Handbook of Discrete and Computational Geometry

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.

"Discrete and computational geometry is a thriving field, and the collective knowledge represented in this third edition of the Handbook is evidence of this. Twenty years have passed since the publication of the first edition [Handbook of discrete and compu-tational geometry, CRC Press Ser. Discrete Math. Appl., CRC, Boca Raton, FL, 1997; MR1730156], and since then, the area has seen a tremendous growth due in part to the influence of both the first and second editions [Handbook of discrete and computational geometry, Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2004; MR2082993] of the Handbook. The new edition can be expected to serve a similar purpose for future generations of researchers and practitioners.The third edition has a new co-editor, Csaba D. T´oth, who joins the two editors of the previous editions, Jacob E. Goodman and Joseph O’Rourke.The third edition of the Handbook has been substantially revised. The addition of almost 400 pages has resulted in a volume of well over 1900 pages. Fifty-eight out of the sixty-five chapters of the second edition have been updated (sometimes with a slightly revised title), many of them by the original authors, others in collaboration with new co-authors. Ten new chapters have been added, for an overall count of sixty-eight chapters.Five new chapters on computational topology and its applications have been added, including four in a new section on “Combinatorial and Computational Topology”. These deal with random simplicial complexes, computational topology of graphs on surfaces, persistent homology, high-dimensional topological data analysis, and geometry and topology of genomic, respectively. A new chapter on proximity algorithms, added to the section on “Algorithms and Complexity of Fundamental Geometric Objects”, provides a comprehensive treatment of relative neighborhood graphs and geometric spanners. There also are two new chapters in the section on “Computational Techniques”: on epsilon-approximations and epsilon-nets and on coresets and sketches, respectively. Finally, two more new chapters in the section on “Applications of Discrete and Compu-tational Geometry” describe recent advances in rigidity theory and concern rigidity of symmetric frameworks and global rigidity, respectively.The new version remains an invaluable resource for anyone interested in discrete and computational geometry and in related areas, including specialists and non-specialists. The Handbook continues to be an attractive one-stop reference book."- Egon Schulte - Mathematical Reviews Clippings - May 2019

• 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