An Introduction to NURBS.pdf

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A n I n t r o d u c t i o n t o N U R B S With Historical Perspective

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An I n t r o d u c t i o n t o N U R B S With Historical Perspective Professor of Aerospace Engineering United States Naval Academy, Annapolis, MD An Imprint of Elsevier

Executive Editor Diane D. Cerra Director of Production and Manufacturing Yonie Overton Senior Production Editor Cheri Palmer Editorial Coordinator Belinda Breyer Cover Design Ross Carron Design Cover Images (1. to r.) Top row: Courtesy of John C. Dill and David F. Rogers; (~) Dunwell Photography, Steve/The Image Bank; (~)David Chasey/PhotoDisc Center row: Q Rob Magiera/The Image Bank/Picture Quest Bottom row: Courtesy of CGI: R/GA Digital Studios Back Cover Images Courtesy of John C. Dill and David F. Rogers Back Cover Photo Courtesy of Ken Mierzejewski Copyediting, Text Design and Indexing NAR Associates Composition N AR Associates using TEX Illustration Windfall Software and the author Proofreader Jennifer McClain Printer Courier Corporation ACADEMIC PRESS An Imprint of Elsevier 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA http : //www. academicpress, com Academic Press Harcourt Place, 32 Jamestown Road, London, NW1 7BY, United Kingdom http ://www. academicpress, corn Morgan Kaufmann Publishers 340 Pine Street, Sixth Floor, San Francisco, CA 94104-3205, USA h ttp://www, mkp. corn (~) 2001 by Academic Press All rights reserved Printed in the United States of America 07 08 09 10 11 No part of this publication may be reproduced, stored in a retrieval system, or trans- mitted in any form or by any means--electronic, mechanical, photocopying, recording, or otherwise--without the prior written permission of the publisher. 9 8 7 6 Permissions may be sought directly from Elsevier's Science and Technology Rights Department in Oxford, UK. Phone: (44) 1865 843830, Fax: (44) 1865 853333, e-mail: permissions@elsevier.co.uk. You may also complete your request on-line via the Elsevier homepage: http://www.elsevier.com by selecting "Customer Support" and then "Obtaining Permissions". Library of Congress Cataloging-in-Publication D a t a Rogers, David F., date An introduction to NURBS : with historical perspective / David F. Rogers. p. cm. Includes bibliographical references and index. ISBN-13:978-1-55860-669-2 ISBN- 10:1-55860-669-6 1. Spline theory--Data processing. 2. Computer-aided design. I. Title. QA224.R64 2001 511'.42--dc21 This book is printed on acid-free paper. 00-039119

Dedicated to Pierre B&ier for his fundamental contribution Robin Forrest for his insight Bill Gordon for his mathematical contributions Carl de Boor and Maurice Cox for the Cox-de Boor algorithm Steve Coons for his mathematical genius Rich Riesenfeld for B-splines Elaine Cohen, Tom Lyche and Rich Riesenfeld for the Oslo Algorithms Lewie Knapp for rational B-splines Ken VerspriUe for NURBS and all those great and small who advance the state of the art

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C o n t e n t s Chapter 1 1.1 1.2 1.3 1.4 1.5 Chapter 2 2.1 2.2 Preface Curve and Surface Representation Introduction Parametric Curves Extension to Three Dimensions Parametric Line Parametric Surfaces Piecewise Surfaces Continuity Geometric Continuity Parametric Continuity Historical Perspective B~zier Curves" Robin Forrest B~zier Curves B4zier Curve Definition B4zier Curve Algorithm Matrix Representation of Bdzier Curves XV 2 2 3 4 6 8 10 10 11 13 17 18 24 25

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