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Springer Undergraduate Mathematics Series Advisory Board M.A.J Chaplain University of Dundee K Erdmann Oxford University L.C.G Rogers University of Cambridge E Suăli Oxford University J.F Toland University of Bath Other books in this series A First Course in Discrete Mathematics I Anderson Analytic Methods for Partial Differential Equations G Evans, J Blackledge, P Yardley Basic Linear Algebra, Second Edition T.S Blyth and E.F Robertson Basic Stochastic Processes Z Brzez´niak and T Zastawniak Complex Analysis J.M Howie Elementary Differential Geometry A Pressley Elementary Number Theory G.A Jones and J.M Jones ´ Searco´id Elements of Abstract Analysis M O Elements of Logic via Numbers and Sets D.L Johnson Essential Mathematical Biology N.F Britton Fields, Flows and Waves: An Introduction to Continuum Models D.F Parker Further Linear Algebra T.S Blyth and E.F Robertson Geometry R Fenn Groups, Rings and Fields D.A.R Wallace Hyperbolic Geometry J.W Anderson Information and Coding Theory G.A Jones and J.M Jones Introduction to Laplace Transforms and Fourier Series P.P.G Dyke Introduction to Ring Theory P.M Cohn Introductory Mathematics: Algebra and Analysis G Smith Linear Functional Analysis B.P Rynne and M.A Youngson Mathematics for Finance: An Introduction to Financial Engineering M Capin´ski and T Zastawniak Matrix Groups: An Introduction to Lie Group Theory A Baker Measure, Integral and Probability, Second Edition M Capin´ski and E Kopp Multivariate Calculus and Geometry, Second Edition S Dineen Numerical Methods for Partial Differential Equations G Evans, J Blackledge, P Yardley Probability Models J Haigh Real Analysis J.M Howie Sets, Logic and Categories P Cameron Special Relativity N.M.J Woodhouse Symmetries D.L Johnson Topics in Group Theory G Smith and O Tabachnikova Vector Calculus P.C Matthews Duncan Marsh Applied Geometry for Computer Graphics and CAD Second Edition With 127 Figures Cover ilustration elements reproduced by kind permission of: Aptech Systems, Inc., Publishers of the GAUSS Mathematical and Statistical System, 23804 S.E Kent-Kangley Road, Maple Valley, WA 98038, USA Tel: (206) 432-7855 Fax (206) 432-7832 email: info@aptech.com URL: www.aptech.com American Statistical Association: Chance Vol No 1, 1995 article by KS and KW Heiner ‘Tree Rings of the Northern Shawangunks’ page 32 fig Springer-Verlag: Mathematica in Education and Research Vol Issue 1995 article by Roman E Maeder, Beatrice Amrhein and Oliver Gloor ‘Illustrated Mathematics: Visualization of Mathematical Objects’ page fig 11, originally published as a CD Rom ‘Illustrated Mathematics’ by TELOS: ISBN 0-387-14222-3, German edition by Birkhauser: ISBN 3-7643-5100-4 Mathematica in Education and Research Vol Issue 1995 article by Richard J Gaylord and Kazume Nishidate ‘Traffic Engineering with Cellular Automata’ page 35 fig Mathematica in Education and Research Vol Issue 1996 article by Michael Trott ‘The Implicitization of a Trefoil Knot’ page 14 Mathematica in Education and Research Vol Issue 1996 article by Lee de Cola ‘Coins, Trees, Bars and Bells: Simulation of the Binomial Process’ page 19 fig Mathematica in Education and Research Vol Issue 1996 article by Richard Gaylord and Kazume nishidate ‘Contagious Spreading’ page 33 fig Mathematica in Education and Research Vol Issue 1996 article by Joe Buhler and Stan Wagon ‘Secrets of the Madelung Constant’ page 50 fig British Library Cataloguing in Publication Data Marsh, Duncan Applied geometry for computer graphics and CAD — 2nd ed — (Springer undergraduate mathematics series) Geometry — Data processing Computer graphics — Mathematics Computer-aided design — Mathematics I Title 516′.0028566 ISBN 1852338016 Library of Congress Cataloging-in-Publication Data Marsh, Duncan Applied geometry for computer graphics and CAD / Duncan Marsh.—2nd ed p cm — (Springer undergraduate mathematics series) Includes bibliographical references and index ISBN 1-85233-801-6 (alk paper) Computer graphics Computer-aided design Geometry—Data processing I Title II Series T385.M3648 2004 516—dc22 2004054958 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted 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 licences issued by the Copyright Licensing Agency Enquiries concerning reproduction outside those terms should be sent to the publishers Springer Undergraduate Mathematics Series ISSN 1615-2085 ISBN 1-85233-801-6 2nd edition Springer-Verlag London Berlin Heidelberg ISBN 1-85233-080-1 1st edition Springer-Verlag London Berlin Heidelberg Springer Science+Business Media springeronline.com © Springer-Verlag London Limited 2005 Printed and bound in the United States of America First published 1999 Second edition 2005 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 Typesetting: Camera ready by the author 12/3830-543210 Printed on acid-free paper SPIN 10946442 To Tine and Emma Preface to the Second Edition The second edition of Applied Geometry for Computer Graphics and CAD features three substantial new sections and an additional chapter The new topics, which include discussions of quaternions, surfaces, solid modelling and rendering, give further insight into the applications of geometry in computer graphics and CAD The text has been revised throughout, and supplemented with further examples and exercises: the second addition contains more than 300 exercises and over 120 illustrations In Chapter 3, a new section introduces quaternions, an important method of representing orientation that is used in computer graphics animation Chapter has been expanded to provide two new sections that focus on the applications of surfaces in CAD: Section 9.6 describes skin and loft surfaces (including Gordon–Coons surfaces), and Section 9.7 discusses geometric modelling The chapter also benefits from additional examples of applications of surfaces; for example, offset and blend surfaces, and shelling and thickening operations A new final chapter addresses rendering methods in computer graphics and CAD, and presents an introduction to silhouettes and shadows There is a web site for the book which contains additional information and further web links: www.springeronline.com/1-85233-801-6/ Cambridgeshire, UK Duncan Marsh vii Preface to the First Edition Applied Geometry for Computer Graphics and CAD explores the application of geometry to computer graphics and computer-aided design (CAD) The textbook considers two aspects: the manipulation and the representation of geometric objects The first three chapters describe how points and lines can be represented by Cartesian (affine) and homogeneous coordinates Planar and spatial transformations are introduced to construct objects from geometric primitives, and to manipulate existing objects Chapter describes the method of rendering three-dimensional objects on a computer screen by application of a linear projection, and the construction of the complete viewing pipeline The material then develops into a study of planar and spatial curves Conics are described in some detail, but the main emphasis is a discussion of the two main curve representations used in CAD packages and in computer graphics, namely, B´ezier and B-spline curves The techniques of the earlier chapters are applied to these curves in order to manipulate and view them The important de Casteljau and de Boor algorithms, for (integral and rational) B´ezier and B-spline curves respectively, are derived and applied The representations of curves lead naturally into surface representations, namely B´ezier, B-spline and NURBS surfaces The transition is relatively painless since many properties of the curve representations correspond to similar surface properties The final chapter introduces curvature for curves and surfaces The book includes more than 250 exercises Some exercises encourage the reader to implement a number of the techniques and algorithms which are discussed These exercises can be carried out using a computer algebra package in order to avoid the complexity of computer programming Certainly this is the most accessible route to obtaining quality graphics Alternatively, the algorithms can be implemented using the reader’s favourite programming language together with a library of graphics routines (e.g PHIGS, OpenGL, or ix x Preface to the First Edition GKS) The two approaches can be mixed as some computer algebra packages can make use of procedures written in programming languages such as C and FORTRAN A number of exercises indicate investigations which would be suitable for coursework, labs or projects The book assumes a knowledge of vectors, matrices, and calculus However, the course has been taught to engineering and computing students with only a little knowledge of these topics; with some additional material, these topics can be taught on a need to know basis Indeed, the material in the book provides a source of motivation for teaching elementary algebra and calculus to non-mathematics students Prerequisite reading on vectors, matrices and continuity of functions can be found in Chapters and of the SUMS series text Introductory Mathematics: Algebra and Analysis by Geoff Smith The author would like to thank a number of people First, the mathematics, computing and engineering students at Napier University who took the modules on which this book is based Second, my colleagues at Napier University; in particular, Dr Winston Sweatman who shares an office with me (need I say more!) Finally, my wife Tine and daughter Emma for their continuing love and support Edinburgh, UK Duncan Marsh Contents Transformations of the Plane 1.1 Introduction 1.2 Translations 1.3 Scaling about the Origin 1.4 Reflections 1.5 Rotation about the Origin 1.6 Shears 1.7 Concatenation of Transformations 1.8 Applications 1.8.1 Instancing 1.8.2 Robotics 1 11 13 15 15 17 Homogeneous Coordinates and Transformations of the Plane 2.1 Introduction 2.1.1 Homogeneous Coordinates 2.2 Points at Infinity 2.3 Visualization of the Projective Plane 2.3.1 Line Model of the Projective Plane 2.3.2 Spherical Model of the Projective Plane 2.4 Transformations in Homogeneous Coordinates 2.4.1 Translations 2.4.2 Scaling about the Origin 2.4.3 Rotation about the Origin 2.5 Concatenation of Transformations 2.5.1 Inverse Transformations 19 19 21 23 24 24 26 27 27 28 29 30 31 xi xii Contents 2.5.2 Rotation about an Arbitrary Point 2.5.3 Reflection in an Arbitrary Line 2.6 Applications 2.6.1 Instancing 2.6.2 Device Coordinate Transformation 2.7 Point and Line Geometry in Homogeneous Coordinates 33 34 36 36 37 38 Homogeneous Coordinates and Transformations of Space 3.1 Homogeneous Coordinates 3.2 Transformations of Space 3.2.1 Translations 3.2.2 Scalings and Reflections 3.2.3 Rotations about the Coordinate Axes 3.2.4 Rotation about an Arbitrary Line 3.2.5 Reflection in an Arbitrary Plane 3.3 Applications 3.3.1 Computer-aided Design 3.3.2 Orientation of a Rigid Body 3.4 Geometric Methods for Lines and Planes in Space 3.5 Quaternions 41 41 42 42 43 43 45 47 49 49 50 52 56 Projections and the Viewing Pipeline 4.1 Introduction 4.2 Projections of the Plane 4.3 Projections of Three-dimensional Space 4.4 The Viewplane Coordinate Mapping 4.5 The Viewing Pipeline 4.6 Classification of Projections 4.6.1 Classification of Parallel Projections 4.6.2 Classification of Perspective Projections 67 67 68 72 76 80 85 85 90 Curves 95 5.1 Introduction 95 5.2 Curve Rendering 98 5.3 Parametric Curves 99 5.4 Arclength and Reparametrization 102 5.5 Application: Numerical Controlled Machining and Offsets 107 5.6 Conics 109 5.6.1 Classification of Conics 112 5.6.2 Conics in Standard Form 116 5.6.3 Intersections of a Conic with a Line 121 5.6.4 Parametrization of an Irreducible Conic 124 Solutions 335 6.24 When the control points are collinear the convex hull is a line segment implying that the B´ezier curve is contained in a line segment 6.25 (a) b0 (3, 4), b1 (5, 5), b2 (6, 3), b3 (4, 2) (b) b0 (0, 0), b1 (−1, 2), b2 (1, 3), b3 (2, 1) (c) b0 (0, 0), b1 (1, 2), b2 (−1, 3), b3 (−2, 1) 6.27 b10 (1.5, 0.75), b11 (3.5, 3.5), b12 (5.5, 4.25), b20 (2, 1.4375), b21 (4, 3.6875), b30 (2.5, 2.0) B(0.25) = (2.5, 2.0) 6.28 (3.456, 1.3776) 6.32 (a) b10 (0.4, 0.1), b11 (1.2, 0.6), b12 (2.2, 0.9), b20 (0.6, 0.225), b21 (1.45, 0.675), b30 (0.8125, 0.3375) B(0.25) = (0.8125, 0.3375) (b) Bleft : (0.2, 0.0), (0.4, 0.1), (0.6, 0.225), (0.8125, 0.3375); Bright : (0.8125, 0.3375), (1.45, 0.675), (2.2, 0.9), (3.4, 0.0) 6.34 (a) B(1/3) = (0.7, 0.275) Chapter 7.1 b10 (2.6, 6.7, 4.3), b11 (4.3, 6.6, 4.7), b12 (4.4, 7.1, 3.7), b20 (3.11, 6.67, 4.42), b21 (4.33, 6.75, 4.40), b30 (3.476, 6.694, 4.414) B(0.3) = (3.476, 6.694, 4.414) 7.2 First derivative: ((4, 3) − (6, 3)) = (−6, 0), ((1, 2) − (4, 3)) = (−9, −3), ((−1, 2) − (1, 2)) = (−6, 0) Second derivative: ((−9, −3) − (−6, 0)) = (−6, −6) 7.4 n (n − 1) (b2 − 2b1 + b0 ) and n (n − 1) (bn − 2bn−1 + bn−2 ) 7.5 Bi,n (t) n n i(1 − t)n−i ti−1 − (n − i) (1 − t)n−i−1 ti i i n−1 n−1 = n (1 − t)n−i ti−1 − n (1 − t)n−i−1 ti i−1 i = n (Bi−1,n−1 (t) − Bi,n−1 (t)) = 7.7 b0 (1, 4), b1 (1, 3), b2 (0, 5) 7.8 (−3 + 8t − 6t2 , −3 + 4t + t2 ) 7.14 C since c0 = b3 = (3, 6) For visual continuity µ3(b3 − b2 ) = 3(−2, 2) and µ3(c1 − c0 ) = 3(−1, 1) Then take µ = 1/2 Change b2 to (4, 5) to obtain C 336 Solutions 7.20 (a)–(c) follow from formulae in the text with w0 = w2 = b0 (1−0.5)2 +w1 b1 2(0.5)(1−0.5)+b2 (0.5)2 (1−0.5)2 +w1 2(0.5)(1−0.5)+(0.5)2 = 1− w1 1+w1 b0 +b2 w1 1+w1 + = b0 +2w1 b1 +b2 2(1+w1 ) b1 = S 7.21 w10 = 2.6, w11 = 4.2, w12 = 2.6, w20 = 3.56, w21 = 3.24, w30 = 3.38, and b10 (5.769, 4.769), b11 (5.571, 3.857), b12 (4.538, 2.308), b20 (5.629, 4.124), b21 (5.074, 3.111), b30 (5.309, 3.539) ⎞ ⎛ ⎞ ⎛ − 29 −33 21 15 − 18 ⎟ ⎜ 27 −39 ⎜ 17 15 ⎟ ⎟ VC = ⎜ 18 − ⎟ 7.24 M = ⎜ ⎝ ⎠ ⎝ ⎠ 0 −60 −9 108 84 60 −48 ⎛ ⎞ ⎛ ⎞ −2 54 330 −78 ⎜ 3/2 5/2 4/2 1/2 ⎟ ⎜ −30 −12 ⎟ ⎜ ⎟ M · VC = ⎜ −24 ⎟ ⎝ −4 ⎠ ⎝ 12 12 −216 −360 −168 ⎠ 135 −135 −135 The projected control points are (−0 692, −4 231), (2, 2.5), (1.286, 2.143), (−1, 1) and weights −78, −12, −168, −135 7.25 An integral curve is obtained when weights are equal Rewrite the expression for the weights wi = (n1 , n2 , n3 ) · bi v4 − (n1 , n2 , n3 ) · (v1 , v2 , v3 ) Weights are equal if and only if either (i) v4 = 0, projection is parallel and wi = −(n1 , n2 , n3 ) · (v1 , v2 , v3 ) for all i, or (ii) v4 = 0, projection is perspective, (n1 , n2 , n3 ) · bi = 0, and the control points lie in a plane parallel to the viewplane Chapter 8.1 B0 (t) = −t+ (t − 2) for t ∈ [2, 3], B1 (t) = (t 2 b0 + − 23 + t − (t − 2) −t+ (t − 3) 2 b1 + 12 (t − 2)2 b2 b1 + − 52 + t − (t − 3) b2 + − 3) b3 for t ∈ [3, 4] 8.2 B0 (t) = 53 − 12 t + 12 (t − 3)2 − 16 (t − 3)3 b0 + 23 − (t − 3)2 + 12 (t − 3)3 b1 + − 43 + 12 t + 12 (t − 3)2 − 12 (t − 3)3 b2 + 16 (t − 3)3 b3 for t ∈ [3, 4] 8.3 B0 (t) = (t − 1) b0 + 2t − 32 t2 b1 + 12 t2 b2 , B1 (t) = −t+ (t − 1) b1 + − 12 + t − (t − 1) 2 b2 + 12 (t − 1) b3 , Solutions 337 B2 (t) = −t+ (t − 2) b2 + − 32 + t − (t − 2) 2 b3 + (t − 2) b4 8.5 B(2.5) = (6.25, −0.25), B(4.2) = (4.28, 4.0) 8.6 b5 (0, 0), b6 (2, 0), b7 (4, 2), t6 = 6, t7 = 7, t8 = 8, t9 = 9, t10 = 10, t11 = 11 8.12 B(2.5) = (6.25, −0.25), B(4.2) = (4.28, 4.0) 8.13 B(2.4) = (4.04, 2.48) (1) (1) 8.18 B (2.8) = (0.8022, −3.3045) b0 (1.875, 9.375), b1 (6.0, 0.0), (1) (1) b2 (1.579, −3.158), b3 (−2.857, −4.286) (1) (1) (1) 8.19 b0 (4/3, 2), b1 (2, −2), b2 (2, 4/3), B (6.2) = (1.733, −0.4), B (7.4) = (2, −0.667) 8.24 wi Ni,d (t) n j=0 wj Nj,d (t) n i=0 = n i=0 n j=0 wi Ni,d (t) wj Nj,d (t) knots 4, 5, 7, 8, 10 = 8.27 B (2.2) = (1.715, −11.853) 8.28 B (0.5) = (0, −4) B (0.8) = (7.101, 2.959) 8.31 B(0.65) = (−0.6897, −0.7241) Chapter 9.2 First derivative with respect to s: (1,0) p0,0 = ((4, 3, 1) − (2, 2, 0)) = (4, 2, 2) , (1,0) p1,0 = ((6, 2, 0) − (4, 3, 1)) = (4, −2, −2) , (1,0) p0,1 (1,0) (1,0) = ((4, 5, 3) − (2, 4, 1)) = (4, 2, 4) , similarly p1,1 = (4, −4, −4) , (1,0) p0,2 = (4, 0, 2) , p1,2 = (4, −2, −2) 9.4 (a) The tangent vectors at S(0, 0): n (p1,0 − p0,0 ) and p (p0,1 − p0,0 ); S(0, 1): n (p0,p − p0,p−1 ) and p (p1,p − p0,p ); S(1, 0): n (pn,0 − pn−1,0 ) and p (pn,1 − pn,0 ); S(1, 1): n (pn,p − pn−1,p ) and p (pn,p − pn,p−1 ) (b) The normal at S(0, 0) is np (p1,0 − p0,0 ) × (p0,1 − p0,0 ); at S(0, 1) is np (p0,p − p0,p−1 ) × (p1,p − p0,p ) etc 9.5 (All the points listed in the order p0,0 , p1,0 , etc.) (2, 0, 1), (1, 0, 2), (3, 0, 3), (1, 0, 4), (1, 0, 5); (2, 2, 1), (1, 1, 2), (3, 3, 3), (1, 1, 4), (1, 1, 5); 338 Solutions (−2, 2, 1), (−1, 1, 2), (−3, 3, 3), (−1, 1, 4), (−1, 1, 5); (−2, 0, 1), (−1, 0, 2), (−3, 0, 3), (−1, 0, 4), (−1, 0, 5); (−2, −2, 1), (−1, −1, 2), (−3, −3, 3), (−1, −1, 4), (−1, −1, 5); (2, −2, 1), (1, −1, 2), (3, −3, 3), (1, −1, 4), (1, −1, 5); (2, 0, 1), (1, 0, 2), (3, 0, 3), (1, 0, 4), (1, 0, 5) 9.6 b0,0 (2, 3, 0), b1,0 (1, 5, 2), b2,0 (1, 7, −1), b3,0 (2, 9, −3), b0,1 (4, 7, −4), b1,1 (3, 9, −2), b2,1 (3, 11, −5), b3,1 (4, 13, −7) 9.7 Control points as for Exercise 9.6, weights w0,0 = w0,1 = 1, w1,0 = w1,1 = 2, w2,0 = w2,1 = 3, w3,0 = w3,1 = 9.9 b0,0 (0, 0, 0), b1,0 (0, a, 0), b2,0 (a, 2a, 0), b0,1 (0, 0, 1), b1,1 (0, a, 1), b2,1 (a, 2a, 1) Parabolic cylinder can be obtain by sweeping a line segment along a quadratic B´ezier curve 9.10 NURBS sphere has control points: (Listed in the order p0,0 , p1,0 , etc) (1, 0, 0), (1, 0, 1), (−1, 0, 1), (−1, 0, 0), (−1, 0, −1), (1, 0, −1), (1, 0, 0); (1, 1, 0), (1, 1, 1), (−1, −1, 1), (−1, −1, 0), (−1, −1, −1), (1, 1, −1), (1, 1, 0); (−1, 1, 0), (−1, 1, 1), (1, −1, 1), (1, −1, 0), (1, −1, −1), (−1, 1, −1), (−1, 1, 0); (−1, 0, 0), (−1, 0, 1), (1, 0, 1), (1, 0, 0), (1, 0, −1), (−1, 0, −1), (−1, 0, 0); (−1, −1, 0), (−1, −1, 1), (1, 1, 1), (1, 1, 0), (1, 1, −1), (−1, −1, −1), (−1, −1, 0); (1, −1, 0), (1, −1, 1), (−1, 1, 1), (−1, 1, 0), (−1, 1, −1), (1, −1, −1), (1, −1, 0); (1, 0, 0), (1, 0, 1), (−1, 0, 1), (−1, 0, 0), (−1, 0, −1), (1, 0, −1), (1, 0, 0) Weights: wi,0 = {1, 0.5, 0.5, 1, 0.5, 0.5, 1}, wi,1 = wi,2 = {0.5, 0.25, 0.25, 0.5, 0.25, 0.25, 0.5}, wi,3 = {1, 0.5, 0.5, 1, 0.5, 0.5, 1}, wi,4 = wi,5 = {0.5, 0.25, 0.25, 0.5, 0.25, 0.25, 0.5}, wi,6 = {1, 0.5, 0.5, 1, 0.5, 0.5, 1} Knots for s and t 0, 0, 0, 0.25, 0.5, 0.5, 0.75, 1, 1, Solutions 339 9.13 Control points: (−1, 0, 1), (0, 0, 0), (1, 0, 1); (−1, −1, 1), (0, 0, 0), (1, 1, 1); (1, −1, 1), (0, 0, 0), (−1, 1, 1); (1, 0, 1), (0, 0, 0), (−1, 0, 1); (1, 1, 1), (0, 0, 0), (−1, −1, 1); (−1, 1, 1), (0, 0, 0), (1, −1, 1); (−1, 0, 1), (0, 0, 0), (1, 0, 1) Weights: wi,0 = wi,2 = {1, 0.5, 0.5, 1, 0.5, 0.5, 1}, wi,1 = {3, 1.5, 1.5, 3, 1.5, 1.5, 3} 9.19 (1 − s)(3t2 + 4, 2t2 , −t) + s(2t, −t4 , 2t + 4) = (4 − 4s + 2st + 3t2 − 3st2 , 2t2 − 2st2 − st4 , 4s − t + 3st) 9.20 (a) (3(1 − 3s2 + 2s3 )t2 + (3s2 − 2s3 )(2t + 10) + s − 2s2 + s3 , 2(1 − 3s2 + 2s3 )t2 + 3(3s2 − 2s3 )t + s − s2 , (1 − 3s2 + 2s3 )t + 2(3s2 − 2s3 )t2 ) = (3t2 − 9t2 s2 + 6t2 s3 + 6ts2 + 28s2 − 4ts3 − 19s3 + s, 2t2 − 6t2 s2 + 4t2 s3 + 9ts2 − 6ts3 + s − s2 , t − 3ts2 + 2ts3 + 6t2 s2 − 4t2 s3 ) (b) 3(1 − 3s2 + 2s3 )t2 + (3s2 − 2s3 )(2t + 10) + (s − 2s2 + s3 )s, 2(1 − 3s2 + 2s3 )t2 + 3(3s2 − 2s3 )t − 2s + 4s2 − 2s3 − (−s2 + s3 )s, (1 − 3s2 + 2s3 )t + 2(3s2 − 2s3 )t2 + (s − 2s2 + s3 )s + 2(−s2 + s3 )s) = (3t2 − 9t2 s2 + 6t2 s3 + 6ts2 + 31s2 − 4ts3 − 22s3 + s4 , 2t2 − 6t2 s2 + 4t2 s3 + 9ts2 − 6ts3 − 2s + 4s2 − s3 − s4 , t − 3ts2 + 2ts3 + 6t2 s2 − 4t2 s3 + s2 − 4s3 + 3s4 340 Solutions 9.21 (1 − s) (1 − t) − 20 (1 − s) t + s (1 − t) − 27 st + (1 − s) (−5 + 25 t) + s −8 (1 − t) + 20 (1 − t) t + 27 t2 + (1 − t) −5 (1 − s) − 18 (1 − s) s − 21 (1 − s) s2 − s3 + t 20 (1 − s) + 44 (1 − s) s + 27 s2 , (1 − s) (1 − t) − (1 − s) t − st + (1 − s) (−5 + t) + s 32 (1 − t) t + t2 + (1 − t) −5 (1 − s) − 12 (1 − s) s − (1 − s) s2 + t (1 − s) + 10 (1 − s) s + s2 , − 10 s (1 − t) − 10 st + s 10 (1 − t) + (1 − t) t + 10 t2 + (1 − t) (1 − s) s + 21 (1 − s) s2 + 10 s3 + t (1 − s) s + 10 s2 = (−5 + 25 t − s + st − st2 + ts2 , − + t + s + 26 st − 25 st2 + s2 − s3 − ts2 + ts3 , s − 13 st + 12 st2 + s2 − s3 − ts2 + ts3 ) Chapter 10 10.1 (a) C (t) = (1, sinh (t/c)), C (t) = 0, 1c cosh (t/c) , t(t) = (1/ cosh (t/c) , (t/c)), n(t) = (− (t/c) , 1/ cosh (t/c)) κ = 1/c cosh2 (t/c) √ 10.3 (a) θ(s) = arcsin s, and x = 12 s − s2 + 12 arcsin s, y = 12 s2 (b) √ √ √ √ √ θ(s) = s, and x = s sin ( s) cos ( s) + cos2 ( s) − 1, y = √ √ √ √ √ sin ( s) cos ( s) − s cos2 ( s) + s √ 2 (c) √ θ(s) = − arctan(s/a), and x = a ln((s + a + s )/a), y = a − 2 a +s 10.4 Reparametrize the curve so that C(s) is unit speed If κ(s) = for all s, then C (s) = κ(s)N(s) implies that C (s) = (0, 0) Thus C (s) = (a1 , a2 ) for some constants a1 and a2 Finally, integrating gives C(s) = (a1 s+b1 , a2 s+b2 ) The result can also be deduced from the fundamental theorem of plane curves: θ = κ(u) du = du etc Solutions 341 10.5 In polar coordinates: (x, y) = (r(θ) cos θ, r(θ) sin θ) Then (x , y ) = (r (θ) cos θ − r(θ) sin θ, r (θ) sin θ + r(θ) cos θ) So the arclength is b 2 (r (θ) cos θ − r(θ) sin θ) + (r (θ) sin θ + r(θ) cos θ) dθ a (r(θ)2 + (r (θ))2 )dθ = Further, (x , y ) = (r (θ) cos θ − 2r (θ) sin θ − r(θ) cos θ, r (θ) sin θ + 2r (θ) cos θ − r(θ) sin θ) and substitution into the formula for curvature gives κ = 10.6 (a) κ = 2(2+2 cos t)1/2 2(r )2 −rr +r (r +(r )2 )3/2 E(t) = (t − sin t, + cos t) ă (b) C(t) = (−a sin t, b cos t), C(t) = (−a cos t, −b sin t) √ −b cos t , a2 sin2 t+b2 cos2 t n(t) = κ(t) = √ −a sin t a2 sin2 t+b2 cos2 t , ab (a2 sin2 t+b2 cos2 t)3/2 a2 −b2 a Evolute is E(t) = cos3 t, − a 10.7 (a) Offset is (−3 sin t, cos t) + (b) κ(t) = , (4 sin2 t+9 cos2 t)3/2 −b2 b sin3 t d (9 sin2 t+4 cos2 t)1/2 κ(t) ˙ = (−2 cos t, −3 sin t) 90 sin t cos t (4 sin2 t+9 cos2 t)5/2 Maxima and min- ima occur when κ(t) ˙ = Hence t = 0, π/2, π, 3π/2 corresponding to the points (3, 0), (0, 2), (−3, 0), (0, −2) (c) Maximum and minimum values of curvature are 34 and 29 Therefore the maximum and minimum radii of curvature are 92 and 43 Hence the maximum radius the ball cutter can be is 92 , otherwise the cutter will be too large to cut the ellipse at the points (0, 2) and (0, −2) ˙ ˙ = 10.10 C(t) = (−4 sin t, −5 cos t, sin t), C(t) t = − 45 sin t, − cos t, 35 sin t n = − 45 cos t, sin t, 35 cos t b = − 35 , 0, − 45 κ = 1/5, τ = Curve is a circle radius sin t − 10.11 t = − 10 n = − 35 cos t + √ b= 3 10 5 sin t, sin t, − 35 sin t − cos t, , √ sin t + √ 3 cos t, 10 cos t − √ cos t, − 3103 cos t + √ , sin t, 12 342 Solutions κ = 1/20, τ = 10.12 κ = √ √ 3/20 − cos2 t −3/2 , τ = Curve is an ellipse 10.13 (a) κ = τ = 1/3(1 + t2 )2 (b) κ = τ = 1/ + cos2 t + 1 2 − 2t2 1/2 + cos2 t + (c) κ = 3 2t 1/2 /(3 + cos t)3/2 , −3/2 + t2 + t4 , τ = −1/2 √ √ √ 26 10.17 (a)√a √= 17, b = 13 2, c = 39 so κ = 867 17 = 175, τ = 17 = 171 (b) κ = 0.185, τ = 0.334 34 10.14 κ = τ = √ 10.24 (a) 2(1 + 4u2 + 4v )−1/2 and 2(1 + 4u2 + 4v )−3/2 (c) r and cos u(R + r cos u)−1 10.25 (a) K = −(1 + u2 + v )−2 , H = −uv(1 + u2 + v )−3/2 10.26 (b) (0, 0, 0) 10.32 π π π kmax cos2 θ + kmin sin2 θ dθ = (θ + cos θ sin θ) kmax + (θ − cos θ sin θ) kmin π = (kmax + kmin ) Chapter 11 11.2 N = √1+4s12 +4t2 (2s, 2t, 1) So at (0.5, 0.5, −0.5), N = √13 (1, 1, −1) The incident ray has direction (0, 10, 20) − (0.5, 0.5, −0.5), giving L = √ √ √ √ √ √ 227 227 19 227 41 227 227 The an, 681 , 681 R = − 432043 , − 1032043227 , − 772043 − 681 gle of incidence is 1.272609737 radians 11.4 V · N = (0, 1, 0) · (−t − 2s, −s + t2 , 1) = −s + t2 So the silhouette points satisfy s = t2 giving the curve t2 , t, 23 t3 + t4 11.5 V · N = (1, 0, 0) · (1 − t2 − s2 , −2st, 1) = − s2 − t2 So the silhouette points satisfy s2 + t2 = defining the unit circle in the (s, t)-plane The circle may be parametrized as (s, t) = (cos θ, sin θ) and substituting into the parametric equation of the surface gives the silhouette curve cos θ, sin θ, cos θ sin2 θ − cos θ + 1/3 cos3 θ 11.6 V · N = (0, 1, 0) · (−18st, − 9s2 + 9t2 , 1) = − 9s2 + 9t2 = The substitution (s, t) = ( 13 cosh θ, 13 sinh θ) gives the silhouette curve 1 1 cosh θ, sinh θ, cosh θ sinh θ − sinh θ − sinh θ 11.7 The plane ax + by + cz + d = has normal N = (a, b, c) For a parallel projection in the direction V(v0 , v1 , v2 ), V · N = av0 + bv1 + cv2 In Solutions 343 general, av0 + bv1 + cv2 = and there are no silhouette points When av0 + bv1 + cv2 = 0, every point is a silhouette point (The condition corresponds to when V is perpendicular to N) For a perspective projection from the viewpoint V(v0 , v1 , v2 ), the condition for a silhouette is (v0 − x, v1 − y, v2 − z) · N = av0 + bv1 + cv2 − (ax + by + cz) = giving av0 + bv1 + cv2 + d = The condition is satisfied if and only if V is a point on the plane and every point of the plane is a silhouette point 11.11 If the viewpoint V lies inside the sphere centred at C, radius r, then |V − C| < r Using Exercise 11.7.(a), d > r and r1 , given by 11.7(b), has no solution (λ1 − 1)/λ, t 11.14 The silhouettes are two lines 1/λ, ± 11.15 V = (10, 0, 0, 1) and ⎛ ⎜ ⎜ ⎜ Q=⎜ ⎜ −1 ⎝ 0 −1 0 −1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ −4 The silhouette plane is: 40x−10z −4 = Solving for x and substituting 2 into the equation of the quadric gives the conic: − 99 25 − 5/4 z + y = which can be parametrized by (y, z) = ± 99/25 cosh t, 396/125 sinh t A parametrization for the silhouette follows 11.17 V = (0, 0, 1), N = (−f (u)g (u) cos v, −f (u)g (u) sin v, f (u)f (u)) and V · N = f (u)f (u) = Since f (u) = silhouette points satisfy f (u) = For each u0 such that f (u0 ) = there is a silhouette circle given by (f (u0 ) cos v, f (u0 ) sin v, g(u0 )) 11.18 Since VQPT is a × matrix it is equal to its own transpose, namely, PQVT 11.20 ⎛ VQPT = λ Therefore x = λ 1 ⎜ ⎜ ⎝ 0 and y = ± √ λ2 −1 λ 0 ⎞⎛ 0 x ⎜ y 0 ⎟ ⎟⎜ 0 ⎠⎝ z −1 ⎞ ⎟ ⎟ = λx − = ⎠ giving two lines ( λ1 , ± √ λ2 −1 , z) λ References [1] Abhyankar, S S and Bajaj, C, ‘Automatic parametrization of rational curves and surfaces I: Conics and conicoids Computer-Aided Design Vol 19, pp11–14, 1987 [2] B´ezier, P, ‘Style, mathematics and NC’ Computer-Aided Design Vol 22 No 9, pp523–526, 1990 [3] Boehm, W and Prautzsch, H, ‘The insertion algorithm’ Computer-Aided Design Vol 17 No 2, pp58–59, 1985 [4] Braid, I C, Hillyard, R C, and Stroud I A, ‘Stepwise construction of polhedra in geometric modelling’ in Mathematical Methods in Computer Graphics and Design, ed K W Brodlie, pp123-141, Academic Press, 1980 [5] Coolidge, J L, A History of the Conic Sections and Quadric Surfaces OUP, 1945 [6] Davis, P, ‘B-splines and geometric design’, SIAM News Vol 29 No 5, 1996 [7] Dill, J ‘An application of colour graphics to the display of surface curvature’ Computer Graphics Vol 15, pp153–161, 1981 [8] Do Carmo, M P, Differential Geometry of Curves and Surfaces PrenticeHall, 1976 [9] Farin, G, Curves and Surfaces for Computer-Aided Geometric Design Third Edition Academic Press, 1993 [10] Forrest, A R, ‘Interactive interpolation and approximation by B´ezier polynomials’, Computer-Aided Design Vol 22 No 9, pp527–537, 1990 Originally published in The Computer Journal Vol 15 No 1, pp71–79, 1972 345 346 References [11] Gibson, C.G., Elementary Geometry of Algebraic Curves Cambridge University Press, 1998 [12] Haralick, R M and Shapiro, L G, Computer and Robot Vision AddisonWesley, 1992 [13] Hoschek, J and Lasser, D, Fundamentals of Computer Aided Geometric Design A K Peters, 1993 [14] Howard, T L J, Hewitt, W T, Hubbold, R J, and Wyrwas, K M, A Practical Introduction to PHIGS and PHIGS PLUS Addison-Wesley, 1991 [15] Lane, J and Riesenfeld, R, ‘A geometric proof for the variation diminishing property of B-spline approximation’ J of Approximation Theory Vol 37, pp1–4, 1983 [16] Mă antylă a, M, An Introduction to Solid Modeling, Computer Science Press, Maryland, 1988 [17] Munchmeyer, F, ‘On surface imperfections’ In R.Martin, editor, The Mathematics of Surfaces II, pp459–474 OUP, 1987 [18] Munchmeyer, F, ‘Shape interrogation: a case study’ In G.Farin, editor, Geometric Modelling: Algorithms and New Trends, pp291–301 SIAM, Philadelphia, 1987 [19] Phong, B-T, ‘Illumination for computer-generated pictures’ Comm ACM, Vol 18, No 6, pp311–317, June 1975 [20] Piegl, L and Tiller, W, The NURBS Book Springer-Verlag, 1995 [21] Rogers, D F and Adams, J A, Mathematical Elements for Computer Graphics Second Edition McGraw-Hill, 1990 [22] Schoenberg, I, ‘Contributions to the problem of approximation of equidistant data by analytic functions’, Quart Appl Math Vol 4, pp45–99, 1946 [23] Sederberg, Th W, Anderson, D C and Goldman, R N, ‘Implicit representation of parametric curves and surfaces’ Computer Vision, Graphics and Image Processing Vol 28, pp72–84, 1984 [24] Semple, J G and Kneebone, G T, Algebraic Projective Geometry OUP, 1952 [25] Smith, G, Introductory Mathematics: Algebra and Analysis SpringerVerlag, 1998 [26] Sommerville, D M Y, Analytical Conics Bell and Sons, 1945 [27] Spivak, M, Calculus W.A.Benjamin, 1967 Index affine invariance 147, 177, 195, 214, 236 ambient light 299, 304 apparent contour 310 apparent cusps 319 attenuation 305 axonometric projection see projection B-rep 263 B-spline 194 – basis 187, 192 – closed 200, 235 – curve 188 – derivatives 207, 216, 238 – integral 188 – NURBS 212 – open 196, 235 – open uniform 202 – periodic 200, 235 – rational 213 – surface 234 – uniform 198 Bernstein – polynomial 141, 144 B´ezier – control point 135 – – homogeneous 175 – control polygon 135–137, 141 – cubic 137 – curve 136, 141, 161 – – curvature 283 – – torsion 283 – derivatives 162 – endpoint-interpolation 139, 147, 236 – integral 141 – linear 136 – piecewise 168 – properties 147 – quadratic 136 – rational 175 – rendering 157 – subdivision 154 – surface 234 binomial 142 blend 233 Boehm algorithm 221 breakpoints 169, 192 C k -continuity 99, 170, 226 CAD see computer-aided design, 260 Cartesian plane catenary 273 cavalier projection see projection centre of perspectivity 68 clip 76 clothoid 273 CMY 299 computer-aided design 49, 135 concatenation see transformation conic 109 – applications 132 – central 113 – conversion 127 – degenerate 109 – directrix 110 – discriminant 112 – eccentricity 110 – ellipse 109, 116, 177 – focus 110 – hyperbola 109, 116, 134, 177 – irreducible 112, 114 347 348 Index – parabola 109, 116, 132, 177 – parametrization 124 – reducible 112, 114 – spatial 130 continuity 99, 192, 195, 214, 226, 253 control point 187 conversion – to B´ezier form 166 convex hull 146, 147, 177, 195, 214, 236 Coons surface 256 coordinate – functions 96 coordinate curve 226 coordinates – Cartesian – homogeneous 14, 20, 41 Plă ucker 54 viewplane see viewplane Cornu spiral 273 CSG 261 curvature 267, 275 – B´ezier curves 283 – normal 286 – principal 286 – vector 275 curve – algebraic 96 – curvature 267, 275 – implicit 96 – non-parametric explicit 96 – parametric 96 – polynomial 96 – rational 96 – regular 99 – segment 96 cusp 138 cycloid 273 equivalence relation 21 Euler angles 51 Euler–Poincar´e formula 265 evolute 274 flat point 291 flat shading 307 font design 203 foreshortening ratio 85 Frenet frame 276 Frenet–Serret formulae 277 Gk -continuity 172, 253 geometric continuity 172 geometric modelling 260 gimbal lock 51, 64 Gordon–Coons surface 256 Gouraud shading 307 graphical primitive 1, 15 Hamilton 56 helix 279 Hermite 254 hidden line 318 homogeneous – control point 213 – coordinates 19, 21 – equation 24, 38 Horner’s method 98 hotspot 303 HSV 298 hue 298 hyperbolic point 291 identity see transformation image implicit 2, 225 incident ray 300 inflection 138 instancing 1, 15, 36 intensity 301, 305 intersection – line and B´ezier curve 158 – line and conic 121 – three planes 53 – two B´ezier curves 159 – two lines 39 inverse see transformation isometric projection see projection de Boor algorithm 205 – rational 218 de Casteljau 151, 152 – rational 180 deformation 204 degree 96, 187 degree raising 146 Denavit–Hartenberg 17 device coordinate transformation 37, 80 device window 76 diffuse reflection 300 dimetric projection see projection dual 40 knot insertion 221 knot vector 187 elliptic point Lambert’s Law 291 301 Index 349 Lambertian surfaces 301 light – ambient 299 – attenuation 305 – directional 299 – distributive 299 – intensity 301 – point source 299 – specular 299 line – through two points 39 line coordinates 54 line vector 39, 52 local control 195, 214 local support 192 lofting 254 logarithmic spiral 273 Monge patch 227 monomial form 166 morphing 203 natural equation 272 normal – line 101 – vector 100 normal plane 276 numerically controlled machining 232 NURBS see B-spline 107, – through three points 52 plane vector 52 point at infinity 23, 25, 26, 41 positivity 192 principal curvature 286 principal direction 286 projection – centre of perspectivity 72 – line 68 – of B´ezier curve 181 – of NURBS curve 214 – parallel 69, 72 – – axonometric 86 – – cavalier 88 – – dimetric 87 – – isometric 87 – – oblique 87 – – orthographic 86 – – trimetric 87 – perspective 68, 72, 90 – – one-point 91 – – three-point 91 – – two-point 91 – viewpoint 72 projective invariance 178, 236 projective plane 19, 23, 24 P2 see projective plane projective space 41 object oblique projection see projection offset 107, 232, 296 order 187 orientation 50 orthogonal change of coordinates 33 orthographic projection see projection quaternions 51, 56 – algebraic properties – animation 65 – conjugate 59 – interpolation 65 – inverse 59 – polar form 60 – rotations 62 – unit 59 P3 see projective space parabolic point 291 parallel curve 107 parallel projection see projection parameter curve 226 parametric 2, 226 parametrization 96 partition of unity 192 perspective projection see projection Phong 303, 309 picture elements 15 piecewise polynomial 170, 187, 192 plane – Cartesian – projective 19, 23, 24 R2 see Cartesian plane rational 175, 213 rectifying plane 276 reflected ray 300 reflection see transformation – ambient 304 – diffuse 300 – specular 302 regular 99, 226 relation 21 rendering 98 reparametrization 104 RGB 298 right inverse 77 robotics 17 58 350 rolling-ball blend 233 rotation see transformation saturation 298 scaling see transformation self-occluding 302 shade 298 shading – flat 307 – Gouraud 307 – Phong 309 shadow 320 shear see transformation shelling 232 silhouette 309 skinning 251 specular light 299 specular reflection 302 speed 99, 275 subdivision 154, 248 surface – B-spline 235 – B´ezier 234 – bilinear 243 – constructions 241 – curvature 285 – – Gaussian 291 – – mean 291 – extruded 241 – Gordon–Coons 256 – implicit 225 – loft 254 – non-parametric explicit 227 – normal 227 – NURBS 235 – of revolution 49, 245 – parametric 226 – quadric 228 – regular 226 – ruled 242 – singular 226 – skin 251 – subdivision 248 – tangent vector 227 – translational swept 244 Index tangent – line 100, 101 – vector 100, 275 thickening 233 tint 298 tone 298 torsion 275 – vector 277 torus 232, 246, 295 trace 96 transformation – affine 21, 42 – concatenation 13, 30 – identity 6, 31 – inverse 6, 31, 32 – non-singular 4, 32 – projective 20, 42 – reflection – – in arbitrary line 34 – – in arbitrary plane 47 – rotation 9, 29, 43 – – about arbitrary line 45 – – about arbitrary point 33 – scaling 7, 28, 43 – shear 11 – singular – translation 5, 27, 42 translation see transformation trimetric projection see projection umbilic point 291 variation diminishing property 147, 177 viewing pipeline 80 viewplane – coordinates 76 – window 76, 80 viewpoint see centre of perspectivity viewport window see device window visual tangent continuity 172 weight 175, 179 ... graphics and CAD — 2nd ed — (Springer undergraduate mathematics series) Geometry — Data processing Computer graphics — Mathematics Computer-aided design — Mathematics I Title 516′.0028566 ISBN... (Springer undergraduate mathematics series) Includes bibliographical references and index ISBN 1-85233-801-6 (alk paper) Computer graphics Computer-aided design Geometry—Data processing I Title II Series. .. Transforms and Fourier Series P.P.G Dyke Introduction to Ring Theory P.M Cohn Introductory Mathematics: Algebra and Analysis G Smith Linear Functional Analysis B.P Rynne and M.A Youngson Mathematics for