| Co-Investigator(Kenkyū-buntansha) |
TOMIOKA Yutaka University of Tokyo, Department of Mathematical Engineering and Information Phys, 工学部, 助手 (30188776)
SUGIHARA Kokichi University of Tokyo, Department of Mathematical Engineering and Information Phys, 工学部, 教授 (40144117)
IRI Masao University of Tokyo, Department of Mathematical Engineering and Information Phys, 工学部, 教授 (40010722)
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| Research Abstract |
Conventionally, geometric algorithms have been designed on the assumption that numerical errors do not take place, and consequently they are not necessarily valid in real computation where numerical errors are inevitable. We searched for a method for designing geometric algorithms that are robust against numerical errors, and established a new design methodology in which the highest priority is placed on the topological consistency of geometric objects and numerical results are used as lower-priority information. Algorithms thus designed does not come across inconsistency and hence carries out its task, giving some output that is at least topologically consistent. This design methodology was applied to design robust algorithms for the problems : (1) the incremental construction of two-dimensional Voronoi diagrams for points, (2) the divide-and-conquer construction of two-dimensional Voronoi diagrams for points, (3) the incremental construction of two-dimensional Voronoi diagrams for li
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ne segments, (4) the incremental construction of three-dimensional Voronoi diagrams for points, (5) the extraction of intersections of line segments, (6) the construction of three-dimensional convex hulls of points, (7) the set-theoretic operations of simple polygons, (8) the intersection of convex polyhedra, and (9) the construction of Laguerre Voronoi diagrams in the plane. In particular, the new algorithms for the problems (1)-(5) were implemented into computer programs, and computational experiments were corried out. The experiments show that, no matter how poor the precision in computation may be, the programs do not fail, that the outputs converge to the true solutions as the precision becomes higher, that the new algorithms achieve robustness without increasing time complexity achieved by conventional algorithms, and that the new algorithms are simple because exceptional processing for degenerate cases is not necessary.
We also investigated numerical methods required in geometric algorithms. The fast automatic differentiation method were used to obtain the tight bound of numerical errors, and the method was applied to many problems such as geographic optimization based on Voronoi diagrams. A new method for tracing abgebraic curves was also constructed, which can trace both global and micro structures in reasonable time. We could also improve the performance of the programs for the problems (1)-(5) bytuning the ways of numerical computations. Less
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