T.K. Dey, K. Sugihara, and C.L. Bajaj. Delaunay triangulations in three dimensions with finite precision arithmetic. Computer Aided Geometric Design, 9:457--470, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as are needed to determine its sign. This will slow down the computation, but techniques have been developed to keep the performance penalty relatively small [73, 143] Besides these general approaches, there have been a number papers dealing with robust computation in specific problems [10, 13, 34, 60, 71, 72, 84, 102]. Dealing with degeneracies. Most algorithms described in the computational geometry literature make the assumption that the input is in general position. For example, for computing the intersections in a set of line segments it is often assumed that no three segments meet in a common point, that ....

T. K. Dey, K. Sugihara, and C. L. Bajaj. Delaunay triangulations in three dimensions with finite precision arithmetic. Cornput. Aided Geom. Design, 9:457-470, 1992.

....a warning and reports a wide facet. In R 2 , there are several robust convex hull and Delaunay triangulation algorithms [Fortune 1989] Guibas et al. 1993] Li and Milenkovic 1990] In R 3 , Sugihara and Dey et al. produce a topologically robust convex hull and Delaunay triangulation [Dey et al. 1992] [Sugihara 1992] Their algorithms are a variation of BeneathBeyond with steps to prevent topological anomalies such as in Figure 5. The output may contain unbounded geometric faults. 12 Delta There are several implementations for computing the convex hull with precise arithmetic. The output is ....

Dey, T. K., Sugihara, K., and Bajaj, C. L. 1992. Delaunay triangulations in three dimensions with finite precision arithmetic. Computer Aided Geometric Design 9, 457-- 470.

.... test case a problem area, geometric proximity, which plays a major role in applications and has recently attracted considerable attention because, due to its demands of high precision for exact computation, it is particularly appropriate in assessing effectiveness of robust approaches (see, e.g. [8, 10, 19, 28, 30, 26, 54, 31]) While Voronoi diagrams are interesting in their own right, the main reason for con1 structing and storing them is to efficiently answer fundamental proximity queries such as nearest neighbor and circular range queries. In this paper, we use the notion of degree to illustrate the drawbacks of ....

T. K. Dey, K. Sugihara, and C. L. Bajaj. Delaunay triangulations in three dimensions with finite precision arithmetic. Comput. Aided Geom. Design, 9:457--470, 1992.

....field of geometric and solid modeling, as an area where many of these problems have been seen and these techniques applied. There are several possible definitions for what makes an algorithm robust. Some have even devised systems for specifying the type of robustness an algorithm achieves ([DSB92]) A number of approaches which are outlined here claim to be robust. Other approaches only claim 1 to increase robustness. Rather than try to examine only approaches which meet a certain robustness criteria, this paper will examine techniques which claim to at least increase robustness in some ....

Tamal K. Dey, Kokichi Sugihara, and Chandrajit L. Bajaj. Delaunay triangulations in three dimensions with finite precision arithmetic. Computer Aided Geometric Design, 9:457--470, 1992.

....sensitive to numerical inaccuracies produced due to fixed precision arithmetic. Many approaches have been proposed in the literature to handle this problem. These include the design of geometric algorithms such that robust implementations can be obtained using only the fixed precision hardware [DSB92, For95, Hof89, Mil89, Sug89] However, designing such algorithms is quite involved and has been restricted to only a few problems. A second approach advocates the use of exact real arithmetic. However, a naive implementation of exact arithmetic can be quite slow and a number of techniques have ....

T. K. Dey, K. Sugihara, and C. L. Bajaj. Delaunay triangulations in three dimensions with finite precision arithmetic. Comput. Aided Geom. Design, 9:457--470, 1992.

....on which side it lies, then crack AB into AC and CB. Advantages: Uses hardware floating point and generates explicit geometric structures. Disadvantages: Requires modification of geometric algorithms and has unbounded geometrical and combinatorial error. Consistent (Stable) Computation [24, 28, 22, 1, 4, 9, 10, 12, 20, 22, 38]: Use hardware floating point and make consistent symbolic decisions in the case of an ambiguous numerical tests. Advantages: uses hardware floating point and sometimes has better bounds on error than data normalization. Disadvantages: Decisions have implicit rather than explicit realizations ....

T. K. Dey, K. Sugihara, and C. L. Bajaj. Delaunay triangulations in three dimensions with finite precision arithmetic. Comput. Aided Geom. Design, 9:457--470, 1992.

....above a new facet. If this occurs, Quickhull generates a warning and reports a wide facet. In R 2 , there are several robust convex hull and Delaunay triangulation algorithms [23] 29] 34] In R 3 , Sugihara and Dey et al. produce a topologically robust convex hull and Delaunay triangulation [18] [42] Their algorithms are a variation of Beneath Beyond with steps to prevent topological anomalies such as in Figure 3. The output may contain unbounded geometric faults. There are several implementations for computing the convex hull with precise arithmetic. The output is a triangulation. If ....

T. K. Dey, K. Sugihara, and C. L. Bajaj. Delaunay triangulations in three dimensions with finite precision arithmetic. Computer Aided Geometric Design, 9:457--470, 1992.

....algorithms for two dimensional Voronoi diagrams. Building three dimensional Voronoi diagrams efficiently is a popular topic in the literature, as exemplified by the recent work of Fang and Piegl [4] but to our knowledge, the only algorithms that guarantee robustness are the work of Dey et al. [2] and of Inagaki et al. 10] We prefer the latter algorithm because its implementation seems more straightforward. This algorithm has the advantage that it uses only combinatorial computations, avoiding inexact numerical computations that compromise robustness. Its disadvantage is that it does not ....

Tamal K. Dey, Kokichi Sugihara, and Chandrajit L. Bajaj. Delaunay triangulations in three dimensions with finite precision arithmetic. Computer Aided Geometric Design, 9:457--470, 1992.

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T.K. Dey, K. Sugihara, and C.L. Bajaj. Delaunay triangulations in three dimensions with finite precision arithmetic. Computer Aided Geometric Design, 9:457--470, 1992.

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T. K. Dey, K. Sugihara, and C. L. Bajaj. Delaunay triangulations in three dimensions with finite precision arithmetic. Comput. Aided Geom. Design, 9:457--470, 1992.

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