Applied Geometry for Computer Graphics and CAD

361 Pages·2005·2.608 MB·English

by  Duncan Marsh

Springer Undergraduate Mathematics Series
Advisory Board
M.A.J.Chaplain UniversityofDundee
K.Erdmann OxfordUniversity
L.C.G.Rogers UniversityofCambridge
E.Su¨li OxfordUniversity
J.F.Toland UniversityofBath
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Duncan Marsh
Applied Geometry for
Computer Graphics
and CAD
Second Edition
With 127 Figures
Coverilustrationelementsreproducedbykindpermissionof:
AptechSystems,Inc.,PublishersoftheGAUSSMathematicalandStatisticalSystem,23804S.E.Kent-KangleyRoad,MapleValley,WA
98038,USA.Tel:[phone]Fax[phone]email:[email protected]:www.aptech.com.
AmericanStatisticalAssociation:ChanceVol8No1,1995articlebyKSandKWHeiner‘TreeRingsoftheNorthernShawangunks’page
32fig2.
Springer-Verlag:MathematicainEducationandResearchVol4Issue31995articlebyRomanEMaeder,BeatriceAmrheinandOliver
Gloor‘IllustratedMathematics:VisualizationofMathematicalObjects’page9fig11,originallypublishedasaCDRom‘Illustrated
Mathematics’byTELOS:ISBN0-387-14222-3,GermaneditionbyBirkhauser:ISBN3-7643-5100-4.
MathematicainEducationandResearchVol4Issue31995articlebyRichardJGaylordandKazumeNishidate‘TrafficEngineering
withCellularAutomata’page35fig2.MathematicainEducationandResearchVol5Issue21996articlebyMichaelTrott‘The
ImplicitizationofaTrefoilKnot’page14.
MathematicainEducationandResearchVol5Issue21996articlebyLeedeCola‘Coins,Trees,BarsandBells:Simulationofthe
BinomialProcess’page19fig3.MathematicainEducationandResearchVol5Issue21996articlebyRichardGaylordandKazume
nishidate‘ContagiousSpreading’page33fig1.MathematicainEducationandResearchVol5Issue21996articlebyJoeBuhlerand
StanWagon‘SecretsoftheMadelungConstant’page50fig1.
BritishLibraryCataloguinginPublicationData
Marsh,Duncan
AppliedgeometryforcomputergraphicsandCAD.—2nded.—
(Springerundergraduatemathematicsseries)
1. Geometry—Dataprocessing 2. Computergraphics—
Mathematics 3. Computer-aideddesign—Mathematics
I. Title
516′.0028566
ISBN1852338016
LibraryofCongressCataloging-in-PublicationData
Marsh,Duncan
AppliedgeometryforcomputergraphicsandCAD/DuncanMarsh.—2nded.
p.cm.—(Springerundergraduatemathematicsseries)
Includesbibliographicalreferencesandindex.
ISBN1-85233-801-6(alk.paper)
1.Computergraphics. 2.Computer-aideddesign. 3.Geometry—Dataprocessing. I.
Title. II.Series.
T385.M3648 2004
516—dc22 2004054958
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theCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced,storedortransmitted,inany
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ductioninaccordancewiththetermsoflicencesissuedbytheCopyrightLicensingAgency.Enquiriesconcerning
reproductionoutsidethosetermsshouldbesenttothepublishers.
SpringerUndergraduateMathematicsSeriesISSN1615-2085
ISBN1-85233-801-6 2ndedition Springer-VerlagLondonBerlinHeidelberg
ISBN1-85233-080-1 1stedition Springer-VerlagLondonBerlinHeidelberg
SpringerScience+BusinessMedia
springeronline.com
©Springer-VerlagLondonLimited2005
PrintedandboundintheUnitedStatesofAmerica
Firstpublished1999
Secondedition2005
Theuseofregisterednames,trademarksetc.inthispublicationdoesnotimply,evenintheabsenceofaspecific
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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 9 has been expanded to provide two new sections that focus on
the applications of surfaces in CAD: Section 9.6 describes skin and loft sur-
faces (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.
Anewfinalchapteraddressesrenderingmethodsincomputergraphicsand
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
AppliedGeometryforComputerGraphicsandCAD explorestheapplicationof
geometry to computer graphics and computer-aided design (CAD). The text-
book considers two aspects: the manipulation and the representation of geo-
metric 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,andtomanipulateexistingobjects.Chapter4describesthemethod
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 pack-
age 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 lan-
guage 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.Anumberofexercisesindicateinvestigationswhichwouldbesuit-
able for coursework, labs or projects.
Thebookassumesaknowledgeofvectors,matrices,andcalculus.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 pro-
vides a source of motivation for teaching elementary algebra and calculus to
non-mathematics students. Prerequisite reading on vectors, matrices and con-
tinuity of functions can be found in Chapters 4 and 7 of the SUMS series text
Introductory Mathematics: Algebra and Analysis by Geoff Smith.
Theauthorwouldliketothankanumberofpeople.First,themathematics,
computingandengineeringstudentsatNapierUniversitywhotookthemodules
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,mywifeTineanddaughterEmmafortheircontinuingloveand
support.
Edinburgh, UK Duncan Marsh
Contents
1. Transformations of the Plane .............................. 1
1.1 Introduction ............................................. 1
1.2 Translations ............................................. 5
1.3 Scaling about the Origin .................................. 6
1.4 Reflections .............................................. 8
1.5 Rotation about the Origin ................................ 9
1.6 Shears .................................................. 11
1.7 Concatenation of Transformations ......................... 13
1.8 Applications ............................................ 15
1.8.1 Instancing......................................... 15
1.8.2 Robotics .......................................... 17
2. Homogeneous Coordinates and Transformations
of the Plane ................................................ 19
2.1 Introduction ............................................. 19
2.1.1 Homogeneous Coordinates........................... 21
2.2 Points at Infinity ........................................ 23
2.3 Visualization of the Projective Plane ....................... 24
2.3.1 Line Model of the Projective Plane ................... 24
2.3.2 Spherical Model of the Projective Plane ............... 26
2.4 Transformations in Homogeneous Coordinates................ 27
2.4.1 Translations ....................................... 27
2.4.2 Scaling about the Origin ............................ 28
2.4.3 Rotation about the Origin........................... 29
2.5 Concatenation of Transformations ......................... 30
2.5.1 Inverse Transformations ............................ 31
xi