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DOI: 10.4230/DagSemProc.06021.9
Transfinite interpolation for well-definition in error analysis in solid modelling

Authors: Neil Stewart and Malika Zidani

Published in: Dagstuhl Seminar Proceedings, Volume 6021, Reliable Implementation of Real Number Algorithms: Theory and Practice (2006)

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 Schröder. 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.
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)

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@InProceedings{stewart_et_al:DagSemProc.06021.9,
  author =	{Stewart, Neil and Zidani, Malika},
  title =	{{Transfinite interpolation for well-definition in error analysis in solid modelling}},
  booktitle =	{Reliable Implementation of Real Number Algorithms: Theory and Practice},
  pages =	{1--12},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6021},
  editor =	{Peter Hertling and Christoph M. Hoffmann and Wolfram Luther and Nathalie Revol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06021.9},
  URN =		{urn:nbn:de:0030-drops-7195},
  doi =		{10.4230/DagSemProc.06021.9},
  annote =	{Keywords: Forward/backward error analysis, robustness, well-definition, trimmed NURBS, combined subdivision, trimming, bounds on normals}
}
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