Table Of ContentJ I M B L I N NÕ S C O R N E R
N O T A T I O N , N O T A T I O N , N O T A T I O N
TheMorganKaufmannSeriesinComputerGraphicsandGeometricModeling
series editor: brian a.barsky,university Andrew Glassner’s Notebook:Recreational Computer
of california,berkeley Graphics
Andrew S. Glassner
Jim Blinn’s Corner: Notation, Notation, Notation
Jim Blinn Warping and Morphing of Graphical Objects
Jonas Gomes, Lucia Darsa, Bruno Costa, and Luiz
Texturing & Modeling:A Procedural Approach,
Velho
third edition
David S. Ebert, F. Kenton Musgrave, Darwyn Peachey, Jim Blinn’s Corner: Dirty Pixels
Ken Perlin, and Steven Worley Jim Blinn
Geometry Tools for Computer Graphics Rendering with Radiance:The Art and Science of Lighting
Philip Schneider and David Eberly Visualization
Greg Ward Larson and Rob Shakespeare
Understanding Virtual Reality
William Sherman and Alan Craig Introduction to Implicit Surfaces
Edited by Jules Bloomenthal
Level of Detail for 3D Graphics:Application and Theory
David Luebke, Martin Reddy, Jonathan D. Cohen, Jim Blinn’s Corner:A Trip Down the Graphics Pipeline
Amitabh Varshney, Benjamin Watson, and Robert Jim Blinn
Huebner
Interactive Curves and Surfaces:A Multimedia Tutorial on
Digital Video and HDTV Algorithms and Interfaces CAGD
Charles Poynton Alyn Rockwood and Peter Chambers
Pyramid Algorithms: A Dynamic Programming Approach to Wavelets for Computer Graphics:Theory and Applications
Curves and Surfaces for Geometric Modeling Eric J. Stollnitz, Tony D. DeRose, and David H. Salesin
Ron Goldman
Principles of Digital Image Synthesis
Non-Photorealistic Computer Graphics:Modeling, Andrew S. Glassner
Rendering, and Animation
Radiosity & Global Illumination
Thomas Strothotte and Stefan Schlechtweg
François X. Sillion and Claude Puech
Curves and Surfaces for CAGD: A Practical Guide,
Knotty: A B-Spline Visualization Program
fifth edition
Jonathan Yen
Gerald Farin
User Interface Management Systems:Models and
Subdivision Methods for Geometric Design:A Constructive
Algorithms
Approach
Dan R. Olsen, Jr.
Joe Warren and Henrik Weimer
Making Them Move: Mechanics, Control, and Animation of
Computer Animation: Algorithms and Techniques
Articulated Figures
Rick Parent
Edited by Norman I. Badler, Brian A. Barsky, and David
The Computer Animator’s Technical Handbook Zeltzer
Lynn Pocock and Judson Rosebush
Geometric and Solid Modeling: An Introduction
Advanced RenderMan:Creating CGI for Motion Pictures Christoph M. Hoffmann
Anthony A. Apodaca and Larry Gritz
An Introduction to Splines for Use in Computer Graphics
Curves and Surfaces in Geometric Modeling:Theory and and Geometric Modeling
Algorithms Richard H. Bartels, John C. Beatty, and Brian A. Barsky
Jean Gallier
Jim BlinnÕs Corner
Notation, Notation, Notation
Jim Blinn
Microsoft Research
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Kaufmann Publishers is aware of a claim, the product names appear in initial
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Morgan Kaufmann Publishers
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Library of Congress Control Number: 2002104303
ISBN: 1-55860-787-0
This book is printed on acid-free paper.
Contents
Contents
Preface vii
ColorPlates Followingpage
CHAPTER ZERO Notation 1
CHAPTER ONE HowtoDrawaSpherePartI,BasicMath 7
JANUARY 1995
CHAPTER TWO HowtoDrawaSpherePartII,
CoordinateSystems 23
MARCH 1995
CHAPTER THREE HowtoDrawaSpherePartIII,
TheHyperbolicHorizon 41
SEPTEMBER 1995
CHAPTER FOUR TheTruthaboutTextureMapping 59
MARCH 1990
CHAPTER FIVE ConsidertheLowly2×2Matrix 69
MARCH 1996
CHAPTER SIX CalculatingScreenCoverage 97
MAY 1996
CHAPTER SEVEN FugueforMMX 107
MARCH–APRIL 1997
CHAPTER EIGHT Floating-PointTricks 123
JULY–AUGUST 1997
CHAPTER NINE AGhostinaSnowstorm 133
JANUARY–FEBRUARY 1998
CHAPTER TEN WPleasure,WFun 147
MAY–JUNE 1998
CHAPTER ELEVEN TenMoreUnsolvedProblemsin
ComputerGraphics 159
SEPTEMBER–OCTOBER 1998
CHAPTER TWELVE TheCrossRatio 169
NOVEMBER–DECEMBER 1998
CHAPTER THIRTEEN InferringTransforms 179
MAY–JUNE 1999
CHAPTER FOURTEEN HowManyDifferentRationalParametricCubic
CurvesAreThere?PartI,InflectionPoints 195
JULY–AUGUST 1999
CHAPTER FIFTEEN HowManyDifferentRationalParametricCubic
CurvesAreThere?PartII,The“Same”Game 207
NOVEMBER–DECEMBER 1999
CHAPTER SIXTEEN HowManyDifferentRationalParametricCubic
CurvesAreThere?PartIII,TheCatalog 219
MARCH–APRIL 2000
CHAPTER SEVENTEEN ABright,ShinyFuture 231
JANUARY 2000
CHAPTER EIGHTEEN OptimizingC++VectorExpressions 237
JULY–AUGUST 2000
CHAPTER NINETEEN PolynomialDiscriminantsPartI,MatrixMagic 255
NOVEMBER–DECEMBER 2000
CHAPTER TWENTY PolynomialDiscriminantsPartII,
TensorDiagrams 269
JANUARY–FEBRUARY 2001
CHAPTER TWENTY-ONE TensorContractioninC++ 293
MARCH–APRIL 2001
Appendix
Index
AbouttheAuthor
Preface
Preface
This book is the third collection of articles originally published in the
IEEEComputerGraphicsandApplicationsjournal.Allofthesearticles,of
course,havesomethingtodowithcomputergraphics.Thefirstfourwere
originally written at Caltech and describe how I drew planets for the JPL
flybymovies.TherestofthemwerewrittenafterIcametoMicrosoftRe-
search and cover a pretty wide range of topics, from assembly language
optimization for parallel processors (Chapter 7) through exotic usage of
C++ template instantiation (Chapter 18) to theoretical mathematics
(Chapter 20). There should be something in here for everyone.
The previous two collections reproduced the original columns with
onlyafewupdatesandfixes.Forthisvolume,however,Isuccumbedtothe
urgetodomajorsurgeryonmanyofthecolumns.Ichangedsomeofthe
mathematicalnotationtosomethingIlikedbetter.Iaddedalotmoredia-
grams (as befits a graphics book). I added answers to questions that I
hadn’t found when the original columns were written. So even if you’ve
read the original magazine articles, go ahead and read this book. It’s a lot
better. (I’m especially proud of the improvements to Chapters 5, 18, 20,
and 21, if anybody wants to compare them with the originals.)
One of the things people say they like about these articles is their ir-
reverentstyle.Thismakesthemalotmorefuntoreadthantowrite.Iusu-
allysuffergreatlyinwritingthem,goingthroughsixorsevenmajordrafts.
Only after I get the exposition and mathematics right (usually involving
chasing down rogue minus signs) do I make the “joke pass.” For the joke
passoverthisbookIhadthefollowingidea.Haveyounoticedthatwhena
movie is re-released on DVD they usually enhance it with a collection of
deleted scenes and hilarious outtakes? How about setting it up so that I
could advertise this book as containing “deleted scenes” and “hilarious
outtakes”?Thisjokewouldonlywork,however,iftheseactuallycontained
meaningful content instead of just fluff. I have, therefore, included some
deleted scenes, stuff that was somewhat interesting but not on the main
topic or some half-formed ideas that I never pursued further. But what
should I do about outtakes? My model would be the wonderful fake
outtakes at the end of the recent Pixar movies. I have to admit, though,
viii
Preface
thatIcameupabitdry.Howabout“Andthenweseethatthedeterminant
ofthematrixisplusone,erno,minusone,er(dissolveintohelplesslaugh-
ter).”Orhowabout“AsweseeinFigure2.5...heywhodrewthemous-
tache on Figure 2.5?” Maybe this concept doesn’t translate well to this
medium.Youcan,though,becharitableandgivemeachuckleasthoughI
actually did it.
When looking over these columns and searching for some common
themetouseasabooksubtitle,IrealizedthatoneofthethingsthatIdida
lotinthesecolumnswastoexperimentwithmathematicalnotation.Ihave
some general comments to make on this, but if you’re like me you don’t
readtheprefacestobooks.SoIputthesecommentsintoanewfirstchap-
ter.SinceoneofthenotationalconventionsIuseistostartvectorcompo-
nent indexing from 0, I have named this Chapter 0.
Acknowledgments
Iwant to thank Bobby Bodenheimer, Rick Szeliski, and Jim Kajiya for
helpful comments about matrices; Steve Gabriel and Gideon Yuval for
giving me some new insights into floating point; and Matt Klaasen for
turning me on to Hilbert’s book, referenced in Chapter 19. Thanks to
KirkOlynykandPaulHeckbertforprovidingtheinspirationforChapter
13; to Todd Veldhuizen and William Baxter for inspiring Chapter 18 and
encouraging me to make it better; and to Jon Paul Schelter for finding
typos.
A long time ago (before these columns got started), I attempted to
writeacomputergraphicstextbook,butInevergotveryfarwithit.Writ-
ing a column for CG&A seems to be the only way that I can get enough
stuff finished to make a whole book. The series of short deadlines breaks
the problem down into manageable chunks. Once again I would like to
thankmyeditorsattheIEEE,RobinBaldwin,NancyHays,LindaWorld,
AlkeniaWinston,andAnneLear,fortheirinfinitepatienceincopingwith
mylateness.Ifitwerenotforthem,thesecolumnswouldneverhavegot-
ten to press.
Finally,IwouldliketothankMicrosoftResearchforgivingmeanen-
vironment where I could do all the thinking and tinkering that produced
the articles in this book.
Chapter Zero: Notation C H A P T E R Z E R O
Notation
A P R I L 2 0 0 2
Acommon thread running through the original articles that became this
book is experimentation with mathematical notation. You might think
mathematical notation would be the most consistent and systematic form
of representation possible. You’d be wrong! Mathematics is a natural lan-
guage,madeupbyhumans,andcontainsjustasmanyinconsistenciesand
arbitraryrulesasanyothernaturallanguage.Inthischapter,Iwanttobe-
ginbymakingafewobservationsonthislanguage,andtoexplainsomeof
the conventions I use in this book.
Mathematical Symbols
Mathematical Symbols
Whensomeonesays“Considerthex,y,zvalues,”youimmediatelythink
they are talking about 3D coordinates. When someone says “the
quantity θ,” you usually guess that they are talking about an angle. When
someonewrites αP +(1−α)P, itislikelythattheyareinterpolatingtwo
0 1
points.Wheneveryouwriteamathematicalexpression,thereisoftenalot
ofsubtleandunstatedmeaningpackedintothesymbolsyouuse.Let’stake
abrieflookatsomeofthechoiceswehaveinbuildingupamathematical
symbol and the meanings these choices imply.
TheBaseSymbol
The first choice we make is which letter to use for the base symbol. The
alphabeticpositionoftheletterisourfirstopportunitytopackextrainfor-
mationintothesymbol.Coordinatesofpointsusuallycomefromtheend
