Interval arithmetic yields efficient dynamic filters for computational geometry

Brönnimann, Hervé; Burnikel, Christoph; Pion, Sylvain

doi:10.1016/s0166-218x(00)00231-6

Cited by 68 publications

(50 citation statements)

References 18 publications

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“…Remark 1. The exact determination of the sign of a sum by Algorithm 4.5 is critical in the evaluation of geometrical predicates [20,4,45,9,27,8,14]. Rewriting a dot product as a sum by splitting products in two parts (Dekker's and Veltkamp's [12] algorithms Split and TwoProduct, see also [37]), we can determine the exact sign of a dot product as well, which in turn decides whether a point is exactly on some plane, or on which side it is.…”

Section: Then Res Is a Faithful Rounding Of Smentioning

confidence: 99%

“…For example, it allows accurate calculation of the residual, the key to the accurate solution of linear systems. Or it allows to compute sign(s) with rigor, a significant problem in the computation of geometrical predicates [10,20,45,9,27,8,14,38], where the sign of the value of a dot product decides whether a point is exactly on a plane or on which side it is.…”

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confidence: 99%

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Accurate Floating-Point Summation Part I: Faithful Rounding

Rump¹,

Ogita²,

Oishi³

2008

SIAM J. Sci. Comput.

162

199

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Abstract. Given a vector of floating-point numbers with exact sum s, we present an algorithm for calculating a faithful rounding of s, i.e. the result is one of the immediate floating-point neighbors of s. If the sum s is a floating-point number, we prove that this is the result of our algorithm. The algorithm adapts to the condition number of the sum, i.e. it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number. All statements are also true in the presence of underflow. The algorithm does not depend on the exponent range. Our algorithm is fast in terms of measured computing time because it allows good instruction-level parallelism, it neither requires special operations such as access to mantissa or exponent, it contains no branch in the inner loop, nor does it require some extra precision: The only operations used are standard floating-point addition, subtraction and multiplication in one working precision, for example double precision. Certain constants used in the algorithm are proved to be optimal.

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Section: Then Res Is a Faithful Rounding Of Smentioning

confidence: 99%

mentioning

confidence: 99%

Accurate Floating-Point Summation Part I: Faithful Rounding

Rump¹,

Ogita²,

Oishi³

2008

SIAM J. Sci. Comput.

162

199

View full text Add to dashboard Cite

show abstract

“…But there is a more efficient way to achieve robustness, which is called arithmetic filtering: The inexact kernel Simple Cartesian<double> is plugged into a Filtered kernel [Fabri and S.Pion 2006], which uses interval arithmetic [Brönnimann et al 2001] to detect if a predicate, evaluated with the inexact number type, is possibly incorrect.…”

Section: Using Filtered Exact Arithmeticmentioning

confidence: 99%

Industrial application of exact Boolean operations for meshes

Schifko¹,

Jüttler

Kornberger

2010

Proceedings of the 26th Spring Conference on Computer Graphics

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We present an algorithm for robust Boolean operations of triangulated solids, which is suitable for real-word industrial applications involving meshes with large numbers of triangles. In order to avoid potential robustness problems, which may be caused by (almost) degenerate triangles or by intersections of nearly co-planar triangles, we use filtered exact arithmetic, based on the libraries CGAL and GNU Multi Precision Arithmetic Library. The method consists of two major steps: First we compute the exact intersection of the meshes using a sweep plane algorithm. Second we apply mesh cleaning methods which allow us to generate output which can safely be represented using floating point numbers. The performance of the method is demonstrated by several examples which are taken from applications at ECS Magna Powertrain.

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“…A highly successful approach for achieving robust algorithms is Exact Geometric Computation (EGC) [23]. To speed up EGC computations, we need floating point filters [2,15]. A basis for such filters is exactly rounded arithmetic at machine-level.…”

Section: Fig 1 Elementary Functions In the Lia-2mentioning

confidence: 99%

Foundations of Exact Rounding

Yap

2009

WALCOM: Algorithms and Computation

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Abstract. Exact rounding of numbers and functions is a fundamental computational problem. This paper introduces the mathematical and computational foundations for exact rounding. We show that all the elementary functions in ISO standard (ISO/IEC 10967) for Language Independent Arithmetic can be exactly rounded, in any format, and to any precision. Moreover, a priori complexity bounds can be given for these rounding problems. Our conclusions are derived from results in transcendental number theory.

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Interval arithmetic yields efficient dynamic filters for computational geometry

Cited by 68 publications

References 18 publications

Accurate Floating-Point Summation Part I: Faithful Rounding

Accurate Floating-Point Summation Part I: Faithful Rounding

Industrial application of exact Boolean operations for meshes

Foundations of Exact Rounding

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