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Simple floating-point filters for the two-dimensional orientation problem

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Abstract

This paper is concerned with floating-point filters for a two dimensional orientation problem which is a basic problem in the field of computational geometry. If this problem is only approximately solved by floating-point arithmetic, then an incorrect result may be obtained due to accumulation of rounding errors. A floating-point filter can quickly guarantee the correctness of the computed result if the problem is well-conditioned. In this paper, a simple semi-static floating-point filter which handles floating-point exceptions such as overflow and underflow by only one branch is developed. In addition, an improved fully-static filter is developed.

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Notes

  1. There are further related crucial problems in computational geometry which require to determine signs of determinants of larger matrices than in (1.1). For example, the point-in-circle problem: “Decide whether a point in the plane lies inside or outside or on the boundary of a circle.” leads to a 4-by-4 matrix. Such problems are not considered in this paper even though we believe that our techniques can be adapted but calculations will certainly become much more involved.

  2. We introduce the algorithm given in [7] The main purpose of the paper by Melquiond and Pion is to develop an automatic constructor of the floating-point filter. Therefore, their approach can be applied not only to the two dimensional orientation problem but also to other problems in computational geometry.

  3. Since there are products of only two floating-point numbers in the evaluation of (1.1), overflow and underflow rarely occur. However, if we handle the InSphere problem which aims to judge whether a point is inside or outside a sphere in the three-dimensional space, then a product of five floating-point numbers appears in the evaluation. Therefore, it is not so rare that floating-point exceptions occur in this problem.

  4. Here “valid” means that condition (3.29) evaluated according to IEEE 754 rules yields the result “true”.

  5. If this is not true, a suitable scaling by multiplying a large number with a power of two for all floating-point coordinates should be applied.

References

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Acknowledgments

The authors wishes to thank the anonymous referee for constructive and valuable comments. This research was partially supported by the CREST program, Japan Science and Technology Agency (JST).

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Authors and Affiliations

  1. College of Systems Engineering and Science, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama-shi, Saitama, 337-8570, Japan

    Katsuhisa Ozaki

  2. Institute for Reliable Computing, Hamburg University of Technology, Schwarzenbergstr. 95, 21073, Hamburg, Germany

    Florian Bünger & Siegfried M. Rump

  3. School of Arts and Sciences, Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo, 167-8585, Japan

    Takeshi Ogita

  4. Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjyuku-ku, Tokyo, 169-8555, Japan

    Shin’ichi Oishi

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  1. Katsuhisa Ozaki
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  2. Florian Bünger
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  3. Takeshi Ogita
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  4. Shin’ichi Oishi
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  5. Siegfried M. Rump
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Correspondence to Katsuhisa Ozaki.

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Communicated by Lars Eldén.

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Ozaki, K., Bünger, F., Ogita, T. et al. Simple floating-point filters for the two-dimensional orientation problem. Bit Numer Math 56, 729–749 (2016). https://doi.org/10.1007/s10543-015-0574-9

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  • DOI: https://doi.org/10.1007/s10543-015-0574-9

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