The Handbook of Discrete and Computational Geometry is intended as a reference book fully accessible to nonspecialists as well as specialists, covering all major aspects of both fields.
The book offers the most important results and methods in discrete and computational geometry to those who use them in their work, both in the academic world—as researchers in mathematics and computer science—and in the professional world—as practitioners in ﬁelds as diverse as operations research, molecular biology, and robotics.
Discrete geometry has contributed signiﬁcantly to the growth of discrete mathematics in recent years. This has been fueled partly by the advent of powerful computers and by the recent explosion of activity in the relatively young ﬁeld of computational geometry. This synthesis between discrete and computational geometry lies at the heart of this Handbook.
A growing list of application fields includes combinatorial optimization, computer-aided design, computer graphics, crystallography, data analysis, error-correcting codes, geographic information systems, motion planning, operations research, pattern recognition, robotics, solid modeling, and tomography.
Table of Contents
Finite point configurations. Packing and covering. Tilings. Helly-type theorems and geometric transversals. Pseudoline arrangements. Oriented matroids.Lattice points and lattice polytopes. Low-distortion embeddings of finite metric spaces. Geometry and topology of polygonal linkages. Geometric graph theory. Euclidean Ramsey theory. Discrete aspects of stochastic geometry. Geometric discrepancy theory and uniform distribution. Polyominoes. Basic properties of convex polytopes. Subdivisions and triangulations of polytopes. Face numbers of polytopes and complexes. Symmetry of polytopes and polyhedra. Polytope skeletons and paths. Polyhedral maps.Topological methods in discrete geometry.Random simplicial complexes. Computational topology of graphs on surfaces. Persistent homology. High-dimensional topological data analysis. Convex hull computations. Voronoi diagrams and Delaunay triangulations. Arrangements.Triangulations and mesh generation. Polygons.Shortest paths and networks. Proximity algorithms. Visibility. Geometric reconstruction problems. Curve and surface reconstruction. Computational convexity. Computational and quantitative real algebraic geometry.Point location. Collision and proximity queries. Range searching. Ray shooting and lines in space. Geometric intersection. Nearest neighbors in high-dimensional spaces.Randomizaton and derandomization. Robust geometric computation. Parallel algorithms in geometry. Epsilon-nets and epsilon-approximations Coresets and sketches. Linear programming. Algorithmic motion planning. Robotics. Computer graphics. Modeling motion. Pattern recognition. Graph drawing.Splines and geometric modeling. Solid modeling. Computation of robust statistics: Depth, median, and related measures. Geographic information systems.Geometric applications of the Grassmann-Cayley algebra. Rigidity of symmetric frameworks. Global rigidity. Crystals, periodic and aperiodic..Applications to structural molecular biology. Geometry and topology of genomics. Software. Two computational geometry libraries: LEDA and CGAL
Csaba D. Toth is a professor of mathematics at Cal State Northridge, located in the city of Los Angeles, and a visiting professor of computer science at Tufts University in the Boston metro area. He is the author of more than 90 papers in discrete and computational geometry. His main research interests are in hierarchical subdivisions in low-dimensional spaces, topological graph theory, and geometric optimization.
Joseph O'Rourke is professor of Computer Science and of Mathematics at Smith College, and director of the Statistical and Data Science program. His research is in computational geometry, developing algorithms for geometric computations. He has authored or coauthored five books besides this Handbook. His most recent book, "How To Fold It," is written for high-school students.
Jacob E. Goodman is, along with Richard Pollack, a Founding Editor of the journal "Discrete & Computational Geometry." He retired from City College, C.U.N.Y. in 2008. The author of over 60 papers in algebraic geometry, combinatorics, and discrete geometry, he has spent most of his time since his retirement composing music and working on a mystery novel. Goodman received the Lester R. Ford Award from the MAA in 1990, and is a Fellow of the AMS.