Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms Abel J.P. Gomes ² Irina Voiculescu Joaquim Jorge ² Brian Wyvill ² Callum Galbraith Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms ABC Abel J.P. Gomes Brian Wyvill Universidade da Beira Interior University of Victoria Covilha Victoria BC Portugal Canada Irina Voiculescu Callum Galbraith Oxford University Computing University of Calgary Laboratory (OUCL) Calgary Oxford Canada United Kingdom Joaquim Jorge Universidade Tecnica de Lisboa Lisboa Portugal ISBN 978-1-84882-405-8 e-ISBN 978-1-84882-406-5 DOI 10.1007/978-1-84882-406-5 Springer Dordrecht Heidelberg London New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009926285 °c Springer-Verlag London Limited 2009 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permit- ted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Li- censing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of 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 laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: KuenkelLopka GmbH Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface This book presents the mathematics, computational methods and data struc- tures, as well as the algorithms needed to render implicit curves and surfaces. Implicit objects have gained an increasing importance in geometric modelling, visualisation, animation, and computer graphics due to their nice geometric properties which give them some advantages over traditional modelling meth- ods. For example, the point membership classification is trivial using implicit representations of geometric objects|a very useful property for detecting col- lisions in virtual environments and computer game scenarios. The ease with which implicit techniques can be used to describe smooth, intricate, and ar- ticulatable shapes through blending and constructive solid geometry show us how powerful they are and why they are finding use in a growing number of graphics applications. The book is mainly directed towards graduate students, researchers and developers in computer graphics, geometric modelling, virtual reality and com- puter games. Nevertheless, it can be useful as a core textbook for a graduate- level course on implicit geometric modelling or even for general computer graphics courses with a focus on modelling, visualisation and animation. Fi- nally, and because of the scarce number of textbooks focusing on implicit geometric modelling, this book may also work as an important reference for those interested in modelling and rendering complex geometric objects. Abel Gomes Irina Voiculescu Joaquim Jorge Brian Wyvill Callum Galbraith March 2009 V Acknowledgments The authors are grateful to those who have kindly assisted with the editing of this book, in particular Helen Desmond and Beverley Ford (Springer-Verlag). We are also indebted to Adriano Lopes (New University of Lisbon, Portu- gal), Afonso Paiva (University of S~aoPaulo, Brazil), Bruno Ara´ujo(Technical University of Lisbon, Portugal), Ron Balsys (Central Queensland University, Australia) and Kevin Suffern (University of Technology, Australia) who gen- erously have contributed beautiful images generated by their algorithms; also to Tamy Boubekeur (Telecom ParisTech, France) for letting us to use the datasets of African woman and Moai statues (Figures 8.7 and 8.10). Abel Gomes thanks the Computing Laboratory, University of Oxford, Eng- land, and CNR-IMATI, Genova, Italy, where he spent his sabbatical year writ- ing part of this book. In particular, he would like to thank Bianca Falcidieno and Giuseppe Patan`efor their support and fruitful discussions during his stage at IMATI. He is also grateful to Foundation for Science and Technology, Institute for Telecommunications and University of Beira Interior, Portugal. Irina Voiculescu acknowledges the support of colleagues at the Universi- ties of Oxford and Bath, UK, who originally enticed her to study this field and provided a stimulating discussion environment; also to Worcester College Oxford, which made an ideal thinking retreat. Joaquim Jorge is grateful to the Foundation for Science and Technology, Portugal, and its generous support through project VIZIR. Brian Wyvill is grateful to all past and present students who have con- tributed to the Implicit Modelling and BlobTree projects; also to the Natural Sciences and Engineering Research Council of Canada. Callum Galbraith acknowledges the many researchers from the Graphics Jungle at the University of Calgary who helped shape his research. In particu- lar, he would like to thank his PhD supervisor, Brian Wyvill, for his excellent experience in graduate school, and Przemyslaw Prusinkiewicz for his expert guidance in the domain of modelling plants and shells; also to the University of Calgary and the Natural Sciences and Engineering Research Council of Canada for their support. VII Contents Preface ........................................................V Acknowledgments .............................................VII Part I Mathematics and Data Structures 1 Mathematical Fundamentals ............................... 7 1.1 Introduction . 7 1.2 Functions and Mappings . 8 1.3 Differential of a Smooth Mapping. 9 1.4 Invertibility and Smoothness . 10 1.5 Level Set, Image, and Graph of a Mapping . 13 1.5.1 Mapping as a Parametrisation of Its Image . 13 1.5.2 Level Set of a Mapping . 15 1.5.3 Graph of a Mapping . 20 1.6 Rank-based Smoothness . 24 1.6.1 Rank-based Smoothness for Parametrisations . 25 1.6.2 Rank-based Smoothness for Implicitations . 27 1.7 Submanifolds . 30 1.7.1 Parametric Submanifolds . 30 1.7.2 Implicit Submanifolds and Varieties. 35 1.8 Final Remarks . 40 2 Spatial Data Structures .................................... 41 2.1 Preliminary Notions . 41 2.2 Object Partitionings . 43 2.2.1 Stratifications . 43 2.2.2 Cell Decompositions . 45 2.2.3 Simplicial Decompositions . 49 2.3 Space Partitionings . 51 IX X Contents 2.3.1 BSP Trees . 52 2.3.2 K-d Trees . 55 2.3.3 Quadtrees . 58 2.3.4 Octrees . 60 2.4 Final Remarks . 62 Part II Sampling Methods 3 Root Isolation Methods.................................... 67 3.1 Polynomial Forms . 67 3.1.1 The Power Form . 68 3.1.2 The Factored Form . 68 3.1.3 The Bernstein Form . 69 3.2 Root Isolation: Power Form Polynomials . 72 3.2.1 Descartes' Rule of Signs . 73 3.2.2 Sturm Sequences . 74 3.3 Root Isolation: Bernstein Form Polynomials . 78 3.4 Multivariate Root Isolation: Power Form Polynomials . 81 3.4.1 Multivariate Decartes' Rule of Signs . 81 3.4.2 Multivariate Sturm Sequences . 82 3.5 Multivariate Root Isolation: Bernstein Form Polynomials . 82 3.5.1 Multivariate Bernstein Basis Conversions . 83 3.5.2 Bivariate Case . 83 3.5.3 Trivariate Case . 84 3.5.4 Arbitrary Number of Dimensions . 86 3.6 Final Remarks . 87 4 Interval Arithmetic ........................................ 89 4.1 Introduction . 89 4.2 Interval Arithmetic Operations . 91 4.2.1 The Interval Number . 91 4.2.2 The Interval Operations . 91 4.3 Interval Arithmetic-driven Space Partitionings . 93 4.3.1 The Correct Classification of Negative and Positive Boxes............................................ 94 4.3.2 The Inaccurate Classification of Zero Boxes . 96 4.4 The Influence of the Polynomial Form on IA . 98 4.4.1 Power and Bernstein Form Polynomials . 99 4.4.2 Canonical Forms of Degrees One and Two Polynomials . 101 4.4.3 Nonpolynomial Implicits . 104 4.5 Affine Arithmetic Operations . 105 4.5.1 The Affine Form Number . 105 4.5.2 Conversions between Affine Forms and Intervals . 106 4.5.3 The Affine Operations . 107 Contents XI 4.5.4 Affine Arithmetic Evaluation Algorithms . 108 4.6 Affine Arithmetic-driven Space Partitionings . 109 4.7 Floating Point Errors . 111 4.8 Final Remarks . 114 5 Root-Finding Methods ....................................117 5.1 Errors of Numerical Approximations . 118 5.1.1 Truncation Errors . 118 5.1.2 Round-off Errors . 119 5.2 Iteration Formulas . 119 5.3 Newton-Raphson Method . 120 5.3.1 The Univariate Case . 121 5.3.2 The Vector-valued Multivariate Case . 123 5.3.3 The Multivariate Case . 124 5.4 Newton-like Methods . 126 5.5 The Secant Method . 127 5.5.1 Convergence . 128 5.6 Interpolation Numerical Methods . 131 5.6.1 Bisection Method . 131 5.6.2 False Position Method . 133 5.6.3 The Modified False Position Method . 136 5.7 Interval Numerical Methods . 136 5.7.1 Interval Newton Method . 136 5.7.2 The Multivariate Case . 139 5.8 Final Remarks . 139 Part III Reconstruction and Polygonisation 6 Continuation Methods .....................................145 6.1 Introduction . ..