H. Desaulniers and N. F. Stewart. Robustness of numerical methods in geometric computation when problem data is uncertain. Computer-Aided Design Special issue - Uncertainties in geometric design, 25(9):539-545, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....combinatorial part of the computation. This is related to the fact that, while basic arithmetic operators and analytic functions on real numbers are Turing computable, comparison of two real numbers is only semi decidable [30] Computing with uncertain inputs is unavoidable in physical modeling [9, 22]. In applications such as robotics or solid modeling, actual geometric inputs are measurement of physical objects, and as such, they are inherently uncertain and can only be represented for example using intervals of numbers. In brief, one can classify existing approaches to robustness into two ....

H. Desaulniers and N. F. Stewart. Robustness of numerical methods in geometric computation when problem data is uncertain. Computer-Aided Design Special issue - Uncertainties in geometric design, 25(9):539-545, 1993.

....operators is provided. Some consequences in computation with the boundary representation paradigm are sketched that can incorporate existing methods [16, 32, 19, 17, 18] into a general, mathematically well founded theory. Moreover, the model is able to capture the uncertainties of input data [7, 25] in actual CAD situations. We need the following requirements for the mathematical model: 1) the notion of computability of solids has to be well de ned, 2) the model has to re ect the observable properties of real solids, 3) it has to be closed under the Boolean operations and all basic ....

....to equivalent subsets cannot be distinguished by such a machine. Moreover, partial solids, and, more generally, domain theoretically de ned 2 A set is saturated if it is upper closed with respect to the specialisation ordering. 6 data types allow us to capture partial, or uncertain input data [7, 25] encountered in realistic CAD situations. In order to be able to compute the continuous membership predicate on X, we extend it to the upper space UX by de ning 2 : UX SX ftt; g with: C 2 (A; B) 8 : tt if C A if C B otherwise Note that we use the in x notation for predicates ....

H. Desaulniers and N. Stewart. Robustness of numerical methods in geometric computation when problem data is uncertain. Computer Aided Design, special issue on uncertainties in geometrical design, 1993.

....operators is provided. Some consequences in computation with the boundary representation paradigm are sketched that can incorporate existing methods [10, 22, 13, 11, 12] into a general, mathematically well founded theory. Moreover, the model is able to 2 capture the uncertainties of input data [5, 16] in actual CAD situations. We need the following requirements for the mathematical model: 1) the notion of computability of solids has to be well de ned, 2) the model has to re ect the observable properties of real solids, 3) it has to be closed under the Boolean operations and all basic ....

....the idealization of a machine used to measure mechanical parts. Two parts corresponding to equivalent subsets cannot be distinguished by such a machine. Moreover, partial solids, and, more generally, domain theoretically de ned data types allow us to capture partial, or uncertain input data [5, 16] encountered in realistic CAD situations. In order to be able to compute the continuous membership predicate on X, we extend it to the upper space UX by de ning 2 : UX SX ftt; g with: C 2 (A; B) 8 : tt if C A if C B otherwise Note that we use the in x notation for predicates ....

H. Desaulniers and N. Stewart. Robustness of numerical methods in geometric computation when problem data is uncertain. Computer Aided Design, special issue on uncertainties in geometrical design, 1993.

....likely, but we focus our attention on gaps for ease of exposition. Concept Reality Figure 1: Gaps along Boundary Curve Interoperability for Distributed Design Systems 3 3 Related Work The problem of accuracy in geometric modeling for CAD has attracted considerable attention in the last decade [5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 21, 25, 27, 30], with attention both to errors inherent to floating point arithmetic and to numerical approximations in algorithms. However, the use of associated tolerances within modeling implementations has received scant attention in the literature (Please see, for example [26] compared to their ....

Desaulniers, H., and Stewart, N. F., Robustness of numerical methods in geometric computation when problem data is uncertain, CAD, 25, No. 9, 539 - 545, 1993.

....solved exactly a problem close to the given problem, as is considered in backward error analysis 8 . The necessary characterization of ill conditioned problems and the development of responsive methods have not yet been completely developed for CAD systems, but some initial steps have been taken [5, 6, 8, 10, 11, 12, 13, 18, 19, 20, 27, 35], where among these are some surface intersection algorithms which are capable of incorporating user specified tolerances [12, 20] The use of tolerances in modeling algorithms has received scant attention in the literature (Please see, for example [28] compared to their widespread use within ....

....CAD systems [1] 8 This is the crux of backward error analysis, which requires study of the intrinsic condition of problems. Robust Engineering Design and Simulation with Tolerances 4 A seminal insight of J. H. Wilkinson [34] led to what is usually referred to as a backward error analysis [5, 21]. Although it is unreasonable to expect error bounds and or algorithms that give satisfying results in all cases, it may be possible to do something almost as useful, namely, it may be possible to show that the numerical method never introduces more error than is already present due to causes ....

Desaulniers, H., and Stewart, N. F., Robustness of numerical methods in geometric computation when problem data is uncertain, CAD, 25, No. 9, 539 - 545, 1993.

....operators is provided. Some consequences in computation with the boundary representation paradigm are sketched that can incorporate existing methods [13, 28, 16, 14, 15] into a general, mathematically well founded theory. Moreover, the model is able to capture the uncertainties of input data [8, 19] in actual CAD situations. We need the following requirements for the mathematical model: 1) the notion of computability of solids has to be well defined, 2) the model has to reflect the observable properties of real solids, 3) the model has to be closed under the Boolean operations, 4) ....

....of a machine used to measure mechanical parts. Two parts corresponding to equivalent subsets cannot be discriminated by such a machine. Moreover, partial solids, and, more generally, domain theoretically defined data types (cf. Section 4) allow us to capture partial, or uncertain input data [8, 19] encountered in realistic CAD situations. Starting with the continuous membership predicate, the natural definition for the complement would be to swap the values tt and ff. This means that the complement of (A; B) is (B; A) cf. requirement (3) As for requirement (4) the figure below represents ....

H. Desaulniers and N. Stewart. Robustness of numerical methods in geometric computation when problem data is uncertain. Computer Aided Design, special issue on uncertainties in geometrical design, 1993.

....where two entities are adjoint by their topological data, yet their geometric data indicates they are disjoint. New intersection algorithms [2, 3] incorporate tolerance intervals that are adaptive to the numerical errors compounded by extensive geometric computations. Backward error analysis [1] has been performed relative to the errors for certain geometric algorithms. 3 Example Consider the simple illustrative geometry of Figure 1. Vertex V0 serves as a trimming endpoint for edge E0, with V0 being within tolerance of E0. E0 . V0 Figure 1: Simple Illustrative Geometry Now, during ....

Desaulniers, H., and Stewart, N. F., Robustness of numerical methods in geometric computation when problem data is uncertain, CAD, 25 (9), pp. 539 - 545.

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H. Desaulniers and N. F. Stewart, "Robustness of numerical methods in geometric computation when problem data is uncertain" Computer Aided Design, August 1993.

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