Analysis and Design of Univariate Subdivision Schemes
This book covers the theory of subdivision curves in detail, which is a prerequisite for that of subdivision surfaces. The book reports on the currently known ways of analysing a subdivision scheme (i.e. measuring criteria which might be important for the
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Editorial Advisory Board Jean-Daniel Boissonnat Gunnar Carlsson Bernard Chazelle Xiao-Shan Gao Craig Gotsman Leo Guibas Myung-Soo Kim Takao Nishizeki Helmut Pottmann Roberto Scopigno Hans-Peter Seidel Steve Smale Peter Schr¨oder Dietrich Stoyan
For further volumes: www.springer.com/series/7580
Malcolm Sabin
Analysis and Design of Univariate Subdivision Schemes With 76 Figures
Malcolm Sabin Numerical Geometry Ltd, John Amner Close 19 CB6 1DT Ely United Kingdom [email protected]
ISSN 1866-6795 e-ISSN 1866-6809 ISBN 978-3-642-13647-4 e-ISBN 978-3-642-13648-1 DOI 10.1007/978-3-642-13648-1 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010934300 Mathematics Subject Classification (2010): 53A04, 65D05, 65D07, 65D10, 65D17, 65D18, 68U05, 68U07 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
‘Subdivision’ is a way of representing smooth shapes in a computer. A curve or surface (both of which contain an infinite number of points) is described in terms of two objects. One object is a sequence of vertices, which we visualise as a polygon, for curves, or a network of vertices, which we visualise by drawing the edges or faces of the network, for surfaces. The other object is a set of rules for making denser sequences or networks. When applied repeatedly, the denser and denser sequences are claimed to converge to a limit, which is the curve or surface that we want to represent. This book focusses on curves, because the theory for that is complete enough that a book claiming that our understanding is complete is exactly what is needed to stimulate research proving that claim wrong. Also because there are already a number of good books on subdivision surfaces. The way in which the limit curve relates to the polygon, and a lot of interesting properties of the limit curve, depend on the set of rules, and this book is about how one can deduce those properties from the set of rules, and how one can then use that understanding to construct rules which give the properties that one wants. This book therefore has four main
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