During the past few decades, the gradual merger of Discrete Geometry and the newer discipline of Computational Geometry has provided enormous impetus to mathematicians and computer scientists interested in geometric problems. This volume, which contains 32 papers on a broad range of topics of current interest in the field, is an outgrowth of that synergism. It includes surveys and research articles exploring geometric arrangements, polytopes, packing, covering, discrete convexity, geometric algorithms and their complexity, and the combinatorial complexity of geometric objects, particularly in low dimension. There are points of contact with many applied areas such as mathematical programming, visibility problems, kinetic data structures, and biochemistry, as well as with algebraic topology, geometric probability, real algebraic geometry, and combinatorics.

This commemorative book contains the 28 major articles that appeared in the 2008 Twentieth Anniversary Issue of the journal Discrete & Computational Geometry, and presents a comprehensive picture of the current state of the field. Formed during the past few decades by the merger of the classical discipline of combinatorial and discrete geometry with the new field of computational geometry that sprang up in the 1970s, discrete and computational geometry now claims the allegiance of a sizeable number of mathematicians and computer scientists all over the world, whose most important work has been appearing since 1986 in the pages of the journal.

The articles in this volume, a number of which solve long-outstanding problems in the field, were chosen by the editors of DCG for the importance of their results, for the breadth of their scope, and to show the intimate connections that have arisen between discrete and computational geometry and other areas of both computer science and mathematics. Apart from the articles, the editors present an expanded preface, along with a set of photographs of groups and individuals who have played a major role in the history of the field during the past twenty years.

Contributors include:

E. Ackerman

P.K. Agarwal

I. Aliev

I. BArAny

A. Barvinok

S. Basu

L.J. Billera

J.-D. Boissonnat

C. Borcea

E. Boros

K. Borys

B. Braun

K. Buchin

O. Cheong

D. Cohen-Steiner

M. Damian

K. Elbassioni

R. Flatland

T. Gerken

J.E. Goodman

X. Goaoc

P. Gronchi

V. Gurvich

S. Har-Peled

J. Hershberger

A. Holmsen

S.K. Hsiao

A. Hubard

J. JerA3nimo

L. Khachiyan

R. Klein

C. Knauer

S.Langerman

J.-Y. Lee

M. Longinetti

E. Miller

P. Morin

U. Nagel

E. Nevo

P. Niyogi

I. Novik

J. Oa (TM)Rourke

J. Pach

I. Pak

M.J. Pelsmajer

S. Petitjean

F. Pfender

R. Pinchasi

R. Pollack

J.S. Provan

K. Przeslawski

R.M. Richardson

G. Rote

M. Schaefer

Y. Schreiber

M. Sharir

J.R. Shewchuk

S. Smale

B. Solomyak

M. Soss

D. A tefankovic

G. Vegter

V.H. Vu

S. Weinberger

L. Wu

D. Yost

H. Yu

T. Zell

During the past few decades, the gradual merger of Discrete Geometry and the newer discipline of Computational Geometry has provided enormous impetus to mathematicians and computer scientists interested in geometric problems. This 2005 volume, which contains 32 papers on a broad range of topics of interest in the field, is an outgrowth of that synergism. It includes surveys and research articles exploring geometric arrangements, polytopes, packing, covering, discrete convexity, geometric algorithms and their complexity, and the combinatorial complexity of geometric objects, particularly in low dimension. There are points of contact with many applied areas such as mathematical programming, visibility problems, kinetic data structures, and biochemistry, as well as with algebraic topology, geometric probability, real algebraic geometry, and combinatorics.

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.