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# Transfinite interpolation for well-definition in error analysis in solid modelling

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DagSemProc.06021.9.pdf
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## Cite As

Neil Stewart and Malika Zidani. Transfinite interpolation for well-definition in error analysis in solid modelling. In Reliable Implementation of Real Number Algorithms: Theory and Practice. Dagstuhl Seminar Proceedings, Volume 6021, pp. 1-12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)
https://doi.org/10.4230/DagSemProc.06021.9

## Abstract

An overall approach to the problem of error analysis in the context of solid modelling, analogous to the standard forward/backward error analysis of Numerical Analysis, was described in a recent paper by Hoffmann and Stewart. An important subproblem within this overall approach is the well-definition of the sets specified by inconsistent data. These inconsistencies may come from the use of finite-precision real-number arithmetic, from the use of low-degree curves to approximate boundaries, or from terminating an infinite convergent (subdivision) process after only a finite number of steps. An earlier paper, by Andersson and the present authors, showed how to resolve this problem of well-definition, in the context of standard trimmed-NURBS representations, by using the Whitney Extension Theorem. In this paper we will show how an analogous approach can be used in the context of trimmed surfaces based on combined-subdivision representations, such as those proposed by Litke, Levin and SchrÃƒÂ¶der. A further component of the problem of well-definition is ensuring that adjacent patches in a representation do not have extraneous intersections. (Here, "extraneous intersections" refers to intersections, between two patches forming part of the boundary, other than prescribed intersections along a common edge or at a common vertex.) The paper also describes the derivation of a bound for normal vectors that can be used for this purpose. This bound is relevant both in the case of trimmed-NURBS representations, and in the case of combined subdivision with trimming.
##### Keywords
• Forward/backward error analysis
• robustness
• well-definition
• trimmed NURBS
• combined subdivision
• trimming
• bounds on normals

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