We give an overview of the LEDA platform for combinatorial and geometric computing and an account of its development. We discuss our motivation for building LEDA and to what extent we have reached our goals. We also discuss some recent theoretical developments. This paper contains no new technical material. It is intended as a guide to existing publications about the system. We refer the reader also to our web-pages for more information.

Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3-space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.

We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixed-precision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.-Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original mass-produced computers were pocket calculators. Although one's first exposure to computers today is likely to be some non-numerical application, numeri...

CGAL is a Computational Geometry Algorithms Library written in C++, which is being developed by research groups in Europe and Israel. The goal is to make the large body of geometric algorithms developed in the field of computational geometry available for industrial application. We discuss the major design goals for CGAL, which are correctness, flexibility, ease-of-use, efficiency, and robustness, and present our approach to reach these goals. Generic programming using templates in C++ plays a central role in the architecture of CGAL. We give a short introduction to generic programming in C++, compare it to the object-oriented programming paradigm, and present examples where both paradigms are used effectively in CGAL. Moreover, we give an overview of the current structure of the CGAL-library and consider software engineering aspects in the CGAL-project. Copyright c ○ 1999 John Wiley & Sons, Ltd. KEY WORDS: computational geometry; software library; C++; generic programming;

Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...

Geometric algorithms are usually described assuming that arithmetic operations are performed exactly on real numbers. A program implemented using a naive substitution of floating-point arithmetic for real arithmetic can fail, since geometric primitives depend upon sign-evaluation and may not be reliable if evaluated approximately. Geometric primitives are reliable if evaluated exactly with integer arithmetic, but this degrades performance since software extended-precision arithmetic is required. We describe static-analysis techniques that reduce the performance cost of exact integer arithmetic used to implement geometric algorithms. We have used the techniques for a number of examples, including line-segment intersection in two dimensions, Delaunay triangulations, and a three-dimensional boundary-based polyhedral modeller. In general, the techniques are appropriate for algorithms that use primitives of relatively low algebraic total degree, e.g., those involving flat objects (...

real expressions with arbitrary precision. Figure 1 shows (part of) the LEDA manual page for reals. reals provide exact computation in a convenient way. In an implementation of a geometric algorithm in C++, reals can be used like doubles. The following example Max-Planck-Institut fur Informatik, Im Stadtwald, 66123 Saarbrucken, Germany. Supported by the ESPRIT Basic Research Actions Program, under contract No. 7141 (project ALCOM II). y Fachbereich 14, Informatik, Universitat des Saarlandes, 66041 Saarbrucken, Germany. z Martin-Luther-Universitat Halle, Fachbereich Mathematik und Informatik, 06099 Halle, Germany. 0 arises in the computation of Voronoi diagrams of line segments [2]. For i, 1 i 3, let l i : a i x + b i

Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, k-th root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms. We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda real. 1

The assumption of real-number arithmetic, which is at the basis of conventional geometric algorithms, has been seriously challenged in recent years, since digital computers do not exhibit such capability. A geometric predicate usually consists of evaluating the sign of some algebraic expression. In most cases, rounded computations yield a reliable result, but sometimes rounded arithmetic introduces errors which may invalidate the algorithms. The rounded arithmetic may produce an incorrect result only if the exact absolute value of the algebraic expression is smaller than some (small) ", which represents the largest error that may arise in the evaluation of the expression. The threshold " depends on the structure of the expression and on the adopted computer arithmetic, assuming that the input operands are error-free. A pair (arithmetic engine,threshold) is an arithmetic filter. In this paper we develop a general technique for assessing the efficacy of an arithmetic filter. The analysis cons...

In this paper, we give an overview of the theory of residues and its applications. We adopt the algebraic point of view and show how this theory can lead to effective computations. For this purpose, we analyze the properties of Bezoutians, objects which underly many problems related to polynomial systems. We recall the main properties of a Gorenstein Algebra, where we can naturally speak of residues. We show how we can construct a residue, in the case of complete intersection, and we give the algebraic counterparts of some theorems of the analytical theory of residues. We also give some direct applications of this approach to the computations of Chow Forms and show how the information concerning multiple roots can be recovered from the local residues in a very simple way. Finally, we propose a new method for solving zero-dimensional systems of n equations P1 = \Delta \Delta \Delta = Pn = 0 in n variables x1 , ..., xn . This method involves elementary manipulations on the matrices...