Table Of ContentThe Mathematics of Surfaces IX
Springer-Verlag London Ltd.
Roberto Cipolla and Ralph Martin (Eds)
The Mathematics
of Surfaces IX
Proceedings of the Ninth IMA Conference on the Mathematics of Surfaces
, Springer
Roberto Cipolla. BA(Honsl. MSE, MEng. DPhil
Department of Engineering. University of Cambridge. Cambridge CB2 lPZ. UK
Ralph Martin. MA, PhD, FIMA, CMath, MBCS. CEng
Department of Computer Science. CardiffUniversity, PO Box 916. 5 The Parade.
CardiffCF24 3XF, UK
ISBN 978-1-4471-1153-5 ISBN 978-1-4471-0495-7 (eBook)
DOI 10.1007/978-1-4471-0495-7
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Preface
These proceedings collect the papers accepted for presentation at the bien
nial IMA Conference on the Mathematics of Surfaces, held in the University
of Cambridge, 4-7 September 2000. While there are many international con
ferences in this fruitful borderland of mathematics, computer graphics and
engineering, this is the oldest, the most frequent and the only one to concen
trate on surfaces.
Contributors to this volume come from twelve different countries in Eu
rope, North America and Asia. Their contributions reflect the wide diversity
of present-day applications which include modelling parts of the human body
for medical purposes as well as the production of cars, aircraft and engineer
ing components. Some applications involve design or construction of surfaces
by interpolating or approximating data given at points or on curves. Others
consider the problem of 'reverse engineering'-giving a mathematical descrip
tion of an already constructed object.
We are particularly grateful to Pamela Bye (at the Institue of Mathemat
ics and its Applications) for help in making arrangements; Stephanie Harding
and Karen Barker (at Springer Verlag, London) for publishing this volume
and to Kwan-Yee Kenneth Wong (Cambridge) for his heroic help with com
piling the proceedings and for dealing with numerous technicalities arising
from large and numerous computer files. Following this Preface is a listing
of the programme committee who with the help of their colleagues did much
work in refereeing the papers for these proceedings. Due to their efforts, many
of the papers have been considerably improved. Our thanks go to all of them
and particularly to our fellow member of the Organising Committee, Malcolm
Sabin.
Cambridge, Roberto Cipolla
June 2000 Ralph Martin
Programme Committee
A. Blake (Microsoft Research Cambridge, UK)
M. Bloor (University of Leeds, UK)
R. Cipolla (University of Cambridge, UK)
R. Farouki (University of California Davis, USA)
A. Fitzgibbon (University of Oxford, UK)
H. Hagen (University of Kaiserslautern, Germany)
D. Kriegman (University of Illinois, USA)
T. Lyche (University of Oslo, Norway)
R. Martin (Cardiff University, UK)
N. Patrikalakis (MIT, USA)
J. Peters (University of Florida, USA)
J. Ponce (University of Illinois, USA)
M. Pratt (NIST, USA)
M. Sabin (Numerical Geometry Ltd., UK)
L. Schumaker (Vanderbilt University, USA)
C. Taylor (University of Manchester, UK)
W. Triggs (INRIA, France)
W. Wang (University of Hong Kong, China)
Contents
Meshless Parameterization
and B-Spline Surface Approximation 1
Michael S. Floater
Computation of Local Differential
Parameters on Irregular Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19
Peter Csakany, Andrew M. Wallace
Remarks on Meshless Local Construction of Surfaces . . . . . . . .. 34
Robert Schaback
Gaussian and Mean Curvature of Subdivision Surfaces ....... 59
Jory Peters, Georg Umlauf
Best Fit Translational and Rotational Surfaces
for Reverse Engineering Shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70
Pal Benko, Tamas Varady
Problem Reduction to Parameter Space ... . . . . . . . . . . . . . . . . . .. 82
Myung-Soo Kim, Gershon Elber
Higher Order Singularities
in Piecewise Linear Vector Fields..... .... . .... .... . . .. .... ... 99
Xavier Tricoche, Gerik Scheuermann, Hans Hagen
Landmarks of a Surface ............ , .......................... 114
Ian R. Porteous, Mike J. Puddephat
Time-Optimal Paths Covering a Surface ...................... 126
Taejung Kim, Sanjay E. Sarma
Surface Evolution and Representation
using Geometric Algebra ....... .............................. 144
Anthony Lasenby, Joan Lasenby
x
Interactive Design of Complex Mechanical
Parts using a Parametric Representation 169
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Hassan Ugail, Michael Robinson, Malcolm l. Go Bloor,
Michael Jo Wilson
Surfaces in the Mind's Eye 180
00000000000000000000000000000000000
Jan Koenderink, Andrea van Doorn, Astrid Kappers
Shape-from-Texture from Eigenvectors of Spectral
Distortion 194
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Eraldo Ribeiro, Edwin R Hancock
Camera Calibration from Symmetry 214
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Kwan-Yee Ko Wong, Paulo Ro So Mendon~a, Roberto Cipolla
Dynamic Shapes of Arbitrary Dimension: The Vector
Distance Functions 227
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Olivier Faugeras, Jose Gomes
Least-Squares Fitting of Algebraic Spline Curves via Normal
Vector Estimation 263
00000000000000000000000000000000000000000000
Bert Jii.ttler
Use of Reverse Automatic Differentiation
in Ship Hull Optimisation 281
0 0 .. 0 .. 0 .. 0 0 .. 0 0 0 ........ 0 0 0 0 .... 0 0 0
Ro Boudjemaa, Mol.Go Bloor and Mojo Wilson
Symmetry Sets and Medial Axes in Two and Three
Dimensions 306
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Peter Giblin
Symmetry - A Research Direction in Curve
and Surface Modelling; Some Results and Applications 322
0 0 0 0 0 0 0
Ho Eo Bez
Functions and Methods to Analyze and
Construct Developable Hull Surfaces 338
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Salvatore Miranda, Claudio Pensa, Fabrizio Sessa
Bipolar and Multipolar Coordinates 348
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Rida To Farouki, Hwan Pyo Moon
Polar Curves and Surfaces 372
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Kenji Ueda
Boundary Representation Models: Validity
and Rectification 389
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Nicholas Mo Patrikalakis, Takis Sakkalis, Guoling Shen
XI
Interval and Affine Arithmetic for Surface Location of
Power- and Bernstein-Form Polynomials ..................... 410
Irina Voiculescu, Jakob Berchtold, Adrian Bowyer, Ralph R. Martin,
Qijiang Zhang
A Class of Bernstein Polynomials that Satisfy
Descartes' Rule of Signs Exactly .............................. 424
Joab R. Winkler, David L. Ragozin
On Approximation in Spaces
of Geometric Objects ......................................... 438
Helmut Pottmann, Martin Peternell
Representing the Time-Dependent Geometry of the Heart for
Fluid Dynamical Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
C.J. Evans, M.l.G. Bloor and M.J. Wilson
Application of Point-Based Smoothing to the Design of Flying
Surfaces ...................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
R.M. Tookey (3 W. Sargeant
Modelling of Material Property Variation for Layered Manu-
facturing ..................................................... 486
Michael J. Pratt
Meshless Parameterization
and B-Spline Surface Approximation
Michael S. Floater
SINTEF, P.O. Box 124, Blindem, 0314 Oslo, Norway
Summary. This paper proposes a method for approximating unorganized points
in R3 with smooth B-spline surfaces. The method involves: meshless parameteri
zationj triangulationj shape-preserving reparameterizationj and least squares spline
approximation.
1 Introduction
The goal of this paper is to describe a new method for approximating a set
of distinct points
(1)
with a tensor-product B-spline surface
ml m2
8(U, v) = ~ ~ Bi(U)Cj(V)Cij, (2)
i=1 j=1
Here B1, ••• , Bml and C1, ••• , Cm2 are B-splines over non-uniform knot vec
tors with orders (degrees plus 1) K and L respectively. We assume that
the points have been sampled from a (simply connected) patch of the sur
face of some object in R3. Many sets of measured data in practice are in
the form of single patches, even though their geometry can be quite com
plex. The surface generation method we propose consists of several sequen
tial steps: meshless parameterizationj triangulationj reparameterizationj and
least squares approximation. The key ingredient is the meshless parameteri
zation of [11] which we use to parameterize the points Zi without the need
of a given mesh or topological structure. The overall surface approximation
method has performed very well in numerical examples, at least when the
underlying surface is not too far from being developable.
Surface generation from 'single patch' data sets can be viewed as a simple
form of reverse engineering, which, in its widest sense, is usually understood
to be the generation of a full B-rep (boundary representation) surface model
from points measured from the whole surface of a physical object in R3. Since
such surfaces are closed and can have arbitrary topology, building a B-rep
model requires not o~ly topology construction but also segmentation and the
sewing together of surface patches. Thus reverse engineering in general is
R. Cipolla et al. (eds.), The Mathematics of Surfaces IX
© Springer-Verlag London Limited 2000
