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

Nonrobustness is a well-known problem in many areas of computational science. Until now, robustness techniques and the construction of robust algorithms have been the province of experts in this field of research. We describe a new C/C++ library (Core) for robust numeric and geometric computation based on the principles of Exact Geometric Computation (EGC). Through our library, for the first time, any programmer can write robust and efficient algorithms. The Core Library is based on a novel numerical core that is powerful enough to support EGC for algebraic problems. This is coupled with a simple delivery mechanism which transparently extends conventional C/C++ programs into robust codes. We are currently addressing efficiency issues in our library: (a) at the compiler and language level, (b) at the level of incorporating EGC techniques, as well as the (c) the system integration of both (a) and (b). Pilot experimental results are described. The basic library is available at http://cs.nyu.edu...

We discuss floating-point filters as a means of restricting the precision needed for arithmetic operations while still computing the exact result. We show that interval techniques can be used to speed up the exact evaluation of geometric predicates and describe an efficient implementation of interval arithmetic that is strongly influenced by the rounding modes of the widely used IEEE 754 standard. Using this approach we engineer an efficient floating-point filter for the computation of the sign of a determinant that works for arbitrary dimensions. We validate our approach experimentally, comparing it with other static, dynamic and semi-static filters. 1

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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

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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

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Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for implementing robust geometric algorithms is to use exact algebraic number types — yet this may lead to a very slow, inefficient program. In this paper we suggest a simple technique for filtering the computations involved in the arrangement construction: when constructing an arrangement vertex, we keep track of the steps that lead to its construction and the equations we need to solve to obtain its coordinates. This construction history can be used for answering predicates very efficiently, compared to a naïve implementation with an exact number type. Furthermore, using this representation most arrangement vertices may be computed approximately at first and can be refined later on in cases of ambiguity. Since such cases are relatively rare, the resulting implementation is both efficient and robust. 1

This paper presents a theoretical and experimental study on two different methods to evaluate the sign of a determinant with integer entries. The first one is a method based on the Gram-Schmidt orthogonalisation process which has been proposed by Clarkson [5]. We review the analysis of Clarkson and propose a variant of his method. The second method is an extension to n \Theta n determinants of the ABDPY method [1] which works only for 2 \Theta 2 and 3 \Theta 3 determinants. Both methods compute the signs of a n \Theta n determinants whose entries are integers on b bits, by using an exact arithmetic on only b + O(n) bits. Furthermore, both methods are adaptive, dealing quickly with easy cases and resorting to the full-length computation only for null determinants.