1
This chaptercovers
What thisbookisabout
Who thisbookisfor
Three littlepreviewstowhetyourappetite
Thisbookaimstogiveanintroductiontocomputersciencethroughaseriesofsketches highlightingaspectsofonecentraltheme:bookproduction.Ifyouarereadingaphysicalpaperbackcopyofthisbook,everythingfromthewriter’sthoughtsuptotheactof printingitwasdoneautomatically.IfyouarereadingthisbookasaneBook,ofcourse, thebookisstillinsidethecomputer.
Whatiscomputerscience?Isitjustprogramming?Isitreallyasciencelikephysics orchemistry,orisitjustabranchofmathematics?Orisitjustafancywayoftalking abouthowweusecomputers,andnotascienceatall?Wewilltrytoanswerthese questionsthroughtensketchesbasedaroundourtheme.
We’lllookatthescienceandmathematicsbehindeverythingfromfontstophotographs.Fromthetransmissionandstorageofdatatothelayoutofapage.From thelow-levelzeroesandonesofacomputer’smostbasicarithmetictotheinputand processingoftheworld’slanguages.Alongthewaywewilllearnaboutsomeoftheart andsciencebehindtypesettingandprintingfromtheearlycomputerera,andfrom thetimebeforecomputers.Ifthisappealstoyou,readon…
1.1 Whothisbookisfor,andwhatitisandisnot
Thisbookdoesnotpresupposeanythingbutgeneralknowledge.Itiswrittentobecomprehensibletoacuriousteenager,andinterestingtothoseofussailinggentlythrough middleage.Youmighthavenopriorinterestincomputerscience,ornotquiteknow whatitis–oryoumaybeaprofessionalorhobbyistprogrammerwhojustwishesto learnaboutbuildingelectronicdocuments.
Thisisnotatextbook.Itwillnotteachyouanewprogramminglanguageortechnology,andyoumayfinditdoesnotrelatetoyourprofessionalworkatall.Itshould, however,giveyouinsightintothewaysweusecomputerstoautomatehumanprocesses, howwemarshaldataandmathematicstodoourworkforus,andhowthepublishing industryusescomputerstobuildelectronicandphysicalbooks.
Thisbookisalittleunusual,butIhopethatwithanopenmindandasenseof curiosity,youmightfindthesameenjoymentinreadingitthatitsauthorfoundin writingit.Towhetyourappetite,let’slookatthreelittlepreviewsoflaterchapters.For each,we’llintroduceaproblem,andaskhowitmightbesolved.Youcanponderthese questionsbeforefindingouttheanswerslaterinthebook.
1.2Previews
Eachchapterinthisbookisalargelyself-containedexplorationofasingletopic–whilst wewillsometimesintroducewordsinanearlierchapterwhicharethenusedwithout explanationinalaterchapter,mostreaderscantakethechaptersinanyorder.Inthis section,wepreviewthreeofthechapters,givingyouatasteofwhatistocome.
Foreachofthesepreviews,we’llpresentaprobleminthefieldofpublishingandtry tothinkabouthowitmightbesolved.Howdowedrawlinesandcurvesonthegrid ofpixelsonourscreens?Howdoesthecomputerrepresentthelettersandformatting makingupthetextwewishtoprint?Howtoweprintsmoothtonedgreyscaleimages withonlyblackinkandwhitepaper?
1.2.1Pixels,linesandcurves
Whenwedrawapictureonapieceofpaperincrayonorpen,wechoosewhereto drawourlinesandshapes,andhowtofillthemin,andwesimplytranslatethatinto thecontinuousmovementofpenoverpaper.Thelinesaresmoothandunbroken. Whenweholdupourcameratotakeaphotographofalandscape,wearecapturing thereflectedlightofaworldfullofintricatedetail.Butwhenwewanttolookatsucha photographordrawingonacomputerortelevisionscreen,orwhenwewishtotransmit ittoanotherdeviceoranothercountry,wedonothavethecapacityfortheseinfinitelydetailedrepresentations.Wemustinsteadcreateagridofso-calledpixels(picture elements).Thesedays,suchpixelsarequitesmall,butmanyofuscanremembera timewhentheyweremuchlargerandblockier.Theprincipleisthesame,however: eachlineorcurveorfilling-inorphotographmustbeconvertedautomaticallyand painstakinglytogridsofcolouredblocks.Howdowedothis?
Figure1.1Linesofdifferingslope,representedusingpixels
Infigure1.1youcanseehowpatternsofpixelsforlinesofdifferingslopesappear. Doyourecognisetheseblockypatternsfromcomputerscreens–or,perhaps,from knitwearorembroidery?Figure1.2showshowwemightconvertafilledshape,inthis caseanampersandletterfromafont,intoapixelrepresentation.Firstusingblackand whiteonly,andthenshadesofgray.Canyouimaginehowthisprocessworks?
&Figure1.2Afilledshape,madeofjustblackandwhitepixels,andwithpixelsofvaryingshadesofgrey
Wemightalsothinkabouthowthisampersandshapewasformedinthefirstplace. Howdowerepresentthelinesandcurvesofeachletterofafont,andhowdoesthefont designerdrawtheminthefirstplace?Figure1.3showstheindividualcurvesandtheir endpointswhichmakeupthiscomplicatedshape.Suchpointsandcurvesmustbe designedmanuallyforeachofthehundredsofletters,numbersandsymbolsrequired –andagainforeachstylisticvariationofthefontsuchasboldoritalic.
Figure1.3Chainsofcurvesusedtodrawanampersand
Inchapters2and3,weaddressalltheseissues,andmore.
1.2.2 Markup
Whenwewritewithapenonpaper,formattingthedocumentiseasy.Ifwewantto makethetitlelarge,orunderlineit,ormakealistorputaboxaroundanimportant paragraph,wedoitnaturallyandeasily.Buthowdoweinstructacomputertodoit?In aword-processingprogramweusethemousetoselecttextandapplyastylelikebold oritalic.Butwhatisreallygoingoninsidethecomputer?Whatistheinternalrepresentationinamachinewhichcanonlyprocessnumbers,andnotthesehigher-level concepts?Howdowerepresentachangein typeface (font), typeshape, typethickness, orthe sizeofthe text?Let’slookatanexample:
SectionTitle
Thisisthe first paragraph,whichis important. Wecouldencodethesepiecesofformattingintheplaintext:
[!SectionTitle!
Thisisthe$first$paragraph,whichis*important*.]
Herewe’veused $ foritalic, * forbold, ! foratitle,and [ and ] toindicatethe paragraphshouldbeboxed.Buttherenowcomesaproblem.Whatdowedoifwe wishtousea $ or !,or [ or ] inthetextitself?Canyouthinkofasolution?Allwillbe revealedinchapter4.
1.2.3Greyareas
Figure1.4Woodblockprint.DerFormschneider (TheBlockcutter),1568.
Figure1.5Amezzotintprint“SunshineFallingon aDoor”,byDanishartistPeterIlsted(1861–1933)
Sometimes,weonlyhaveblackinkandwhitepaperavailable-eventodaysomenewspapersareinblackandwhite,someeBookreaderslikewise,andplentyofbookstoo,
including thisone.Inthepast,printingwithmultipleshadesofgraywassimplynotpossible.Evenbeforeprintingmachines,artwasreproducedusingprimitivetechniques suchasthewoodblockprint(figure1.4).Thisinvolvedcarvingtheimageintowood, dippingitininkandpressingitontopaper,likeachildwithhalfapotato.Forcenturies,artistsstruggledtorefinethistechniquefromoneinwhichshadesofgrayhad toberepresentedbythecarvingofsmalllinesorcross-hatchpatternsintothewood, intooneinwhichveryfineandconvincinggreyshadescouldbereliablyreproduced, asinthemezzotintoffigure1.5.
Wefacethesameproblemsintheageofmechanicalprinting.Figure1.6shows,at thetop,anidealizedgradientfromwhitetoblack,throughthegreys.Atthebottom isanapproximationproducedusingonlyblackinkandwhitepaperwhichmightbe suitableforprintingoncoarse,porouspaperlikenewsprint.Canyouworkouthow thispatternofdotsmighthavebeencalculated?
Wewilllookathistoricalandpresent-daygreyscaleimagereproductioninchapter9.
Summary
This bookintroducescomputersciencethroughtenlittlesketches,eachrelatedtothethemeofbookpublishing.
Wehavepreviewedsomeofthesesketchesinthischaptertostimulateyour imagination.
Read ontofindtheanswerstothequestionsposedinthischapter.
Figure1.6Agradientandits‘halftone’dots
This chaptercovers
2 Putting marksonpaper
Coordinates andunitsforplacingink onapageorpixelsonascreen
The physicalcharacteristicsofpaper, inkandcomputerscreens
Building linesandfillingshapes,using onlydots
Inthischapter,westartrightfromthebeginning:ablankpage,andsomemeansof placingmarksonit.Thismightbeblackinkonapieceofpaper,orpixelsonaphone, tablet,eBookreader,orcomputerscreen.Wewilllearnhowtochoosewheretoplace thesemarksonthepagetobuildlinesandfilledshapes.Thereby,wewillseehoweven complextextandgraphicsarecomposedofjusttheselittlemarks.
2.1Wherethingsgo
Inthisbook,weshallneedverylittleformalmathematics,butifweareconsideringthe arrangementoflettersandwordsandlinesandpicturesonthepage,weshallneeda wayofdiscussingtheideaofposition–thatistosay, where somethingis,ratherthan just what itis.Thankfully,ourpaperisflatandrectangular,sowecanusethesimple coordinateswelearnedinschool.Inotherwords,wejustmeasurehowfarweareabove thebottomleftcornerofthepage,andhowfartotheright.Wecanwritethisasapair
of numbers;forexample,thecoordinate(6,2)issixlengthsrightandtwolengthsup fromthebottom-leftofthepage.Itisconventiontouse x todenotetheacrosspartof thecoordinate,and y todenotetheuppart.Theseareknownas Cartesian coordinates, namedforfrenchmathematicianandphilosopherRenéDescartes(1596–1659)–the LatinformofhisnameisRenatusCartesius,whichisalittlecloserto“Cartesian”.The ideawasdiscoveredindependently,ataboutthesametime,byPierredeFermat(1607–1665).Figure2.1showsthecoordinate(6,2)drawnonagraph,withaxesfor x and y, andlittlemarksontheaxestomakeiteasiertojudgepositionbyeye.
Wecanassignunitsifwelike,suchascentimetresorinches,todefinewhatthese “lengths”are.Inpublishing,weliketousealittleunitcalleda point or pt,whichis1/72 ofaninch.Thisisconvenientbecauseitallowsustotalkmostlyusingwholenumbers (itiseasiertotalkabout450ptthanabout6.319inches).Weneedsuchsmallunits becausetheitemsonourpagearequitesmallandmustbecarefullypositioned(look atthewritingonthispage,andseehoweachtinylittleshaperepresentingacharacter issocarefullyplaced).Figure2.2showshowanA4page(whichisabout595ptswide andabout842ptstall)mightlook.
Chapter1
LoremIpsum
Loremipsumdolorsitamet,consectetueradipiscingelit.Utpuruselit,vestibulum ut,placeratac,adipiscingvitae,felis.Curabiturdictumgravidamauris.Namarcu libero,nonummyeget,consectetuerid,vulputatea,magna.Donecvehiculaaugue euneque.Pellentesquehabitantmorbitristiquesenectusetnetusetmalesuada famesacturpisegestas.Maurisutleo.Crasviverrametusrhoncussem.Nulla etlectusvestibulumurnafringillaultrices.Phaselluseutellussitamettortor gravidaplacerat.Integersapienest,iaculisin,pretiumquis,viverraac,nunc. Praesentegetsemvelleoultricesbibendum.Aeneanfaucibus.Morbidolornulla, malesuadaeu,pulvinarat,mollisac,nulla.Curabiturauctorsempernulla.Donec variusorciegetrisus.Duisnibhmi,congueeu,accumsaneleifend,sagittisquis, diam.Duisegetorcisitametorcidignissimrutrum. Namduiligula,fringillaa,euismodsodales,sollicitudinvel,wisi.Morbiauctor loremnonjusto.Namlacuslibero,pretiumat,lobortisvitae,ultricieset,tellus. Donecaliquet,tortorsedaccumsanbibendum,eratligulaaliquetmagna,vitae ornareodiometusami.Morbiacorcietnislhendreritmollis.Suspendisseut massa.Crasnecante.Pellentesqueanulla.Cumsociisnatoquepenatibuset magnisdisparturientmontes,nasceturridiculusmus.Aliquamtincidunturna. Nullaullamcorpervestibulumturpis.Pellentesquecursusluctusmauris. Nullamalesuadaporttitordiam.Donecfeliserat,conguenon,volutpatat, tincidunttristique,libero.Vivamusviverrafermentumfelis.Donecnonummy pellentesqueante.Phasellusadipiscingsemperelit.Proinfermentummassaac quam.Seddiamturpis,molestievitae,placerata,molestienec,leo.Maecenas lacinia.Namipsumligula,eleifendat,accumsannec,suscipita,ipsum.Morbi blanditligulafeugiatmagna.Nunceleifendconsequatlorem.Sedlacinianulla vitaeenim.Pellentesquetinciduntpurusvelmagna.Integernonenim.Praesent euismodnunceupurus.Donecbibendumquamintellus.Nullamcursuspulvinar lectus.Donecetmi.Namvulputatemetuseuenim.Vestibulumpellentesquefelis eumassa.
Figure2.2Apage,itscontents,anditsunits
Youcanseethatthechapterheading“Chapter1”beginsatabout(80,630).Notice thatthecoordinatesofthebottomleftofthepage(calledthe origin)are,ofcourse,
Figure2.1PlacingapointusingCartesiancoordinates
(0,0). Thechoiceofthebottomleftasouroriginissomewhatarbitrary–onecould makeanargumentthatthetopleftpoint,withverticalpositionsmeasureddownwards, isamoreappropriatechoice,atleastintheWestwherewereadtoptobottom.Of course,onecouldalsohavetheoriginatthetoprightorbottomright,withhorizontal positionsmeasuringleftward.
Weshallbeusingsuchcoordinatestodescribethepositionandshapeofeachpart ofeachletter,eachword,andeachparagraph,aswellasanydrawingsorphotographs tobeplacedonthepage.Wewillseehowlinescanbedrawnbetweencoordinates,and howtomaketheelegantcurveswhichformthelettersinatypeface(sometimescalled afont).
2.2Placingdots
Oncewehavedeterminedwhatshapeswewishtoputoneachpage,wemustconsider thefinalformofourdocument.Youmaybereadingthisasaphysicalpaperbackbook, printedandboundbyveryexpensiveequipment.Youmaybereadingitasanelectronic document(suchasaPDFfile)onyourcomputer,tablet,orsmartphone.Or,youmay bereadingitonsomesortofspecial-purposeeBookreader.Eachofthesescenarios hasdifferentcharacteristics.Everypageoftheprintedbookismadeupofhundredsof millionsoflittledots,eachofwhichmaybewhite(noink)orblack(ink).Wecannot typicallyseethedotswiththenakedeye.Thenumberofdotsinagivenspaceisknown asthe resolution (fromtheword“resolve”).Alowresolutionimageisonewhereitis easyfortheeyetoresolve(thatis,distinguish)theindividualdots.Ahighresolution imagehasdotssosmallandtightlypackedthatthenakedeyecannotdistinguishthem.
Ahighresolutionprinter,suchastheoneprintingthephysicalcopyofthisbook, mayhaveasmanyas600or1200dotsperinch(dpi);thatistosay,between600 × 600= 360,000and1200 × 1200=1,440,000dotspersquareinch.Thescreenofacomputer ortabletmayonlyhave100to300dpi,butitcandisplaymanyshadesofgreyand colours.Iftheresolutionistoolow,weseeblockyimages.Figure2.3showspartofa capitalletterAinblackandwhiteat60dpi,30dpi,and15dpi.
Wehaveusedsquaredotshere,suchasmaybeusedonamoderncomputerscreen (wecallthem pixels,whichisshortfor“pictureelements”).Forviewingapageona typicaltabletcomputer,wemighthaveonly2048 × 1536=5,193,728dotsonthewhole screen,buttheymaybecoloursorgreys,aswellasblackorwhite.Whenprintingabook likethis,wehavemanymoredots,butonlyblackink.Letussay,forexample,thatwe haveaUSLetterpage(8.5inchesby11inches)andweareprintingataresolutionof
Figure2.3Threedifferentresolutions
1200 dpi.Wehave1200 × 1200=1,440,000dotspersquareinch,sowehave1200 × 1200 × 8.5 × 11,or134,640,000dotsonthepage,eachofwhichmaybeblackorwhite.
Figure2.4showsphotographs,takenunderamicroscope,ofletteringasitappears inhighqualityprinting,andonthemuchlowerquality,cheapernewsprintusedfor thedailynewspaper.
Figure2.4Microscopepicturesofhighqualityprinting(upper)andnewsprint(lower)
Theupperrowshowshigh-resolutionprintingofletteringoncoatedpaper,suchas mightbeusedforaglossypamphlet,underamicroscopeat20xmagnification,and thesameat400xmagnification.ThelowerrowisstandardtextoftheLondonTimes printedonnewsprintat20xmagnificationandthesameat400xmagnification.
Thehomeorofficelaserprinterworksbyusingalasertopreparearollerinsuch awaythatapowderwilladhereonlytoareaswherethelaserhasbeenshone.The powder(calledtoner)isthentransferredfromtherollertopaper,andbondedtoitby heat.Figure2.5showsthattheparticlesoftonerbehaveratherdifferentlyfromink.
Figure2.5Microscopepicturesoflaser-printedtext
On theleftisawordprintedin1pt,2pt,4pt,6pt,and8pttextunderamicroscope, withmagnificationat20x.Ontheright,the2ptwordwithmagnificationat400x(a typefaceofagivensizeisroughlythatnumberofpointstall,say,foritscapitalletters.)
Allthesedotsformahugeamountofinformationwhichiscostlyanddifficultto manipulate.So,wewillnormallystoreourpagesinamorestructuredway–some paragraphs,whicharemadeofwords,whicharemadeofletters,whicharedrawnfrom sometypeface,whichisdefinedusinglinesandcurves.Thehundredsofmillionsof dotswhichwillfinallymakeupthepageonlyexisttemporarilyastheimageisprinted, orplacedontothescreen.(Theexception,ofcourse,iswhenweusephotographs aspartofourpage–thecolourofeachdotiscapturedbythecamera,andwemust maintainitinthatform.)Untilrecentlythestorage,transmission,andmanipulation ofhighresolutionphotographswasasignificantproblem.Thestorage,transmission, andmanipulationofhighresolutionvideostillis–imaginehowmanylittlecoloured dotsmakeupastillimage,thenmultiplyby25or50imagespersecondforthe2hours (7200seconds)afeaturefilmlasts.
2.3Makinglines
Wehavetalkedonlyaboutsingledots.However,weshallneedlines,curves,andfilled shapestobuildourpage.Supposethatwewishtodrawaline.Howcanweworkout whichdotstopaintblacktorepresenttheline?Horizontalandverticallinesseemeasy –wejustputinkoneachdotinthatroworcolumn,forthewholelengthoftheline.If wewantathickerline,wecaninkmultiplerowsorcolumnseithersideoftheoriginal line.Buttherearemanyusefullinesotherthanthehorizontalandverticalones.To begin,weshallneedawaytodefinealine.Wecanjustusetwocoordinates–thoseof thepointsateitherend.Forexample,Figure2.6showstheline(1,1)—(6,3).
Figure2.6Alinedefinedbythecoordinatesofitstwoend-points
Inmathematics,wewouldusuallyconsideralinetobeofinfinitelength,andsothis isreallyaline segment,butweshalljustcallitaline.Noticethatthislinecouldequally bedefinedas(6,3)—(1,1).
Asafirststrategy,letustrycolouringinonedotineachcolumnfromcolumn1to column6,wherethelineispresent.Figure2.7showstheresultofchoosingthedot whosecentreisclosesttothelineineachcase.
Figure2.7Approximatingalinewithsixdots
Admittedly,thisdoesnotlookmuchlikealine.Butifwechooseahigherresolutionforalineofthesameslope,andsodrawmoreandsmallerdots,weseeabetter approximation,asshowsinFigure2.8.
Figure2.8Approximatingalinewitheighteendots
Now,youmaywonderwhywechosetodrawonedotineachcolumninsteadofone dotineachrow.Forexample,insteadofputtingonedotineachofthecolumnsfrom column1tocolumn6,wemightputonedotineachoftherowsfromrow1torow3, againchoosingtheoneinthatrownearesttheactualline.Forthisshallowline,doing sowouldleadtothemostunpleasantresultshowninFigure2.9.
Figure2.9Thewrongapproach:onedotperrow
Ifthelineissteeperthan45°,theconverseistrue(drawitonpapertosee).So, wechoosetoputoneblackdotineachrowinsteadofineachcolumninthiscase. Horizontalandverticallinesaresimplyspecialcasesofthisgeneralmethod–forthe verticalcasewedrawonedotineachrow;forthehorizontalcaseonedotineach column.Forthelineatexactly45°,thetwomethods(rowandcolumn)producethe sameresult.Figure2.10illustratesthesortsofpatternsofdotsweseeforlinesofvarious slopesusingthisimprovedprocedure.
Figure2.10Linesofdifferingslope
Thisimageis100dotstallandwide.Theresultsarenotterriblygood,fortworeasons.First,atlowresolutions,theindividualdotsareclearlyvisible.Moreover,the densityofthelinesvaries.Ahorizontalorverticallineoflength100willhave100dots overitslength,butthe45° linehas100dotsoveralengthofabout141(thediagonal ofasquarewithsidesoflength100is √2 × 100), andsoitsdensityofdotsislower,and itappearslessdark.
Whenweareusingascreen,ratherthanpaper,todisplayourline,wecantake advantageoftheabilitytousemorethanjustblackandwhite.Andso,wecanuse varyingshadesofgrey:dotswhicharerightonthelineareverydarkgrey,dotswhich arejustclosearelightergrey.Figure2.11showsalinedrawninthismanner,atthree scales.
Figure2.11Usingshadesofgreyforsmootherlines
Wecanseethatthelineissmootherthanwouldotherwisebethecase.Ifyouare readingthisbookonanelectronicdevice,youmaybeabletoseethiseffectonthetext orimageswithamagnifyingglass.Figure2.12showsanotherexample,withamore complex,filledshape,whichmightbeusedtorepresentanampersandcharacter.
&Figure2.12Ashapefilledfirstwithblackandwhitedots,andthenwithdotsofdifferingshadesofgrey
On theleftisanidealisedhighresolutionshape.Inthemiddle,justblackandwhite atalowerresolution.Ontheright,preparedfordisplayonascreensupportinggreyas wellasblackandwhite,atthesamelowerresolution.Thisuseofgreysisknownas antialiasing,sincethejaggededgesinlowresolutionlinesareknownas aliasing.Thisterm originatedinthefieldofsignalprocessingandisusedtodescribeproblemsstemming fromlow-resolutionversionsofhigh-resolutionsignals.Figure2.13isaphotograph, takenunderamicroscope,ofsuchanantialiasedlineonamoderncomputerscreen.
Theleftimageismagnified20x;therightimage400x.Therectangularshapesyou canseeintheimagesaretheseparateRed,Green,andBluesub-pixels,whichamodern LCDmonitorusestobuildupallthedifferentcoloursandgreysitmayneed(the monitormakesapicturebyemittinglightandRed,Green,andBluearetheprimary coloursoflight.)
Whatmightareasonableminimumresolutionbe?Tosimplify,let’sreturntothe scenariowhereweonlyhaveblackandwhitedots–noantialiasing.Theresolution requiredtomakethepagelooksmoothdependsonthedistanceatwhichthetypical viewerseesit.Foracomputerscreen,thismightbetwentyinches.Forasmartphone, eightinches.Forabillboard,twentyorfiftyfeet(ifyouhaveneverwalkedrightupto abillboardandlookedattheprinting,doso–itissurprisinglycoarse.)Thelimitof thehumanopticalsystem’sabilitytodistinguishthecolourofadjacentdots,ortheir existenceorabsence,isthedensityoflightsensitivecellsontheretina.Atadistance of12inches,adensityof600dotsperinchontheprintedpagemayberequired.For abillboard,wemayonlyneed20or50dotsperinch.Onascreen,antialiasingallows ustousealowerresolutionthanwemightotherwiseneed.
2.4Buildingshapes
Wehaveseenhowtodrawlinesbetweenpoints,andsowecanbuildshapesbychainingtogethermultiplelines.Forexample,thelines(1,1)—(10,1),(10,1)—(10,10), (10,10)—(1,10),and(1,10)—(1,1)formasquare(youcandrawitonpaperifyou wish).Wemightdefinethismoreconciselyas(1,1)—(10,1)—(10,10)—(1,10)—(1,1). However,ifwewishtoproduceafilledshape(suchasaletterinaword)wewould havetomakeitupfromlotsoflittlehorizontallinesorlotsoflittleverticalones,to
Figure2.13Anantialisedlineunderamicroscopeat20xand400xmagnifications
make surethateverydotwewantedtobecoveredwascovered.Weshouldliketoautomatethisprocess,soastoavoidmanuallyspecifyingeachpartofthefilledsection.
Forexample,Figure2.14showsachild’spictureofahouse,madefromseverallines.
Noticethatwehavebuiltthreedifferentsetsofjoined-uplines:onefortheoutline ofthehouse,andtwomore,oneforeachwindow.Consideringthebottom-leftdotto beat(0,0),theyare,infact,thesesetsoflines:
forthehouseoutline
(1,1)—(1,10)—(9,18)—(13,14)—(13,16)—(14,16)—(14,13)—(17,10)—(17,1)— (11,1)—(11,5)—(7,5)—(7,1)—(1,1)
fortheleftwindow
(3,10)—(6,10)—(6,7)—(3,7)—(3,10)
fortherightwindow
(12,10)—(15,10)—(15,7)—(12,7)—(12,10)
Now,wecanproceedtodesignamethodtofilltheshape.Foreachrowoftheimage, webeginontheleft,andproceedrightwardpixel-by-pixel.Ifweencounterablackdot, weremember,andenterfillingmode.Infillingmode,wefilleverydotblack,untilwe hitanotherdotwhichwasalreadyblack–thenweleavefillingmode.Seeinganother already-blackdotputsusbackintofillingmode,andsoon.
Figure2.14Asimpleoutlinepicture,builtfromlines
Figure2.15Fillingindividuallines
Figure2.15shows twohighlightedlines.Inthefirst,weentertheshapeonceatthe sideoftheroof,fillacross,andthenexititattherighthandsideoftheroof.Inthe second,wefillasection,exittheshapewhenwehitthedoorframe,enteritagainat theotherdoorframe–fillingagain–andfinallyexitit.Ifwefollowthisprocedurefor thewholeimage,thehouseisfilledasexpected.TheresultisshowninFigure2.16.
Theimageontheleftshowsthenewdotsingrey;thatontherightthefinalimage.Noticethatthewindowsanddoordidnotcauseaproblemforourmethod.In thischapter,wehaveconverteda vector graphic(thelistofcoordinatesmakingupthe house)intoa raster graphic(asetofcoloursforeachdotonthescreenorpaper).
Inthischapter,wehavelookedattheverybasicsofhowtoconvertdescriptionsof shapesintopatternsofdotssuitableforaprinterorscreen.Wehavealsoexplored someofthephysicalcharacteristicsofscreens,paper,andink.Inthenextchapter,we willconsiderthemorecomplicatedshapesneededtodrawlettersandnumbers,which consistnotonlyofstraightlines,butalsocurves.
Summary
Everypageofabooksuchasthisoneismadeupofahugenumberoftiny blackinkmarksplacedontowhitepaper,orahugenumberofpixels,each chosentobewhiteorblackorgrey.
Tomakesuchapage,wehavetoconvertvectordescriptionsoftheshapes (lettersanddrawings)intorasterform.
There aremethodsforconvertinglinestorasterform,andforfillingshapes madefromsuchlines.
Figure2.16Thewholefilledhouse
Problems
Solutions onpage139. Gridsforyoutophotocopyorprintouthavebeenprovidedon page161.Alternatively,usegraphpaperordrawyourowngrids.
1 Givesequencesofcoordinateswhichmaybeusedtodrawthesesetsoflines.
2 Draw thesetwosequencesofcoordinatesonseparate20x20grids,withlines betweenthepoints.Whatdotheyshow?
(5,19)—(15,19)—(15,16)—(8,16)—(8,12)—(15,12)—(15,9)—(8,9)— (8,5)—(15,5)—(15,2)—(5,2)—(5,19)
(0,5)—(10,10)—(5,0)—(10,3)—(15,0)—(10,10)—(20,5)—(17,10)— (20,15)—(10,10)—(15,20)—(10,17)—(5,20)—(10,10)—(0,15)—(3,10)— (0,5)
3 Giventhefollowinglineson20x20grids,selectpixelstoapproximatethem.
4 On 20x20grids,choosepixelstofillintoapproximatethefollowingcharacters.Keeptheminproportiontooneanother.
3Letter forms
This chaptercovers
The mathematicsofcomputer-generated curves
Building letteroutlinesfromsuchcurves
Drawingfilledshapesfromtheseoutlines
Wehavelearnedhowtobuildshapesfromstraightlinesandhowtodrawthoselines ontoascreenbuiltfrompixelsorontopaperusingink.Ifweonlyeverdrewourshapes atonesize,wecouldjustuseenoughtinylittlestraightlinestobuildupanyshape, includingcurvedones.However,wemightliketodrawourshapesatdifferentsizes,or displaythemondeviceswithdifferentso-calledresolutions(theremaybe1200dotsper inchonacommercialprinter,forexample,butonly200dotsperinchonacomputer screen).Wedon’treallywanttostoreahundredlittlelinesandtheircoordinatesto describeacircle–weshouldjustliketosay“acircleofradius20ptatcoordinates(200, 300)”,forexample.Inthischapter,wewilllearnhowtoconvertdescriptionslikethis intothesetsofinkdotsorpixelsneededtomakeupanimageonpaperorscreen.
3.1Straightandcurvedlines
Considertheshapeswhichmakeupaletterofatypeface– Palatino,forexample,which isdescribedinChapter10.Atypefaceisacollectionoflittleshapes,oneforeachletter, whicharearrangedonthepageinlines.Wemustbeabletoscalethemtodifferent
sizes (large,forexample,forchapterheadings,orsmallerforfootnotes).Theymost oftencontaincurvesaswellasstraightlines.ConsidertheletterC,whichhascurved parts.Figure3.1showstheeffectofincreasingthesizeofaletterCdesignedusingonly lines.
lines Figure3.2AletterCcomposedofonlystraight linesandcirculararcs
Thestraightlinessurvivescalingintact,butthecurvesarespoiled.Almostassimple asdrawinglines(atleastonpaper)isdrawingcircles–wecandothiswithapairof compasses.Figure3.2showshowwecouldusethesesectionsofcircles(orarcs)to buildourletterCinstead.Thissufficesonlyforsimpleletterforms.Imaginetryingto drawcomplicatedcurves,suchastheampersandcharactershowninFigure3.3,using justapairofcompasses.
&Circulararcsdon’thelpmuchhere(noteverycurveispartofacircle).Wewould needdozensifnotahundredpiecesofcirclestoaccuratelyfollowtheshapeofthe letter.Beforecomputers,thesolutionwasasetof frenchcurves,whichareshapedpieces ofwood,rubber,orplasticwhosecontoursarecomposedofcurvesofvaryingtightness. Figure3.4showsatypicalsetoffrenchcurves.Thecurvesaremanipulatedbymoving androtatingthem,untilanappropriatecontourisfoundforpartofthedesiredshape. Sometimesthisisdonebystartingwithpointsthroughwhichthecurvemustpass, makingiteasiertolineuptheforms.Acomplexcurvewillbemadeupbyusingseveral differentpartsofthefrenchcurves,makingsurethatthejoinsbetweenthechosen sectionsaresmooth.
Figure3.1AletterCcomposedofonlystraight
Figure3.3Alettershaperequiringmanycurves
Figure3.4Frenchcurves
Thecurvesectionsusedandpointspassedthroughcanberecorded,sothatthe shapemaybereproduced.Suchcurvesarenolongerusedbydraughtsmen,whouse computersinstead,buttheyarestillusedby,forexample,dressmakers.Alarger,less tightlycurvedsetofshapeswereformerlyusedforthedesignofboathulls.Theyare knownasshipcurves.
3.2Computerisedcurves
Howmightweapplythesetechniquestocomputeriseddrawing?TwoFrenchmenin thecarindustry,PierreBézier(1910–1999)atRenaultandPauldeCasteljau(1930–2022)atCitroën,areresponsibleforthedevelopmentofakindofcurvewhichiseasy andpredictableforthedesignertoworkwith,canbeusedtodescribemanyuseful sortsofshapes,andisamenabletomanipulationanddisplaybycomputer.Theyare knownasBéziercurves.
Wehaveseenthatastraightlinemaybedefinedsimplybyitstwoend-points,for example(1,3)—(6,5).ABéziercurveisdefinedbyfourpoints:itstwoend-points, andtwootherpointscalled controlpoints,oneforeachend-point.Thesecontrolpoints, whichmaybepositionedanywhere,areusedtopullthecurveawayfromthestraight linebetweentheend-points.Thefurtherawaythecontrolpoints,themorethecurve deviatesfromthestraightlinebetweenitsend-points.Figure3.5exhibitsseveralBézier
curves.(Wehaveshownthecontrolpointslinkedtotheirrespectiveend-pointswith dottedlines.)
NotethataBéziercurvemaybendonewayortwo,andthatitmayalsobestraight. Thedesignercanmovethecontrolpointsinteractivelytobuildawiderangeofcurved lines.Ifyouhaveaccesstoaninteractivegraphicsprogramonyourcomputer,tryusing ittoplaywiththecontrolpointsofaBéziercurve.
Thecurveswehavemadearestilltoosimpletobuildacomplicatedshape,such asourampersand.Tomakesuchshapes,westitchtogetheranumberofsuchcurves, makingsurethatanend-pointofonecurvecoincideswithanend-pointofthenext, formingachain.Figure3.6showsasetofsuchchainsforourampersand.Thereare threechainsofcurves,whichwecall paths,intheampersand:oneforthemainpart andoneeachforthetwo“holes”.Thereare58curves,ofwhich5arestraightlines.
Figure3.5SeveralBéziercurvesandtheircontrolpoints
Figure3.6Béziercurvechainstodrawanampersandcharacter
Note thatseveralofthesecurvesjoinverysmoothlytooneanother.Thisisamatter ofgettingthecontrolpointsforbothcurvesintherightplace:ifthecontrol-point associatedwiththeend-pointofonecurveisarrangedat180°tothecontrolpoint associatedwiththeend-pointofthenext,thejoinwillbesmooth,withnochangein slopeatthatpoint.Otherwise,therewillbeadefinitecorner.Iftheend-pointsoftwo adjacentcurvesinthesequencedonotcoincideatall,therewillbeagap.(Wecall suchcurvesdiscontinuous.)Figure3.7givesexamplesofdiscontinuous,continuous, andsmoothcontinuousBéziercurvepairs.
Figure3.7Discontinuous(upper),continuous(lowerleft),andsmoothcontinuous(lowerright)pairsof Béziercurves.Noticethecontrolpointsinthesmoothcontinuouscaseareat180°tooneanother.
HowcanwedrawaBéziercurveonthescreenorprintitonpaper?Itseemsmuch morecomplicatedthanthestraightlineswehavealreadydrawn.Itturnsout,though, thatthereisasimpleway:werepeatedlysplitupthecurveintosmallerandsmaller sections,untileachoneisalmostasflatasaline.Then,wecandraweachofthoselittle linesusingthemethodwedevelopedinChapter2.Howdoweperformsuchasubdivision?Wecouldpickequally-spacedpointsalongthelengthofthecurve.However, thiswouldusetoomanylineswherethecurveisopen,andtoofewwhenthecurveis tight.
Analgorithm(method),devisedbyDeCasteljau,issimpleenoughtodoonpaper, andproducesasubdivisionofthecurvewhichisappropriatetothetightnessofeach part.Considerthecurvewithend-pointsAandDandcontrolpointsBandCshown atthetopleftofFigure3.8.
