D. P. Dobkin and D. Silver. Applied computational geometry: Towards robust solutions of basic problems. J. Comput. Syst. Sci., 40(1):70--87, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....patterns. A variety of techniques have been designed to make geometric algorithms robust in the presence of high precision numerical computations (e.g. involving square roots) and degenerate geometric configurations (e.g. more than two collinear points or more than three cocircular points) [3, 14, 24, 28, 33, 34, 43, 47, 48, 49, 57, 76, 78, 85, 86, 87]. GeomLib adopts the paradigm of exact computation (see, e.g. Refs. 3, 14, 86] and uses the concept of degree [57] to characterize the arithmetic precision requirement of a geometric algorithm. Namely, a geometric algorithm of degree d requires in its computations a precision that is, in the ....

....the encapsulation of the geometric information within the basic geometric objects allows the implementation of a geometric algorithm to be independent from the arithmetic used. However, the problem of the correctness of the arithmetic computations has to be considered, as indicated, e.g. in Refs. [3, 14, 24, 28, 33, 34, 43, 47, 48, 49, 57, 76, 78, 85, 86, 87]. The assumption of real number arithmetic has proved unrealistic, since digital computers do not exhibit such capability natively, i.e. in hardware. On the other hand, exact rational arithmetic via software may excessively slow down computations. In light of these problems, the equivalent ....

D. P. Dobkin and D. Silver. Applied computational geometry: Towards robust solutions of basic problems. J. Comput. Syst. Sci., 40(1):70--87, 1989.

....for the representations of points and of line equations and so to guarantee that all intersections can be represented without error. The problem of robustness and topological correctness of geometric computation has of course also been addressed in the computational geometry literature [DS90, For85] One can distinguish perturbation free approaches (e.g. KM83, OTU87] where the idea is to perform geometric computations with suOEciently high precision such that no errors occur; the task is to determine how many digits are needed to represent the result of the used geometric ....

....approaches (e.g. KM83, OTU87] where the idea is to perform geometric computations with suOEciently high precision such that no errors occur; the task is to determine how many digits are needed to represent the result of the used geometric primitives exactly. Perturbation approaches (e.g. DS90, GM95, Mil89, Sch94] allow one to slightly change the input data or the results of computations in order to be able to represent data at a xed level of precision. In this context, the realm based ROSE algebra can be viewed as an application of the perturbation approach within spatial databases, ....

D. Dobkin and D. Silver. Applied Computational Geometry: Towards Robust Solutions of Basic Problems. Journal of Computer and System Sciences, 40:7087, 1990.

....But geometric algorithms are quite sensitive regarding this procedure, and in the end, the task is mostly left to the programmer to close the gap between theory and practice. This leads inevitably not only to numerical rounding errors but also to topological inconsistencies and degeneracies [DS90, For85, GY86, Hof89], since topological information is commonly inferred from coordinate based geometric data. Hence, an applied computational geometry is needed which takes into account the finite representations available in computer systems. In this paper we design numerically robust, topologically consistent, and ....

....and robust numerical results must be obtained. Provided that the input data are exact, the task is to determine how many digits of precision are required by numerical computations so that the algorithm produces correct results and takes into account desired accuracy. Perturbation approaches (e.g. [DS90, EM88, GM95, GSS89, GY86, HHK88, Mil89, NME90, Sch94]) allow to slightly change input data or computed results. Because in many applications the input data are approximate from the beginning, such slight alterations seem to be tolerable. This paper is based on an interesting subclass of perturbation approaches that attempt to transform geometric ....

D. Dobkin & D. Silver. Applied Computational Geometry: Towards Robust Solutions of Basic Problems. Journal of Computer and System Sciences, 40, 70-87, 1990.

....problems, and support libraries for visualization, I O, conversion to standard geometry formats used in industry, etc. will be developed within the project. Although more and more research papers in computational geometry address implementation issues of geometric algorithms (examples are [5, 6, 9, 12, 13, 15, 21]) a lot of theoretical and experimental research is still needed. Such research is also part of the cgal project. 3 The Kernel The kernel has been designed in Summer 1995 by the cgal sites Berlin, Saarbr ucken, Sophia Antipolis and Utrecht, where the latter three already had ample experience ....

D. Dobkin and D. Silver. Applied computational geometry: Towards robust solutions of basic problems. Journal of Computer and System Sciences, 40:70--87, 1990.

....overview of basic methods and provides more detailed references. This dissertation does not address the effect of more sophisticated compilers on algorithm development. Degeneracy and roundoff error. The design of robust geometric algorithms has been the subject of much theoretical research [DS90, For89, For92, Mil88, Yap90] While handling these details is important, a general discussion of these issues is beyond the scope of this thesis. My algorithms assume non degenerate input and are for the most part naive about numerical issues. Generalized abstract models. Finally, this work does ....

D. Dobkin and D. Silver. Applied computational geometry: Towards robust solutions of basic problems. Journal of Computer and System Sciences, 40:70--87, 1990.

.... Here, we determine the precision at which computations must be done given the input precision and the nature of the computation [SSG, FM, Bar] A third approach considers the problem of building a tracker which follows the computation and determines the precision as the computation proceeds [DS]. The idea is to detect when insufficient precision remains and backtrack to a point whence the computation can be redone at a higher precision. 4.2. Decomposition of polygons and polyhedra A recurrent theme in computer graphics is the modeling and rendering of complex objects. These problems ....

Dobkin, D. and Silver, D., "Applied computational geometry: towards robust solutions of basic problems", JCSS, vol. 40, 1990, 70-87.

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D. P. Dobkin and D. Silver. Applied computational geometry: Towards robust solutions of basic problems. J. Comput. Syst. Sci., 40(1):70--87, 1989.

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