Browse Books

Reviewer: Robert Goldberg

This is an applied approach to fundamental concepts in computational geometry and should be read by every serious practitioner. It is an expanded version (by about 50 pages) of the first edition [1] and a sequel to the author's earlier work on “art gallery” theorems [2]. This edition contains many new figures and exercises, and about 80 additional references. The author not only proves theorems elegantly, but also provides algorithms and their implementation in C, in order to concretize the abstract concepts by allowing readers to experience “live” demonstrations by computer. (Both C and Java versions are provided at the Web site http://cs.smith.edu/~orourke.) The book is divided into nine chapters. The first two deal with the polygonal domain and discuss triangulation and sectioning (partitioning). The next four chapters deal with the three main structures used in computational geometry—the convex hull (two-dimensional and three-dimensional, sequential and randomized implementations, and higher dimensionality); Voronoi diagrams; and arrangements (which also covers higher-order dimensionality, including for Voronoi diagrams). The next two chapters present the two main applications in the field: searching (based on intersections) and motion planning (building from shortest path problems to convex polygons to robotic arm motion.) The last chapter collects the resources available for both applied and theoretical researchers and includes references to the relevant literature at all levels (books, journals, and conference proceedings) and software. Though most of the algorithms included in the book are accompanied by a C implementation, implementations are not provided for every algorithm, to avoid burdening readers with programming details that would disrupt the flow of the text. The 12 algorithms for which full implementations are provided were chosen either because of their brevity or because the implementations help in understanding the algorithms. For objects described in two dimensions, full code is provided for the area of a polygon and for triangulation of a polygon (a simple version in section 1.4.3 and a more complex Delaunay in chapter 5.) More sophisticated algorithms are provided for point in polygon testing, intersection of a segment with a segment (the code can easily be extended to allow the segment to be described in three-dimensional space), intersection of convex polygons, and Minkowski convolution with a convex polygon. For objects described in three dimensions, full code is provided for point in polyhedron testing; the convex hull in three dimensions; intersection of a segment and a triangle; and multilink robot arm reachability (section 8.6.3), the most sophisticated application in the book. At the Web site, randomized input routines are provided for generating input test data. These are the generation of random points in a cube, the generation of random points in a sphere, and the uniform distribution of points on a sphere. The material is presented in a pedagogical fashion, which includes incremental development of a number of the algorithms and a broader presentation of the historical development of the current algorithms. It includes both theoretical and practical considerations, with further thought encouraged in the exercises. For example, chapter 3 presents convex hull algorithms in two dimensions. It begins with a context—the applications for which the convex hull is useful: collision avoidance, fitting ranges with a line, smallest enclosing box, and shape analysis (which is extensively explored in exercise 3.9.3 [2] and figures 3.15a–c). Then, section 3.1 compactly provides the necessary definitions and terminology for understanding the chapter. Based on some straightforward lemmas, section 3.2 constructs naive algorithms for solving the problem. Section 3.3 explores an earlier approach to the problem based on “gift wrapping” and examines some of the algorithmic details in order to reduce the complexity of the algorithm. A popular algorithm in the 1970s was the quickhull algorithm (section 3.4), which was based on the popular quicksort sorting routine. Graham presented an “optimal” O n log n algorithm to construct the convex hull (section 3.5) in the early 1970s, but this and the algorithms presented earlier in the chapter do not readily extend to three dimensions. Therefore, the final sections of the chapter present two more approaches: the incremental algorithm (section 3.7) and the divide-and-conquer approach (section 3.8), both of which have direct three-dimensional correlates. This chapter, like the others, presents the material in a fashion that both researchers looking for theorems and practitioners looking for code and implementation details will appreciate. From a pedagogical point of view, this book is an excellent choice for both undergraduate classes (perhaps with more emphasis on the implementations) and graduate classes (considering a number of the exercises) because of the extensive exercises that review, explore details, and encourage further reading. In fact, the author is completing a more extensive solutions manual for the book. In the introduction, the author expresses two of his goals for this book—that he has “chosen a representative set of algorithms” and that “room is left for student projects.” He has been quite successful in accomplishing these.