorithms described in this book from a methodological point of view will distinguish essentially three methods: the incremental method, the divide-and-conquer method, and the sweep method.

The incremental method is perhaps the method which is the most largely emphasized in the book. It is also the most natural method, since it consists of processing the input to the problem one item at a time. The algorithm initiates the process by solving the problem for a small subset of the input, then maintains the solution to the problem as the remaining data are inserted one by one. In some cases, the algorithm may initially sort the input, in order to take advantage of the fact that the data are sorted. In other cases, the order in which the data are processed is indifferent, sometimes even deliberately random. In the latter case, we are dealing with the randomized incremental method, which will be stated and analyzed at length in chapter 5. We therefore will not expand further on the incremental method in this chapter.

The divide-and-conquer method is one of the oldest methods for the design of algorithms, and its use goes well beyond geometry. In computational geometry, this method leads to very efficient algorithms for certain problems.