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Proceedings Paper

Semantic interpretation of inconsistent curve and surface data for the definition of solids
Author(s): Helene Desaulniers; Neil F. Stewart
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Paper Abstract

The problem of numerical robustness in Solid Modelling involves proving theorems about the subset of E3 defined by the output of an algorithm using finite-precision floating point arithmetic. However, both input and output sets are usually specified by symbolic data and geometric data that are possibly inconsistent, and the latter comprise geometric vertices in E3, surface intersection curves, and surface patches defined by approximation techniques that leads to the definition of precise subsets of E3, such that the stored input sets are consistent with the symbolic data, and typically closer to the user's input sets that can be warranted by the inherent uncertainty in the data. This definition will permit rigorous proofs, based on the fundamental equation fl(xoy) equals xoy(1 + (eta) ), of theorems providing a backward error analysis; that is, it will be possible to show that, if a problem is well conditioned, we have the exact solution for a problem defined by slightly perturbed sets. Here, the perturbation is measured by the maximum of the Hausdorff distance between the exact and the perturbed sets, and the Hausdorff distance between the boundaries of these two sets. It will also be possible to show that the error is small as measured by a certain pseudo-distance reflecting relative variation of the boundaries. Furthermore, in circumstances where the piecewise linear Schoenflies theorem in E3 is true, it will follow that there is a space homeomorphism from E3 onto E3 relating the two sets; that is, in a very strong sense, they have the same topological form. The interpolatory technique is described in the curvilinear case, but so far we have used this approach only to prove results about stability of algorithms for sets defined by planar surface patches with inconsistent geometric vertex information. An illustrative example is given.

Paper Details

Date Published: 1 November 1992
PDF: 10 pages
Proc. SPIE 1830, Curves and Surfaces in Computer Vision and Graphics III, (1 November 1992); doi: 10.1117/12.131735
Show Author Affiliations
Helene Desaulniers, Univ. du Quebec a Trois-Rivieres (Canada)
Neil F. Stewart, Univ. de Montreal (Canada)

Published in SPIE Proceedings Vol. 1830:
Curves and Surfaces in Computer Vision and Graphics III
Joe D. Warren, Editor(s)

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