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.

This book covers combinatorial data structures and algorithms, algebraic issues in geometric computing, approximation of curves and surfaces, and computational topology. Each chapter fully details and provides a tutorial introduction to important concepts and results. The focus is on methods which are both well founded mathematically and efficient in practice. Coverage includes references to open source software and discussion of potential applications of the presented techniques.

This state-of-the-art study of the techniques used for designing curves and surfaces for computer-aided design applications focuses on the principle that fair shapes are always free of unessential features and are simple in design. The authors define fairness mathematically, demonstrate how newly developed curve and surface schemes guarantee fairness, and assist the user in identifying and removing shape aberrations in a surface model without destroying the principal shape characteristics of the model. Aesthetic aspects of geometric modeling are of vital importance in industrial design and modeling, particularly in the automobile and aerospace industries. Any engineer working in computer-aided design, computer-aided manufacturing, or computer-aided engineering will want to add this volume to his or her library. Researchers who have a familiarity with basic techniques in computer-aided graphic design and some knowledge of differential geometry will find this book a helpful reference. It is essential reading for statisticians working on approximation or smoothing of data with mathematical curves or surfaces.

Requires only a basic knowledge of mathematics and is geared toward the general educated specialists. Includes a gallery of color images and Mathematica code listings.

This volume contains revised papers that were presented at the international workshop entitled Computational Methods for Algebraic Spline Surfaces (“COMPASS”), which was held from September 29 to October 3, 2003, at Schloß Weinberg, Kefermarkt (A- tria). The workshop was mainly devoted to approximate algebraic geometry and its - plications. The organizers wanted to emphasize the novel idea of approximate implici- zation, that has strengthened the existing link between CAD / CAGD (Computer Aided Geometric Design) and classical algebraic geometry. The existing methods for exact implicitization (i. e. , for conversion from the parametric to an implicit representation of a curve or surface) require exact arithmetic and are too slow and too expensive for industrial use. Thus the duality of an implicit representation and a parametric repres- tation is only used for low degree algebraic surfaces such as planes, spheres, cylinders, cones and toroidal surfaces. On the other hand, this duality is a very useful tool for - veloping ef?cient algorithms. Approximate implicitization makes this duality available for general curves and surfaces. The traditional exact implicitization of parametric surfaces produce global rep- sentations, which are exact everywhere. The surface patches used in CAD, however, are always de?ned within a small box only; they are obtained for a bounded parameter domain (typically a rectangle, or – in the case of “trimmed” surface patches – a subset of a rectangle). Consequently, a globally exact representation is not really needed in practice.

This volume constitutes the thoroughly refereed post-conference proceedings of the 7th International Conference on Mathematical Methods for Curves and Surfaces, MMCS 2008, held in Tønsberg, Norway, in June/July 2008. The 28 revised full papers presented were carefully reviewed and selected from 129 talks presented at the conference. The topics addressed by the papers range from mathematical analysis of various methods to practical implementation on modern graphics processing units.

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.

This volume brings together selected contributed papers presented at the International Conference of Computational Methods in Science and Engineering (ICCMSE 2006), held in Chania, Greece, October 2006. The conference aims to bring together computational scientists from several disciplines in order to share methods and ideas. The ICCMSE is unique in its kind. It regroups original contributions from all fields of the traditional Sciences, Mathematics, Physics, Chemistry, Biology, Medicine and all branches of Engineering. It would be perhaps more appropriate to define the ICCMSE as a conference on computational science and its applications to science and engineering. Topics of general interest are: Computational Mathematics, Theoretical Physics and Theoretical Chemistry. Computational Engineering and Mechanics, Computational Biology and Medicine, Computational Geosciences and Meteorology, Computational Economics and Finance, Scientific Computation. High Performance Computing, Parallel and Distributed Computing, Visualization, Problem Solving Environments, Numerical Algorithms, Modelling and Simulation of Complex System, Web-based Simulation and Computing, Grid-based Simulation and Computing, Fuzzy Logic, Hybrid Computational Methods, Data Mining, Information Retrieval and Virtual Reality, Reliable Computing, Image Processing, Computational Science and Education etc. More than 800 extended abstracts have been submitted for consideration for presentation in ICCMSE 2005. From these 500 have been selected after international peer review by at least two independent reviewers.

Computational geometry is a borderline subject related to pure and applied mathematics, computer science, and engineering. The book contains articles on various topics in computational geometry based on invited lectures and contributed papers presented during the program on computational geometry at the Morningside Center of Mathematics at the Chinese Academy of Sciences (Beijing). The opening article by R.-H. Wang gives a nice survey of various aspects of computational geometry, many of which are discussed in detail in the volume. Topics of the other articles include problems of optimal triangulation, splines, data interpolation, problems of curve and surface design, problems of shape control, quantum teleportation, and more. The book is suitable for graduate students and researchers interested in computational geometry and specialists in theoretical computer science.