Mathematics for Computer Graphics

573 Pages·2022·7.601 MB·English

by  John Vince

Undergraduate Topics in Computer Science
John Vince
Mathematics for
Computer Graphics
Sixth Edition
Undergraduate Topics in Computer
Science
SeriesEditor
IanMackie,UniversityofSussex,Brighton,UK
AdvisoryEditors
SamsonAbramsky ,DepartmentofComputerScience,UniversityofOxford,
Oxford,UK
ChrisHankin ,DepartmentofComputing,ImperialCollegeLondon,London,UK
MikeHinchey ,Lero–TheIrishSoftwareResearchCentre,Universityof
Limerick,Limerick,Ireland
DexterC.Kozen,DepartmentofComputerScience,CornellUniversity,Ithaca,
NY,USA
AndrewPitts ,DepartmentofComputerScienceandTechnology,Universityof
Cambridge,Cambridge,UK
HanneRiisNielson ,DepartmentofAppliedMathematicsandComputerScience,
TechnicalUniversityofDenmark,KongensLyngby,Denmark
StevenS.Skiena,DepartmentofComputerScience,StonyBrookUniversity,Stony
Brook,NY,USA
IainStewart ,DepartmentofComputerScience,DurhamUniversity,Durham,
UK
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Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/7592
John Vince
Mathematics for Computer
Graphics
Sixth Edition
JohnVince
Breinton,UK
ISSN1863-7310 ISSN2197-1781 (electronic)
UndergraduateTopicsinComputerScience
ISBN978-1-4471-7519-3 ISBN978-1-4471-7520-9 (eBook)
https://doi.org/10.1007/978-1-4471-7520-9
1st–5theditions:©Springer-VerlagLondonLtd.2001,2006,2010,2014,2017
6thedition:©Springer-VerlagLondonLtd.,partofSpringerNature2022
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Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
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Thisbookisdedicatedtomywife,Heidi.
Preface
The first edition of this book began life as part of Springer’s Essential series and
containedtenchaptersandapproximately220pages.Thissixthandlasteditionhas
twentychaptersandapproximately600pages.Overtheinterveningeditions,Ihave
revisedandextendedpreviousdescriptionsandintroducednewchaptersonsubjects
that I believe are relevant to computer graphics, such as differential calculus and
interpolation, and new subjects that I had to learn about, such as quaternions and
geometric algebra. Hopefully, this edition explores enough mathematical ideas to
satisfymostpeopleworkingincomputergraphics.
AlthoughthefirsteditionofthisbookwasproducedonahumblePCusingWORD,
subsequenteditionswereproducedonanAppleiMacusingLATEX.Irecommendto
anybuddingauthorsthattheyshouldlearnLATEXanduseSpringer’stemplatesto
createtheirfirstmanuscript.Furthermore,today’scomputersaresofastthatIoften
compiletheentirebookforthesakeofchangingasinglecharacter—itonlytakes5
or6seconds!
Ihaveusedcolourinthetexttoemphasisethepatternsbehindcertainnumbers
andintheillustrationstoclarifythemathematics.
It is extremely difficult to ensure that there are no spelling mistakes, missing
brackets, spurious punctuation marks and, above all, mathematical errors. I truly
havedonemybesttocorrectthetextandassociatedequations,butifIhavemissed
some,thenIapologisenow.
In all of my books, I try to mention the names of important mathematicians
associatedwithaninventionordiscoveryandtheperiodoverwhichtheywerealive.
Inthisbook,Imention50suchpeople,andtherelevantdatesareattachedtothefirst
citation.
Whilst writing this book I have borne in mind what it was like for me when I
wasstudyingdifferentareasofmathematicsforthefirsttime.Inspiteofreadingand
rereadinganexplanationseveraltimes,itcouldtakedaysbefore‘thepennydropped’
andaconceptbecameapparent.Hopefully,thereaderwillfindthefollowingexplana-
tionsusefulindevelopingtheirunderstandingofthesespecificareasofmathematics
andenjoythesoundofvariouspenniesdropping!
vii
viii Preface
IwouldliketothankHelenDesmond,EditorforComputerScience,forallowing
metogiveupholidaysandhobbiesinordertocompleteanotherbook!
Breinton,UK JohnVince
May2022
Contents
1 Introduction .................................................. 1
1.1 MathematicsforComputerGraphics ....................... 1
1.2 UnderstandingMathematics .............................. 1
1.3 WhatMakesMathematicsDifficult? ....................... 2
1.4 BackgroundtoThisBook ................................ 2
1.5 HowtoUseThisBook ................................... 2
1.6 SymbolsandNotation ................................... 3
References .................................................... 4
2 Numbers ..................................................... 5
2.1 Introduction ............................................ 5
2.2 Background ............................................ 5
2.3 Counting ............................................... 5
2.4 SetsofNumbers ........................................ 6
2.5 Zero ................................................... 7
2.6 NegativeNumbers ....................................... 8
2.6.1 The Arithmetic of Positive and Negative
Numbers ..................................... 9
2.7 ObservationsandAxioms ................................ 10
2.7.1 CommutativeLaw ............................. 10
2.7.2 AssociativeLaw ............................... 10
2.7.3 DistributiveLaw .............................. 11
2.8 TheBaseofaNumberSystem ............................ 11
2.8.1 Background .................................. 11
2.8.2 OctalNumbers ................................ 12
2.8.3 BinaryNumbers ............................... 13
2.8.4 HexadecimalNumbers ......................... 13
2.8.5 AddingBinaryNumbers ........................ 16
2.8.6 SubtractingBinaryNumbers .................... 18
ix
x Contents
2.9 TypesofNumbers ....................................... 18
2.9.1 NaturalNumbers .............................. 19
2.9.2 Integers ...................................... 19
2.9.3 RationalNumbers ............................. 19
2.9.4 IrrationalNumbers ............................. 20
2.9.5 RealNumbers ................................. 20
2.9.6 AlgebraicandTranscendentalNumbers ........... 20
2.9.7 ImaginaryNumbers ............................ 21
2.9.8 ComplexNumbers ............................. 24
2.9.9 TranscendentalandAlgebraicNumbers ........... 26
2.9.10 Infinity ....................................... 27
2.10 Summary .............................................. 28
2.11 WorkedExamples ....................................... 28
2.11.1 AlgebraicExpansion ........................... 28
2.11.2 BinarySubtraction ............................. 28
2.11.3 ComplexNumbers ............................. 28
2.11.4 ComplexRotation ............................. 29
3 Algebra ...................................................... 31
3.1 Introduction ............................................ 31
3.2 Background ............................................ 32
3.2.1 SolvingtheRootsofaQuadraticEquation ........ 33
3.3 Indices ................................................ 37
3.3.1 LawsofIndices ............................... 38
3.4 Logarithms ............................................. 38
3.5 FurtherNotation ........................................ 40
3.6 Functions .............................................. 40
3.6.1 ExplicitandImplicitEquations .................. 41
3.6.2 FunctionNotation ............................. 41
3.6.3 Intervals ...................................... 42
3.6.4 FunctionDomainsandRanges .................. 43
3.6.5 OddandEvenFunctions ........................ 44
3.6.6 PowerFunctions .............................. 46
3.7 Summary .............................................. 46
3.8 WorkedExamples ....................................... 46
3.8.1 AlgebraicManipulation ........................ 46
3.8.2 SolvingaQuadraticEquation ................... 47
3.8.3 Factorising ................................... 49
References .................................................... 49
4 Trigonometry ................................................. 51
4.1 Introduction ............................................ 51
4.2 Background ............................................ 51
4.3 UnitsofAngularMeasurement ............................ 51