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**Published in:** Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

Emerging macro-molecular simulations, such as supercoiling
of DNA and protein unfolding, have an opportunity to profit from
two decades of experience with geometric models within computer-aided
geometric design (CAGD). For CAGD, static models are often sufficient,
while form and function are inextricably related in biochemistry, resulting
in greater attention to critical topological characteristics of these
dynamic models. The greater emphasis upon dynamic change in macro-molecular
simulations imposes increased demands for faithful integration
of topology and geometry, as well as much stricter requirements
for computational efficiency. This article presents transitions from the
CAGD domain to meet the greater fidelity and performance demands
for macro-molecular simulations.

Edward L. F. Moore, Thomas J. Peters, David R. Ferguson, and Neil F. Stewart. Integrating Topology and Geometry for Macro-Molecular Simulations. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{moore_et_al:DagSemProc.04351.16, author = {Moore, Edward L. F. and Peters, Thomas J. and Ferguson, David R. and Stewart, Neil F.}, title = {{Integrating Topology and Geometry for Macro-Molecular Simulations}}, booktitle = {Spatial Representation: Discrete vs. Continuous Computational Models}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {4351}, editor = {Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.16}, URN = {urn:nbn:de:0030-drops-1240}, doi = {10.4230/DagSemProc.04351.16}, annote = {Keywords: Computational topology , spline , approximation} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 8021, Numerical Validation in Current Hardware Architectures (2008)

This work was presented in two parts at Dagstuhl seminar 08021.
The two presentations described work in
progress, including a ``backward bound'' for a combined backward/forward
error analysis for the problem mentioned in the title.
We seek rigorous proofs that representations of computed sets, produced by
algorithms to compute Boolean operations, are well formed, and that the
algorithms are correct. Such proofs should eventually take account of the use of
finite-precision arithmetic, although the proofs presented here do not.
The representations studied are based on subdivision surfaces. Such
representations are being used more and more frequently in place of trimmed
NURBS representations, and the robustness analysis for these new representations
is simpler than for trimmed NURBS.
The particular subdivision-surface representation used is based on the Loop
subdivision scheme. The analysis is broken into three parts. First, it is
established that the input operands are well-formed two-dimensional manifolds
without boundary. This can be done with existing methods.
Secondly, we introduce the so-called ``limit mesh'', and view the
limit meshes corresponding to the input sets as defining an approximate problem
in the sense of a backward error analysis. The presentations mentioned above
described a proof of the corresponding error bound. The third part of the
analysis corresponds to the ``forward bound'': this remains to be done.

Di Jiang and Neil Stewart. Robustness of Boolean operations on subdivision-surface models. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{jiang_et_al:DagSemProc.08021.17, author = {Jiang, Di and Stewart, Neil}, title = {{Robustness of Boolean operations on subdivision-surface models}}, booktitle = {Numerical Validation in Current Hardware Architectures}, pages = {1--10}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8021}, editor = {Annie Cuyt and Walter Kr\"{a}mer and Wolfram Luther and Peter Markstein}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08021.17}, URN = {urn:nbn:de:0030-drops-14435}, doi = {10.4230/DagSemProc.08021.17}, annote = {Keywords: Robustness, finite-precision arithmetic, Boolean operations, subdivision surfaces} }

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

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.

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.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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