Abstract
We propose an efficient and exact method for the adaptive sign detection of 4×4 determinants using a standard arithmetic unit. The entities of determinants are variable length integers (integers of arbitrary bit length). The integers are expressed in 16-bit data units, and the sign detection is reduced to the computation of 4×4 determinants of 16-bit integers. To accelerate the computation, the calculation is performed by using a standard arithmetic unit. We have implemented our method and confirmed that it significantly improves the computation time of 4×4 determinants. The method can be applicable to many geometric algorithms that need the exact sign evaluation of 4×4 determinants, especially to construct robust geometric algorithms.
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References
Edelsbruner H, Mücke EP (1990) Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans Graph 9:66–104
Forrest AR (1987) Computational geometry and software engineering: towards a geometric computing environment. In: Rogers DF, Earnshaw RA (eds) Techniques for computer graphics. Springer, Berlin Heidelberg New York, pp 23–37
Fortune S (1995) Polyhedral modelling with exact arithmetic. In: Hoffmann C, Rossignac J (eds) Proceedings of the 3rd symposium on solid modeling and applications. ACM Press, New York, pp 225–233
Hanamitsu H, Yoshida N, Lou L, Suzuki S, Yamaguchi F (1997) Adaptive sign detection method using a floating point processing unit-sign detection for 3×3 determinants. J Jpn Soc Precision Eng 63:657–663
Hoffmann CM (1989) Geometric and solid modeling: an introduction. Morgan Kaufmann, San Francisco
Liu CL (1968) Introduction to combinatorial mathematics. McGraw-Hill, New York
Segal M (1990) Using tolerances to guarantee valid polyhedral modeling results. Comput Graph 24:105–114
Stolfi J (1991) Oriented projective geometry: a framework of geometric processing. Academic, Boston
Sugihara K (1992) Topologically consistent algorithms related to convex polyhedra. In: Ibaraki T, Inagaki Y, Iwama K, Nishizeki T, Yamashita M (eds) Proceedings of the 3rd international symposium on algorithms and computation (ISAAC ’92), Nagoya, Japan, 16–18 December 1992. Lecture notes in computer science, vol 650. Springer, Berlin Heidelberg New York, pp 209–218
Sugihara K, Iri M (1989) A solid modeling system free from topological inconsistency. J Inf Process 12:380–393
Sugihara K, Iri M (1992) Construction of the Voronoi diagram for “one million” generators in single-precision arithmetic. Proc IEEE 80:1471–1484
Yamaguchi F (1998) A shift of playground for geometric processing from euclidean to homogeneous. Vis Comput 14:315–327
Yamaguchi F, Niizeki M (1997) Some basic geometric test conditions in terms of Plücker coordinates and Plücker coefficients. Vis Comput 13:29–41
Yoshida N, Shiokawa M, Yamaguchi F (1994) Solid modeling based on a new paradigm. Comput Graph Forum 13:55–64
Yoshida N, Shiokawa M, Yamaguchi F (1995) Adaptive sign detection in 4×4 determinant method – Adaptive sign detection method for n-dimensional inner products. J Adv Automat Technol 7:128–134
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Yamauchi , T., Yoshida , N., Doi , J. et al. Efficient method of adaptive sign detection for 4×4 determinants using a standard arithmetic processing unit. Visual Comp 20, 37–46 (2004). https://doi.org/10.1007/s00371-003-0224-0
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DOI: https://doi.org/10.1007/s00371-003-0224-0