Table Of ContentComputation of Curves and Surfaces
NATO ASI Series
Advanced Science Institutes Series
A Series presenting the results of activities sponsored by the NA TO Science Committee,
which aims at the dissemination of advanced scientific and technological know/edge,
with a view to strengthening links between scientific communities.
The Series is published by an international board of publishers in conjunction with the
NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics London and New York
C Mathematical Kluwer Academic Publishers
and PhYSical Sciences Dordrecht, Boston and London
D Behavioural and Social Sciences
E Applied Sciences
F Computer and Systems Sciences Springer-Verlag
G Ecological Sciences Berlin, Heidelberg, New York, London,
H Cell Biology Paris and Tokyo
Series C: Mathematical and Physical Sciences - Vol. 307
Computation of Curves
and Surfaces
edited by
Wolfgang Dahmen
Institut fOr Mathematik I,
Freie Universitat, Berlin, FRG
Mariano Gasca
Department of Applied Mathematics,
University of Zaragoza, Zaragoza, Spain
and
Charles A. Micchelli
IBM T.J. Watson Research Center,
Yorktown Heights, NY, USA
Kluwer Academic Publishers
Dordrecht / Boston / London
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Study Institute on
Computation of Curves and Surfaces
Puerto de la Cruz, Tenerife, Spain
July 10-21,1989
Library of Congress Cataloging in Publication Data
NATO Advanced Study Institute on Computation of Curves and Surfaces
(1989 : Tenerife, Canary Islands)
Computation of curves and surfaces: proceedings of the NATO
Advanced Study Institute on Computation of Curves and Surfaces,
Puerto de 1a Cruz, Tenerife, Spain, July 10-21, 1989 I edited by
Wolfgang Dahmen, Mariano Gasca, and Charles A. Micchel1i.
p. em. -- (NATO ASI series. Series C, Mathematical and
physical sciences; vol. 307)
ISBN-13: 978-94-010-7404-9 e-ISBN-13: 978-94-009-2017-0
001: 10.1007/978-94-009-2017-0
1. Curves, A1gebratc--Data processing--Congresses. 2. Surfaces-
-Data processing--Congresses. I. Dahmen, Wolfgang, 1949-
II. Gasca, Mariano, 1943- III. Micchelli, Charles A.
IV. Title. V. Series: NATO ASI series. Series C, Mathematical and
physical sciences; no. 307.
OA567.N37 1989
516.3·52--dc20 90-4240
CIP
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TABLE OF CONTENTS
Preface vii
I. Discrete Methods for Curves and Surface Representation
W. DAHMEN and C.A. MICCHELLI:
Stationary Subdivision, Fractals and Wavelets 3
R.N. GOLDMAN:
Recursive Triangles 27
J. HOSCHEK:
Exact and Approximate Conversion of Spline Curves and Spline
Surfaces 73
T. LYCHE:
Discrete B-splines and Conversion Problems 117
II. Interpolation and Data Fitting
C.K. CHUI:
Vertex Splines and their Applications to Interpolation of Discrete
Data 137
F. FONTANELLA:
Shape Preserving Interpolation 183
M. GASCA:
Multivariate Polynomial Interpolation 215
A. LE MEHAUTE:
A Finite Element Approach to Surface Reconstruction 237
L.L. SCHUMAKER:
Reconstructing 3D Objects from Cross-Sections 275
III. Multivariate Splines and Applications
C. DE BOOR:
Quasiinterpolants and Approximation Power of Multivariate
Splines 313
T.N.T. GOODMAN:
Polyhedral Splines 347
vi
C.R. TRAAS:
Practice of Bivariate Quadratic Simplicial Splines 383
IV. Algebraic and Differential Geometric Techniques
W. BOHM:
Algebraic and Differential Geometric Methods in C.A.G.D. 425
J.A. GREGORY, V.K.H. LAD and J. ZHOD:
Smooth Parametric Surfaces and n-Sided Patches 457
C.M. HOFFMANN:
Algebraic and Numerical Techniques for Offsets and Blends 499
List of Attendants 529
Contributed papers 535
Poster Presentations 535
PREFACE
Assembled here is a collection of articles presented at a NATO ADVANCED STU-
DY INSTITUTE held at Puerto de la Cruz, Tenerife, Spain during the period of
July 10th to 21st, 1989. In addition to the editors of these proceedings Professor
Larry L. Schumaker from Vanderbilt University, Nashville, Tennessee, served as a
member of the international organizing committee. The contents of the contribu-
tions fall within the heading of COMPUTATION OF CURVES AND SURFACES
and therefore address mathematical and computational issues pertaining to the dis-
play, modeling, interrogation and representation of complex geometrical objects in
various scientific and technical environments.
As is the intent of the NATO ASI program the meeting was two weeks in
length and the body of the scientific activities was organized around prominent
experts. Each of them presented lectures on his current research activity. We were
fortunate to have sixteen distinguished invited speakers representing nine NATO
countries: W. Bohm (Federal Republic of Germany), C. de Boor (USA), C.K.
Chui (USA), W. Dahmen (Federal Republic of Germany), F. Fontanella (Italy), M.
Gasca (Spain), R. Goldman (Canada), T.N.T. Goodman (UK), J.A. Gregory (UK),
C. Hoffman (USA), J. Hoschek (Federal Republic of Germany), A. Le Mehaute
(France), T. Lyche (Norway), C.A. Micchelli (USA), 1.1. Schumaker (USA), C.
Traas (The Netherlands). The audience consisted of both young researchers as
well as established scientists from twelve NATO countries and several non-NATO
countries.
The lectures were presented in a "tutorial" style. Therefore, as is apparent
from the contributions, an effort has been made by the authors to provide intro-
ductory material preceding the more recent results. Consequently, we hope that
the proceedings will be a source of useful information and will be accessible to a
wide spectrum of readers.
Computing and manipulating with curves and surfaces has become an increas-
ingly important issue in various areas of applied mathematics and technology. New
perspectives and problems arise that do not typically find satisfactory answers in
traditional mathematical subjects. Rather an interplay of techniques and concepts
from various disciplines such as Approximation Theory, Numerical Analysis, Com-
puter Graphics, Differential and Algebraic Geometry is essential. Accordingly, one
has to draw upon a diversity of techniques and algorithms from different fields of
mathematics and computer science. Bringing these fields closer together is necessar-
ily a rather slow process which is hindered by differences in style and the manner
of analysing problems of common interest. A major objective of this conference
was to assist in removing these obstacles. Our common ground was an interest
vii
viii
in mathematical ideas relating to algorithmic issues in the geometry of curves and
surfaces. In Puerto de la Cruz we had the opportunity to bring together scientists
with strong common interests but rather different technical backgrounds within a
most pleasant, congenial and productive atmosphere.
A diversity of relevant topics of current interest are addressed in the main lec-
tures collected in this volume. New concepts in spline theory, the analysis of their
approximation capability, interpolation schemes and finite element approaches are
developed in the light of surface reconstruction. In addition to the question of ac-
curacy, the preservation of shape characteristics such as monotonicity or convexity
has become an important issue in many applications. An overview of existing meth-
ods and indications of future research directions are given. Ideas from Algebraic
and Differential Geometry and their stable numerical realization play an important
role in connection with parametric and implicit surface representations and with
the computation of offset curves and surfaces. A unified setting is developed for a
number of recursive algorithms for surface computation. Such recursive algorithms
are also closely related to subdivision techniques which have many practical appli-
cations for surface display and interrogation. Surprisingly, subdivision also turns
out to be closely related to multiresolution analysis (a concept mainly developed for
image analysis) and wavelet theory. Further topics of great practical significance
are discussed including data compression, conversion between different curve and
surface representations as well as reconstruction from cross-sectional data arising,
for instance, in medical applications.
We have decided to group these contributions under the subtitles: Discrete
Methods for Curve and Surface Representation, Interpolation and Data Fitting,
Multivariate Splines and Applications, and Differential and Algebraic Geometric
Techniques. We are aware of the fact that any such grouping entails a certain
degree of arbitrariness and does not reflect the various interrelations between some
of the contributions. In spite of this we hope that it provides some guideline for
accessing the material. The proceedings only contain the lectures delivered by the
principal speakers. We mention, however, that we plan to publish at a later date
the contributed talks in a special issue of "Monografias de la Academia de Ciencias
de Zaragoza" of Spain.
Obviously, we owe our major debt of gratitude to the NATO Scientific Affairs
Division whose generous financial support, helpful organizational guidelines, assis-
tance and encouragement made it possible for us to organize a successful meeting.
We would also like to gratefully acknowledge additional financial support of the
U.S. Navy, Office of Naval Research, London, and the Ministerio de Educaci6n y
Ciencia of Spain.
The Universities of La Laguna and Zaragoza, especially their respective depart-
ments, Analisis Matematico and Matematica Aplicada, provided us with valuable
assistance in solving logistical and organizational problems, as well as making their
facilities available to us. In this regard, we single out for special thanks Professor
ix
Nacere Hayek, emeritus professor of the University of La Laguna, for his efforts.
No doubt, each participant would agree that the friendly and generous hospitality
of Cabildo de Tenerife and Ayuntamiento de Puerto la Cruz, as expressed by the
social events they organized at Lagos Martianez, Parador Nacional del Teide and
the Casino de Puerto de la Cruz, added significantly to the pleasant ambiance of
the meeting.
Finally, we would like to mention the timely financial support of the Ministerio
de Educaci6n y Ciencia of Spain under the DGICYT program for a sabbatical visit
of Dr. C.A. Micchelli. This allowed us to coordinate and simplify many of the tasks
we faced in organizing the meeting and preparing these conference proceedings.
I.Discrete Methods for CurlJes lind Surfllce Representlltion
