Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor

Voronoi diagrams—a survey of a fundamental geometric data structure

From the Publisher:
Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization.
This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and
high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces.
Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third book.

Lectures on Discrete Geometry

An $(n,s)$ Davenport--Schinzel sequence, for positive integers $n$ and $s$, is a sequence composed of $n$ symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation $a \cdots b \cdots a \cdots b \cdots$ of length $s+2$ between two distinct symbols $a$ and $b$. The close relationship between Davenport--Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive, because a wide variety of geometric problems can be formulated in terms of lower envelopes. A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems. This paper gives a comprehensive survey on the theory of Davenport--Schinzel sequences and their geometric applications.

http://www.math.tau.ac.il/~michas/dssurvey.pdf

Davenport-Schinzel sequences and their geometric applications

Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

Keywords: Hamiltonian cycle ; algorithm ; planar embedding ; polygonal curves Note: Professor Pach's number: [137]. Also in: Graph Drawing '98 (Sue Whitesides, ed.), Lecture Notes in Computer Science 1547, Springer-Verlag, Berlin, 1998, 263-274. Reference DCG-ARTICLE-2001-010doi:10.1007/PL00007258 Record created on 2008-11-14, modified on 2017-05-12

Embedding planar graphs at fixed vertex locations

Geometric transversal theory has its origins in Helly’s theorem: Theorem 1.1 (Helly’s Theorem) [49]. Suppose A is a family of at least d + 1 convex sets in IR d , and A is finite or each member of A is compact. Then if every d + 1 members of A have a common point, there is a point common to all the members of A.

Geometric Transversal Theory

Helly-type theorems and geometric transversals

This paper addresses the compile-time optimization of a form of nested-loop computation that is motivated by a computational physics application. The computations involve multi-dimensional surface and volume integrals where the integrand is a product of a number of array terms. Besides the issue of optimal distribution of the arrays among the processors, there is also scope for reordering of the operations using the commutativity and associativity properties of addition and multiplication, and the application of the distributive law to significantly reduce the number of operations executed. A formalization of the operation minimization problem and proof of its NP-completeness is provided. A pruning search strategy for determination of an optimal form is developed. An analysis of the communication requirements and a polynomial-time algorithm for determination of optimal distribution of the arrays are also provided.

On Optimizing a Class of Multi-Dimensional Loops with Reduction for Parallel Execution

We present an algorithm for constructing isosurfaces in any dimension. The input to the algorithm is a set of scalar values in a d-dimensional regular grid of (topological) hypercubes. The output is a set of (d-1)-dimensional simplices forming a piecewise linear approximation to the isosurface. The algorithm constructs the isosurface piecewise within each hypercube in the grid using the convex hull of an appropriate set of points. We prove that our algorithm correctly produces a triangulation of a (d-1 )-manifold with boundary. In dimensions three and four, lookup tables with 2/sup 8/ and 2/sup 16/ entries, respectively, can be used to speed the algorithm's running time. In three dimensions, this gives the popular marching cubes algorithm. We discuss applications of four-dimensional isosurface construction to time varying isosurfaces, interval volumes, and morphing.

/pdf/isosurface-construction-in-any-dimension-using-convex-hulls-2ldn6w9g7k.pdf

Rephael Wenger

Papers

Embedding planar graphs at fixed vertex locations

Geometric Transversal Theory

Helly-type theorems and geometric transversals

On Optimizing a Class of Multi-Dimensional Loops with Reduction for Parallel Execution

Isosurface construction in any dimension using convex hulls