Freeform surface characterisation: Theory and practice

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

The specification and characterisation of freeform surfaces is immature: many CAD packages only allow the nominal freeform geometry to be specified. To control manufacturing and function, the allowable geometric variability also needs to be specified. Currently International standards specify allowable geometry through geometrical tolerancing. Geometrical tolerancing has proved to be a very blunt instrument with many cases, particularly in the aerospace and biomedical industries, of failure of the function of freeform surfaces due to inadequate specification. The paper begins by discussing the importance of the decomposition of the surface geometry into different scales for both specification and characterisation. The three main types of geometrical decomposition: linear, morphological and segmentation are briefly discussed. The Laplace-Beltrami operator (LBO) is the generalization of the Laplace operator to manifolds (i.e. freeform surfaces). By taking Eigenfunctions/Eigenvalues of the LBO a spectral decomposition of the freeform geometry can be realized. The Eigenfunctions are called manifold harmonics and are a direct generalization of Fourier harmonics for freeform surfaces. Tolerancing the Eigenfunctions provides a very powerful and flexible system to control geometrical variability of freeform surfaces; extending the geometrical tolerancing system by allowing decomposition of the geometrical shapes within tolerance zones. Further a generalized Gaussian filter can also be realized by Gaussian weighting the LBO spectra and reconstructing the weighted Eigenfunctions.

LanguageEnglish
Article number012005
JournalJournal of Physics: Conference Series
Volume483
Issue number1
DOIs
Publication statusPublished - 2014

Fingerprint

eigenvectors
decomposition
specifications
operators
geometry
harmonics
aerospace industry
Laplace transformation
surface geometry
computer aided design
eigenvalues
manufacturing
industries
filters

Cite this

@article{575f17b8a0cc4e08ae18fadc8e52ac69,
title = "Freeform surface characterisation: Theory and practice",
abstract = "The specification and characterisation of freeform surfaces is immature: many CAD packages only allow the nominal freeform geometry to be specified. To control manufacturing and function, the allowable geometric variability also needs to be specified. Currently International standards specify allowable geometry through geometrical tolerancing. Geometrical tolerancing has proved to be a very blunt instrument with many cases, particularly in the aerospace and biomedical industries, of failure of the function of freeform surfaces due to inadequate specification. The paper begins by discussing the importance of the decomposition of the surface geometry into different scales for both specification and characterisation. The three main types of geometrical decomposition: linear, morphological and segmentation are briefly discussed. The Laplace-Beltrami operator (LBO) is the generalization of the Laplace operator to manifolds (i.e. freeform surfaces). By taking Eigenfunctions/Eigenvalues of the LBO a spectral decomposition of the freeform geometry can be realized. The Eigenfunctions are called manifold harmonics and are a direct generalization of Fourier harmonics for freeform surfaces. Tolerancing the Eigenfunctions provides a very powerful and flexible system to control geometrical variability of freeform surfaces; extending the geometrical tolerancing system by allowing decomposition of the geometrical shapes within tolerance zones. Further a generalized Gaussian filter can also be realized by Gaussian weighting the LBO spectra and reconstructing the weighted Eigenfunctions.",
author = "Scott, {P. J.} and X. Jiang",
year = "2014",
doi = "10.1088/1742-6596/483/1/012005",
language = "English",
volume = "483",
journal = "Journal of Physics: Conference Series",
issn = "1742-6588",
publisher = "IOP Publishing Ltd.",
number = "1",

}

Freeform surface characterisation : Theory and practice. / Scott, P. J.; Jiang, X.

In: Journal of Physics: Conference Series, Vol. 483, No. 1, 012005, 2014.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Freeform surface characterisation

T2 - Journal of Physics: Conference Series

AU - Scott, P. J.

AU - Jiang, X.

PY - 2014

Y1 - 2014

N2 - The specification and characterisation of freeform surfaces is immature: many CAD packages only allow the nominal freeform geometry to be specified. To control manufacturing and function, the allowable geometric variability also needs to be specified. Currently International standards specify allowable geometry through geometrical tolerancing. Geometrical tolerancing has proved to be a very blunt instrument with many cases, particularly in the aerospace and biomedical industries, of failure of the function of freeform surfaces due to inadequate specification. The paper begins by discussing the importance of the decomposition of the surface geometry into different scales for both specification and characterisation. The three main types of geometrical decomposition: linear, morphological and segmentation are briefly discussed. The Laplace-Beltrami operator (LBO) is the generalization of the Laplace operator to manifolds (i.e. freeform surfaces). By taking Eigenfunctions/Eigenvalues of the LBO a spectral decomposition of the freeform geometry can be realized. The Eigenfunctions are called manifold harmonics and are a direct generalization of Fourier harmonics for freeform surfaces. Tolerancing the Eigenfunctions provides a very powerful and flexible system to control geometrical variability of freeform surfaces; extending the geometrical tolerancing system by allowing decomposition of the geometrical shapes within tolerance zones. Further a generalized Gaussian filter can also be realized by Gaussian weighting the LBO spectra and reconstructing the weighted Eigenfunctions.

AB - The specification and characterisation of freeform surfaces is immature: many CAD packages only allow the nominal freeform geometry to be specified. To control manufacturing and function, the allowable geometric variability also needs to be specified. Currently International standards specify allowable geometry through geometrical tolerancing. Geometrical tolerancing has proved to be a very blunt instrument with many cases, particularly in the aerospace and biomedical industries, of failure of the function of freeform surfaces due to inadequate specification. The paper begins by discussing the importance of the decomposition of the surface geometry into different scales for both specification and characterisation. The three main types of geometrical decomposition: linear, morphological and segmentation are briefly discussed. The Laplace-Beltrami operator (LBO) is the generalization of the Laplace operator to manifolds (i.e. freeform surfaces). By taking Eigenfunctions/Eigenvalues of the LBO a spectral decomposition of the freeform geometry can be realized. The Eigenfunctions are called manifold harmonics and are a direct generalization of Fourier harmonics for freeform surfaces. Tolerancing the Eigenfunctions provides a very powerful and flexible system to control geometrical variability of freeform surfaces; extending the geometrical tolerancing system by allowing decomposition of the geometrical shapes within tolerance zones. Further a generalized Gaussian filter can also be realized by Gaussian weighting the LBO spectra and reconstructing the weighted Eigenfunctions.

UR - http://www.scopus.com/inward/record.url?scp=84899020293&partnerID=8YFLogxK

U2 - 10.1088/1742-6596/483/1/012005

DO - 10.1088/1742-6596/483/1/012005

M3 - Article

VL - 483

JO - Journal of Physics: Conference Series

JF - Journal of Physics: Conference Series

SN - 1742-6588

IS - 1

M1 - 012005

ER -