Perspective (graphical) - Wikipedia

The most characteristic features of linear perspective are that objects appear smaller as their distance from the observer increases, and that they are subject ... Perspective(graphical) FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Formofgraphicalprojectionwheretheprojectionlinesconvergetooneormorepoints "Perspectiveprojection"redirectshere.Foramoremathematicaltreatment,seePerspectivetransform. Staircaseintwo-pointperspective ExternalvideoLinearPerspective:Brunelleschi'sExperiment,Smarthistory[1]HowOne-PointLinearPerspectiveWorks,Smarthistory[2]EmpireoftheEye:TheMagicofIllusion:TheTrinity-Masaccio,Part2,NationalGalleryofArt[3] Linearorpoint-projectionperspective(fromLatin:perspicere'toseethrough')isoneoftwotypesofgraphicalprojectionperspectiveinthegraphicarts;theotherisparallelprojection.[citationneeded]Linearperspectiveisanapproximaterepresentation,generallyonaflatsurface,ofanimageasitisseenbytheeye.Themostcharacteristicfeaturesoflinearperspectivearethatobjectsappearsmallerastheirdistancefromtheobserverincreases,andthattheyaresubjecttoforeshortening,meaningthatanobject'sdimensionsalongthelineofsightappearshorterthanitsdimensionsacrossthelineofsight.Allobjectswillrecedetopointsinthedistance,usuallyalongthehorizonline,butalsoaboveandbelowthehorizonlinedependingontheviewused. ItalianRenaissancepaintersandarchitectsincludingMasaccio,PaoloUccello,PierodellaFrancescaandLucaPaciolistudiedlinearperspective,wrotetreatisesonit,andincorporateditintotheirartworks. Contents 1Overview 1.1Aerialperspective 1.2One-pointperspective 1.3Two-pointperspective 1.4Three-pointperspective 1.5Curvilinearperspective 1.6Foreshortening 2History 2.1Earlyhistory 2.2Renaissance 3Limitations 4Seealso 5Notes 6References 6.1Sources 7Furtherreading 8Externallinks Overview[edit] Acubeintwo-pointperspective Raysoflighttravelfromtheobject,throughthepictureplane,andtotheviewer'seye.Thisisthebasisforgraphicalperspective. Perspectiveworksbyrepresentingthelightthatpassesfromascenethroughanimaginaryrectangle(realizedastheplaneofthepainting),totheviewer'seye,asifaviewerwerelookingthroughawindowandpaintingwhatisseendirectlyontothewindowpane.Ifviewedfromthesamespotasthewindowpanewaspainted,thepaintedimagewouldbeidenticaltowhatwasseenthroughtheunpaintedwindow.Eachpaintedobjectinthesceneisthusaflat,scaleddownversionoftheobjectontheothersideofthewindow.[4]Becauseeachportionofthepaintedobjectliesonthestraightlinefromtheviewer'seyetotheequivalentportionoftherealobjectitrepresents,theviewerseesnodifference(sansdepthperception)betweenthepaintedsceneonthewindowpaneandtheviewoftherealscene. Allperspectivedrawingsassumetheviewerisacertaindistanceawayfromthedrawing.Objectsarescaledrelativetothatviewer.Anobjectisoftennotscaledevenly:acirclecanbeflattenedtoaneccentricellipseandasquarecanappearasatrapezoidoranyotherconvexquadrilateral.Thisdistortionisreferredtoasforeshortening. Perspectivedrawingshaveahorizonline,whichisoftenimplied.Thisline,directlyoppositetheviewer'seye,representsobjectsinfinitelyfaraway.Theyhaveshrunk,inthedistance,totheinfinitesimalthicknessofaline.Itisanalogousto(andnamedafter)theEarth'shorizon. Anyperspectiverepresentationofascenethatincludesparallellineshasoneormorevanishingpointsinaperspectivedrawing.Aone-pointperspectivedrawingmeansthatthedrawinghasasinglevanishingpoint,usually(thoughnotnecessarily)directlyoppositetheviewer'seyeandusually(thoughnotnecessarily)onthehorizonline.Alllinesparallelwiththeviewer'slineofsightrecedetothehorizontowardsthisvanishingpoint.Thisisthestandard"recedingrailroadtracks"phenomenon.Atwo-pointdrawingwouldhavelinesparalleltotwodifferentangles.Anynumberofvanishingpointsarepossibleinadrawing,oneforeachsetofparallellinesthatareatananglerelativetotheplaneofthedrawing. Perspectivesconsistingofmanyparallellinesareobservedmostoftenwhendrawingarchitecture(architecturefrequentlyuseslinesparalleltothex,y,andzaxes).BecauseitisraretohaveasceneconsistingsolelyoflinesparalleltothethreeCartesianaxes(x,y,andz),itisraretoseeperspectivesinpracticewithonlyone,two,orthreevanishingpoints;evenasimplehousefrequentlyhasapeakedroofwhichresultsinaminimumofsixsetsofparallellines,inturncorrespondingtouptosixvanishingpoints. Ofthemanytypesofperspectivedrawings,themostcommoncategorizationsofartificialperspectiveareone-,two-andthree-point.Thenamesofthesecategoriesrefertothenumberofvanishingpointsintheperspectivedrawing. Inthisphotograph,atmosphericperspectiveisdemonstratedbyvariouslydistantmountains Aerialperspective[edit] Mainarticle:Aerialperspective Aerial(oratmospheric)perspectivedependsondistantobjectsbeingmoreobscuredbyatmosphericfactors,sofartherobjectsarelessvisibletotheviewer.Asthedistancebetweenanobjectandaviewerincreases,thecontrastbetweentheobjectanditsbackgrounddecreases,andthecontrastofanymarkingsordetailswithintheobjectalsodecreases.Thecoloursoftheobjectalsobecomelesssaturatedandshifttowardsthebackgroundcolour. Aerialperspectivecanbecombinedwith,butdoesnotdependon,oneormorevanishingpoints. One-pointperspective[edit] Adrawinghasone-pointperspectivewhenitcontainsonlyonevanishingpointonthehorizonline.Thistypeofperspectiveistypicallyusedforimagesofroads,railwaytracks,hallways,orbuildingsviewedsothatthefrontisdirectlyfacingtheviewer.Anyobjectsthataremadeupoflineseitherdirectlyparallelwiththeviewer'slineofsightordirectlyperpendicular(therailroadties/sleepers)canberepresentedwithone-pointperspective.Theseparallellinesconvergeatthevanishingpoint. One-pointperspectiveexistswhenthepictureplaneisparalleltotwoaxesofarectilinear(orCartesian)scene—ascenewhichiscomposedentirelyoflinearelementsthatintersectonlyatrightangles.Ifoneaxisisparallelwiththepictureplane,thenallelementsareeitherparalleltothepictureplane(eitherhorizontallyorvertically)orperpendiculartoit.Allelementsthatareparalleltothepictureplanearedrawnasparallellines.Allelementsthatareperpendiculartothepictureplaneconvergeatasinglepoint(avanishingpoint)onthehorizon. Examplesofone-pointperspective Acubedrawingusingtwo-pointperspective Two-pointperspective[edit] Adrawinghastwo-pointperspectivewhenitcontainstwovanishingpointsonthehorizonline.Inanillustration,thesevanishingpointscanbeplacedarbitrarilyalongthehorizon.Two-pointperspectivecanbeusedtodrawthesameobjectsasone-pointperspective,rotated:lookingatthecornerofahouse,orattwoforkedroadsshrinkingintothedistance,forexample.Onepointrepresentsonesetofparallellines,theotherpointrepresentstheother.Seenfromthecorner,onewallofahousewouldrecedetowardsonevanishingpointwhiletheotherwallrecedestowardstheoppositevanishingpoint. Two-pointperspectiveexistswhenthepictureplaneisparalleltoaCartesiansceneinoneaxis(usuallythez-axis)butnottotheothertwoaxes.Ifthescenebeingviewedconsistssolelyofacylindersittingonahorizontalplane,nodifferenceexistsintheimageofthecylinderbetweenaone-pointandtwo-pointperspective. Two-pointperspectivehasonesetoflinesparalleltothepictureplaneandtwosetsobliquetoit.Parallellinesobliquetothepictureplaneconvergetoavanishingpoint,whichmeansthatthisset-upwillrequiretwovanishingpoints. Examplesoftwo-pointperspective Acubeinthree-pointperspective Three-pointperspective[edit] Three-pointperspectiveisoftenusedforbuildingsseenfromabove(orbelow).Inadditiontothetwovanishingpointsfrombefore,oneforeachwall,thereisnowoneforhowtheverticallinesofthewallsrecede.Foranobjectseenfromabove,thisthirdvanishingpointisbelowtheground.Foranobjectseenfrombelow,aswhentheviewerlooksupatatallbuilding,thethirdvanishingpointishighinspace. Three-pointperspectiveexistswhentheperspectiveisaviewofaCartesianscenewherethepictureplaneisnotparalleltoanyofthescene'sthreeaxes.Eachofthethreevanishingpointscorrespondswithoneofthethreeaxesofthescene. One,twoandthree-pointperspectivesappeartoembodydifferentformsofcalculatedperspective,andaregeneratedbydifferentmethods.Mathematically,however,allthreeareidentical;thedifferenceismerelyintherelativeorientationoftherectilinearscenetotheviewer. Examplesofthree-pointperspective Curvilinearperspective[edit] Mainarticle:Curvilinearperspective Bysuperimposingtwoperpendicular,curvedsetsoftwo-pointperspectivelines,afour-or-above-pointcurvilinearperspectivecanbeachieved.Thisperspectivecanbeusedwithacentralhorizonlineofanyorientation,andcandepictbothaworm's-eyeandbird's-eyeviewatthesametime. Additionally,acentralvanishingpointcanbeused(justaswithone-pointperspective)toindicatefrontal(foreshortened)depth.[5] Examplesofcurvilinearperspective Foreshortening[edit] Twodifferentprojectionsofastackoftwocubes,illustratingobliqueparallelprojectionforeshortening("A")andperspectiveforeshortening("B") Foreshorteningisthevisualeffectoropticalillusionthatcausesanobjectordistancetoappearshorterthanitactuallyisbecauseitisangledtowardtheviewer.Additionally,anobjectisoftennotscaledevenly:acircleoftenappearsasanellipseandasquarecanappearasatrapezoid. Althoughforeshorteningisanimportantelementinartwherevisualperspectiveisbeingdepicted,foreshorteningoccursinothertypesoftwo-dimensionalrepresentationsofthree-dimensionalscenes.Someothertypeswhereforeshorteningcanoccurincludeobliqueparallelprojectiondrawings.Foreshorteningalsooccurswhenimagingruggedterrainusingasynthetic-apertureradarsystem.[citationneeded] Inpainting,foreshorteninginthedepictionofthehumanfigurewasimprovedduringtheItalianRenaissance,andtheLamentationovertheDeadChristbyAndreaMantegna(1480s)isoneofthemostfamousofanumberofworksthatshowoffthenewtechnique,whichthereafterbecameastandardpartofthetrainingofartists.(AndreaMantegnaisalsoanauthoroftheFrescoesintheCameradegliSposi;inwhichapartcalled"Theoculus"usesforeshorteningrepresentedbythefigureswhichlookdownuponthewatchers.) History[edit] Thebackgroundbuildingsinthisfirst-century BCfrescofromtheVillaofP.FanniusSynistorshowtheprimitiveuseofvanishingpoints.[6] Rudimentaryattemptstocreatetheillusionofdepthweremadeinancienttimes,withartistsachievingisometricprojectionbytheMiddleAges.VariousearlyRenaissanceworksdepictperspectivelineswithanimpliedconvergence,albeitwithoutaunifyingvanishingpoint.ItiscommonlyacceptedthatthefirsttomasterperspectivewasItalianRenaissancearchitectFilippoBrunelleschi,whodevelopedtheadherenceofperspectivetoavanishingpointintheearlyfifteenthcentury.ItissaidthathisdiscoverywasimmediatelyinfluentialonsubsequentRenaissanceartandwasexploredcontemporaneouslyinmanuscriptsbyLeonBattistaAlberti,PierodellaFrancescaandothers. Thisscenarioisstilldebated,however,becauseBrunelleschi'stavolettaislost,whichdoesnotallowadirectassessmentofthecorrectnessofhisperspectiveconstruction,andbecausetheconditionslistedbyAntoniodiTuccioManettiinhisVitadiSerBrunellescoareinconsistent.[7] Earlyhistory[edit] ASongdynastywatercolorpaintingofamillinanobliqueprojection,12th century ThefloortilesinLorenzetti'sAnnunciation(1344)stronglyanticipatemodernperspective. Theearliestartpaintingsanddrawingstypicallysizedmanyobjectsandcharactershierarchicallyaccordingtotheirspiritualorthematicimportance,nottheirdistancefromtheviewer,anddidnotuseforeshortening.Themostimportantfiguresareoftenshownasthehighestinacomposition,alsofromhieraticmotives,leadingtotheso-called"verticalperspective",commonintheartofAncientEgypt,whereagroupof"nearer"figuresareshownbelowthelargerfigureorfigures;simpleoverlappingwasalsoemployedtorelatedistance.[8]Additionally,obliqueforeshorteningofroundelementslikeshieldsandwheelsisevidentinAncientGreekred-figurepottery.[9] SystematicattemptstoevolveasystemofperspectiveareusuallyconsideredtohavebegunaroundthefifthcenturyBCintheartofancientGreece,aspartofadevelopinginterestinillusionismalliedtotheatricalscenery.ThiswasdetailedwithinAristotle'sPoeticsasskenographia:usingflatpanelsonastagetogivetheillusionofdepth.[10]ThephilosophersAnaxagorasandDemocritusworkedoutgeometrictheoriesofperspectiveforusewithskenographia.Alcibiadeshadpaintingsinhishousedesignedusingskenographia,sothisartwasnotconfinedmerelytothestage.EuclidinhisOptics(c. 300BC)arguescorrectlythattheperceivedsizeofanobjectisnotrelatedtoitsdistancefromtheeyebyasimpleproportion.[11]Inthefirst-century BCfrescoesoftheVillaofP.FanniusSynistor,multiplevanishingpointsareusedinasystematicbutnotfullyconsistentmanner.[6] Chineseartistsmadeuseofobliqueprojectionfromthefirstorsecondcenturyuntilthe18thcentury.Itisnotcertainhowtheycametousethetechnique;DuberyandWillats(1983)speculatethattheChineseacquiredthetechniquefromIndia,whichacquireditfromAncientRome,[12]whileotherscredititasanindigenousinventionofAncientChina.[13][14][15]ObliqueprojectionisalsoseeninJapaneseart,suchasintheUkiyo-epaintingsofToriiKiyonaga(1752–1815).[12][a] VariouspaintingsanddrawingsfromtheMiddleAgesshowamateurattemptsatprojectionsofobjects,whereparallellinesaresuccessfullyrepresentedinisometricprojection,orbynonparalleloneswithoutavanishingpoint. Bythelaterperiodsofantiquity,artists,especiallythoseinlesspopulartraditions,werewellawarethatdistantobjectscouldbeshownsmallerthanthosecloseathandforincreasedrealism,butwhetherthisconventionwasactuallyusedinaworkdependedonmanyfactors.SomeofthepaintingsfoundintheruinsofPompeiishowaremarkablerealismandperspectivefortheirtime.[16]Ithasbeenclaimedthatcomprehensivesystemsofperspectivewereevolvedinantiquity,butmostscholarsdonotacceptthis.Hardlyanyofthemanyworkswheresuchasystemwouldhavebeenusedhavesurvived.ApassageinPhilostratussuggeststhatclassicalartistsandtheoriststhoughtintermsof"circles"atequaldistancefromtheviewer,likeaclassicalsemi-circulartheatreseenfromthestage.[17]TheroofbeamsinroomsintheVaticanVirgil,fromabout400AD,areshownconverging,moreorless,onacommonvanishingpoint,butthisisnotsystematicallyrelatedtotherestofthecomposition.[18]IntheLateAntiqueperioduseofperspectivetechniquesdeclined.TheartofthenewculturesoftheMigrationPeriodhadnotraditionofattemptingcompositionsoflargenumbersoffiguresandEarlyMedievalartwasslowandinconsistentinrelearningtheconventionfromclassicalmodels,thoughtheprocesscanbeseenunderwayinCarolingianart. MedievalartistsinEurope,likethoseintheIslamicworldandChina,wereawareofthegeneralprincipleofvaryingtherelativesizeofelementsaccordingtodistance,butevenmorethanclassicalartwereperfectlyreadytooverrideitforotherreasons.Buildingswereoftenshownobliquelyaccordingtoaparticularconvention.Theuseandsophisticationofattemptstoconveydistanceincreasedsteadilyduringtheperiod,butwithoutabasisinasystematictheory.Byzantineartwasalsoawareoftheseprinciples,butalsousedthereverseperspectiveconventionforthesettingofprincipalfigures.AmbrogioLorenzettipaintedafloorwithconvergentlinesinhisPresentationattheTemple(1342),thoughtherestofthepaintinglacksperspectiveelements.[19]Otherartistsofthegreaterproto-Renaissance,suchasMelchiorBroederlam,stronglyanticipatedmodernperspectiveintheirworksbutlackedtheconstraintofavanishingpoint. Renaissance[edit] MasolinodaPanicale'sSt.PeterHealingaCrippleandtheRaisingofTabitha(c. 1423),theearliestextantartworkknowntouseaconsistentvanishingpoint[20] (detail) ItisgenerallyacceptedthatFilippoBrunelleschiconductedaseriesofexperimentsbetween1415and1420,whichincludedmakingdrawingsofvariousFlorentinebuildingsincorrectperspective.[21]AccordingtoVasariandAntonioManetti,inabout1420,Brunelleschidemonstratedhisdiscoverybyhavingpeoplelookthroughaholeinthebackofapaintinghehadmade.Throughit,theywouldseeabuildingsuchastheFlorenceBaptistery.WhenBrunelleschiliftedamirrorinfrontoftheviewer,itreflectedhispaintingofthebuildingswhichhadbeenseenpreviously,sothatthevanishingpointwascenteredfromtheperspectiveoftheparticipant.[22]Brunelleschiappliedthenewsystemofperspectivetohispaintingsaround1425.[23] Thisscenarioisindicative,butfacesseveralproblems. Firstofall,nothingcanbesaidforcertainabouttheperspectiveofthebaptisteryofSanGiovanni,becauseBrunelleschi'spanelislost. Second,nootherperspectivepaintingbyBrunelleschiisknown. Third,intheaccountwrittenbyAntoniodiTuccioManettiattheendofthe15thcenturyonBrunelleschi'spanel,thereisnotasingleoccurrenceofthewordexperiment. Fourth,theconditionslistedbyAntoniodiTuccioManettiarecontradictorywitheachother.Forexample,thedescriptionoftheeyepiecesetsavisualfieldof15°muchnarrowerthanthevisualfieldresultingfromtheurbanlandscapedescribed.[24] MelozzodaForlì'suseofupwardforeshorteninginhisfrescoes SoonafterBrunelleschi'sdemonstrations,nearlyeveryartistinFlorenceandinItalyusedgeometricalperspectiveintheirpaintingsandsculpture,[25]notablyDonatello,Masaccio,LorenzoGhiberti,MasolinodaPanicale,PaoloUccello,andFilippoLippi.Notonlywasperspectiveawayofshowingdepth,itwasalsoanewmethodofcreatingacomposition.Visualartcouldnowdepictasingle,unifiedscene,ratherthanacombinationofseveral.EarlyexamplesincludeMasolino'sSt.PeterHealingaCrippleandtheRaisingofTabitha(c. 1423),Donatello'sTheFeastofHerod(c. 1427),aswellasGhiberti'sJacobandEsauandotherpanelsfromtheeastdoorsoftheFlorenceBaptistery.[26]Masaccio(d. 1428)achievedanillusionisticeffectbyplacingthevanishingpointattheviewer'seyelevelinhisHolyTrinity(c. 1427),[27]andinTheTributeMoney,itisplacedbehindthefaceofJesus.[28][b]Inthelate15thcentury,MelozzodaForlìfirstappliedthetechniqueofforeshortening(inRome,Loreto,Forlìandothers).[30] Thisoverallstoryisbasedonqualitativejudgments,andwouldneedtobefacedagainstthematerialevaluationsthathavebeenconductedonRenaissanceperspectivepaintings. ApartfromthepaintingsofPierodellaFrancesca,whichareamodelofthegenre,themajorityof15thcenturyworksshowseriouserrorsintheirgeometricconstruction.ThisistrueofMasaccio'sTrinityfresco[31]andofmanyworks,includingthosebyrenownedartistslikeLeonardodaVinci.[32] AsshownbythequickproliferationofaccurateperspectivepaintingsinFlorence,Brunelleschilikelyunderstood(withhelpfromhisfriendthemathematicianToscanelli),[33]butdidnotpublish,themathematicsbehindperspective.Decadeslater,hisfriendLeonBattistaAlbertiwroteDepictura(c. 1435),atreatiseonpropermethodsofshowingdistanceinpainting.Alberti'sprimarybreakthroughwasnottoshowthemathematicsintermsofconicalprojections,asitactuallyappearstotheeye.Instead,heformulatedthetheorybasedonplanarprojections,orhowtheraysoflight,passingfromtheviewer'seyetothelandscape,wouldstrikethepictureplane(thepainting).Hewasthenabletocalculatetheapparentheightofadistantobjectusingtwosimilartriangles.Themathematicsbehindsimilartrianglesisrelativelysimple,havingbeenlongagoformulatedbyEuclid.[c]AlbertiwasalsotrainedinthescienceofopticsthroughtheschoolofPaduaandundertheinfluenceofBiagioPelacanidaParmawhostudiedAlhazen'sBookofOptics.[34]Thisbook,translatedaround1200intoLatin,hadlaidthemathematicalfoundationforperspectiveinEurope.[35] Perspectiveremained,forawhile,thedomainofFlorence.JanvanEyck,amongothers,failedtoutilizeaconsistentvanishingpointfortheconverginglinesinpaintings,asintheArnolfiniPortrait(1434).Gradually,andpartlythroughthemovementofacademiesofthearts,theItaliantechniquesbecamepartofthetrainingofartistsacrossEurope,andlaterotherpartsoftheworld. PietroPerugino'suseofperspectiveinDeliveryoftheKeys (1482),afrescoattheSistineChapel PierodellaFrancescaelaboratedonDepicturainhisDeProspectivapingendiinthe1470s,makingmanyreferencestoEuclid.[36]Albertihadlimitedhimselftofiguresonthegroundplaneandgivinganoverallbasisforperspective.DellaFrancescaflesheditout,explicitlycoveringsolidsinanyareaofthepictureplane.DellaFrancescaalsostartedthenowcommonpracticeofusingillustratedfigurestoexplainthemathematicalconcepts,makinghistreatiseeasiertounderstandthanAlberti's.DellaFrancescawasalsothefirsttoaccuratelydrawthePlatonicsolidsastheywouldappearinperspective.LucaPacioli's1509Divinaproportione(DivineProportion),illustratedbyLeonardodaVinci,summarizestheuseofperspectiveinpainting,includingmuchofDellaFrancesca'streatise.[37]Leonardoappliedone-pointperspectiveaswellasshallowfocustosomeofhisworks.[38] Two-pointperspectivewasdemonstratedasearlyas1525byAlbrechtDürer,whostudiedperspectivebyreadingPieroandPacioli'sworks,inhisUnterweisungdermessung("Instructionofthemeasurement").[39] Perspectivefeaturesheavilyintheresearchofthe17th-centuryarchitect,geometer,andopticianGirardDesarguesonperspective,opticsandprojectivegeometry,aswellasthetheoremnamedafterhim. Limitations[edit] Thissectionhasmultipleissues.Pleasehelpimproveitordiscusstheseissuesonthetalkpage.(Learnhowandwhentoremovethesetemplatemessages) Thissectionmaycontainanexcessiveamountofintricatedetailthatmayinterestonlyaparticularaudience.Pleasehelpbyspinningofforrelocatinganyrelevantinformation,andremovingexcessivedetailthatmaybeagainstWikipedia'sinclusionpolicy.(November2020)(Learnhowandwhentoremovethistemplatemessage) Thissectionpossiblycontainsoriginalresearch.Pleaseimproveitbyverifyingtheclaimsmadeandaddinginlinecitations.Statementsconsistingonlyoforiginalresearchshouldberemoved.(November2020)(Learnhowandwhentoremovethistemplatemessage) (Learnhowandwhentoremovethistemplatemessage) SatireonFalsePerspectivebyWilliamHogarth,1753 Exampleofapaintingthatcombinesvariousperspectives:TheFrozenCity(MuseumofArtAarau,Switzerland)byMatthiasA.K.Zimmermann Perspectiveimagesarecalculatedassumingaparticularvanishingpoint.Inorderfortheresultingimagetoappearidenticaltotheoriginalscene,avieweroftheperspectivemustviewtheimagefromtheexactvantagepointusedinthecalculationsrelativetotheimage.Thiscancelsoutwhatwouldappeartobedistortionsintheimagewhenviewedfromadifferentpoint.Theseapparentdistortionsaremorepronouncedawayfromthecenteroftheimageastheanglebetweenaprojectedray(fromthescenetotheeye)becomesmoreacuterelativetothepictureplane.Inpractice,unlesstheviewerchoosesanextremeangle,likelookingatitfromthebottomcornerofthewindow,theperspectivenormallylooksmoreorlesscorrect.Thisisreferredtoas"Zeeman'sParadox".[40]Ithasbeensuggestedthatadrawinginperspectivestillseemstobeinperspectiveatotherspotsbecausewestillperceiveitasadrawing,becauseitlacksdepthoffieldcues.[41] Foratypicalperspective,however,thefieldofviewisnarrowenough(oftenonly60degrees)thatthedistortionsaresimilarlyminimalenoughthattheimagecanbeviewedfromapointotherthantheactualcalculatedvantagepointwithoutappearingsignificantlydistorted.Whenalargerangleofviewisrequired,thestandardmethodofprojectingraysontoaflatpictureplanebecomesimpractical.Asatheoreticalmaximum,thefieldofviewofaflatpictureplanemustbelessthan180degrees(asthefieldofviewincreasestowards180degrees,therequiredbreadthofthepictureplaneapproachesinfinity). Tocreateaprojectedrayimagewithalargefieldofview,onecanprojecttheimageontoacurvedsurface.Tohavealargefieldofviewhorizontallyintheimage,asurfacethatisaverticalcylinder(i.e.,theaxisofthecylinderisparalleltothez-axis)willsuffice(similarly,ifthedesiredlargefieldofviewisonlyintheverticaldirectionoftheimage,ahorizontalcylinderwillsuffice).Acylindricalpicturesurfacewillallowforaprojectedrayimageuptoafull360degreesineitherthehorizontalorverticaldimensionoftheperspectiveimage(dependingontheorientationofthecylinder).Inthesameway,byusingasphericalpicturesurface,thefieldofviewcanbeafull360degreesinanydirection.Forasphericalsurface,allprojectedraysfromthescenetotheeyeintersectthesurfaceatarightangle. Justasastandardperspectiveimagemustbeviewedfromthecalculatedvantagepointfortheimagetoappearidenticaltothetruescene,aprojectedimageontoacylinderorspheremustlikewisebeviewedfromthecalculatedvantagepointforittobepreciselyidenticaltotheoriginalscene.Ifanimageprojectedontoacylindricalsurfaceis"unrolled"intoaflatimage,differenttypesofdistortionsoccur.Forexample,manyofthescene'sstraightlineswillbedrawnascurves.Animageprojectedontoasphericalsurfacecanbeflattenedinvariousways: Animageequivalenttoanunrolledcylinder Aportionofthespherecanbeflattenedintoanimageequivalenttoastandardperspective Animagesimilartoafisheyephotograph Seealso[edit] Anamorphosis Cameraangle Cutawaydrawing Perspectivecontrol Trompe-l'œil Uki-e Zograscope Notes[edit] ^Inthe18thcentury,Chineseartistsbegantocombineobliqueperspectivewithregulardiminutionofsizeofpeopleandobjectswithdistance;noparticularvantagepointischosen,butaconvincingeffectisachieved.[12] ^Neartheendofthe15thcentury,LeonardodaVinciplacedthevanishingpointinhisLastSupperbehindChrist'sothercheek.[29] ^Inviewingawall,forinstance,thefirsttrianglehasavertexattheuser'seye,andverticesatthetopandbottomofthewall.Thebottomofthistriangleisthedistancefromtheviewertothewall.Thesecond,similartriangle,hasapointattheviewer'seye,andhasalengthequaltotheviewer'seyefromthepainting.Theheightofthesecondtrianglecanthenbedeterminedthroughasimpleratio,asprovenbyEuclid. References[edit] ^"LinearPerspective:Brunelleschi'sExperiment".SmarthistoryatKhanAcademy.Archivedfromtheoriginalon24May2013.Retrieved12May2013. ^"HowOne-PointLinearPerspectiveWorks".SmarthistoryatKhanAcademy.Archivedfromtheoriginalon13July2013.Retrieved12May2013. ^"EmpireoftheEye:TheMagicofIllusion:TheTrinity-Masaccio,Part2".NationalGalleryofArtatArtBabble.Archivedfromtheoriginalon1May2013.Retrieved12May2013. ^D'Amelio,Joseph(2003).PerspectiveDrawingHandbook.Dover.p. 19.ISBN 9780486432083. ^"TheBeginner'sGuidetoPerspectiveDrawing".TheCuriouslyCreative.Retrieved17August2019. ^abHurt,Carla(9August2013)."RomanspaintbetterperspectivethanRenaissanceartists".FoundinAntiquity.Retrieved4October2020. ^Raynaud,Dominique(2014).OpticsandtheRiseofPerspective.Oxford:BardwellPress.pp. 1–2]. ^Calvert,Amy."EgyptianArt(article)|AncientEgypt".KhanAcademy.Retrieved14May2020. ^Regoli,GigettaDalli;Gioseffi,Decio;Mellini,GianLorenzo;Salvini,Roberto(1968).VaticanMuseums:Rome.Italy:Newsweek.p. 22. ^"SkenographiainFifthCentury".CUNY.Archivedfromtheoriginalon17December2007.Retrieved27December2007. ^Smith,A.Mark(1999).PtolemyandtheFoundationsofAncientMathematicalOptics:ASourceBasedGuidedStudy.Philadelphia:AmericanPhilosophicalSociety.p. 57.ISBN 978-0-87169-893-3. ^abcCucker,Felipe(2013).ManifoldMirrors:TheCrossingPathsoftheArtsandMathematics.CambridgeUniversityPress.pp. 269–278.ISBN 978-0-521-72876-8.DuberyandWillats(1983:33)writethat'ObliqueprojectionseemstohavearrivedinChinafromRomebywayofIndiaroundaboutthefirstorsecondcenturyAD.'Figure10.9[Wen-Chireturnshome,anon,China,12thcentury]showsanarchetypeoftheclassicaluseofobliqueperspectiveinChinesepainting. ^"SeeingHistory:Isperspectivelearnedornatural?".EclecticLight.10January2018.Overthesameperiod,thedevelopmentofsophisticatedandhighly-detailedvisualartinAsiaarrivedataslightlydifferentsolution,nowknownastheobliqueprojection.WhereasRomanandsubsequentEuropeanvisualarteffectivelyhadmultipleandincoherentvanishingpoints,Asianartusuallylackedanyvanishingpoint,butalignedrecessioninparallel.Animportantfactorhereistheuseoflongscrolls,whichevennowmakefullycoherentperspectiveprojectionunsuitable. ^MartijndeGeus(9March2019)."ChinaProjections".ArchDaily.Retrieved8July2020. ^Krikke,Jan(2January2018)."WhytheworldreliesonaChinese"perspective"".Medium.com.About2000yearsago,theChinesedevelopeddengjiaotoushi(等角透視),agraphictoolprobablyinventedbyChinesearchitects.ItcametobeknownintheWestasaxonometry.AxonometrywascrucialinthedevelopmentoftheChinesehandscrollpainting,anartformthatarthistorianGeorgeRowleyreferredtoas"thesupremecreationofChinesegenius".Classichandscrollpaintingswereuptotenmetersinlength.Theyareviewedbyunrollingthemfromrighttoleftinequalsegmentsofabout50 cm.Thepaintingtakestheviewerthroughavisualstoryinspaceandtime. ^"Pompeii.HouseoftheVettii.FaucesandPriapus".SUNYBuffalo.Archivedfromtheoriginalon24December2007.Retrieved27December2007. ^Panofsky,Erwin(1960).RenaissanceandRenascencesinWesternArt.Stockholm:Almqvist&Wiksell.p. 122,note1.ISBN 0-06-430026-9. ^VaticanVirgilimage ^HeidiJ.HornikandMikealCarlParsons,IlluminatingLuke:TheinfancynarrativeinItalianRenaissancepainting,p.132 ^"Perspective:TheRiseofRenaissancePerspective".WebExhibits.Retrieved15October2020. ^Gärtner,Peter(1998).Brunelleschi(inFrench).Cologne:Konemann.p. 23.ISBN 3-8290-0701-9. ^Edgerton2009,pp. 44–46. ^Edgerton2009,p. 40. ^DominiqueRaynaud(1998).L'Hypothèsed'Oxford.Essaisurlesoriginesdelaperspective.Paris:PressesuniversitairesdeFrance.pp. 132–141. ^"...andtheseworks(ofperspectivebyBrunelleschi)werethemeansofarousingthemindsoftheothercraftsmen,whoafterwardsdevotedthemselvestothiswithgreatzeal."Vasari'sLivesoftheArtistsChapteronBrunelleschi ^"TheGatesofParadise:LorenzoGhiberti'sRenaissanceMasterpiece".ArtInstituteofChicago.2007.Retrieved20September2020. ^Vasari,TheLivesoftheArtists,"Masaccio". ^Adams,Laurie(2001).ItalianRenaissanceArt.Oxford:WestviewPress.p. 98.ISBN 978-0813349022. ^White,SusanD.(2006).DrawLikeDaVinci.London:CassellIllustrated,p.132.ISBN 9781844034444. ^Harness,Brenda."MelozzodaForli|MasterofForeshortening".FineArtTouch.Retrieved15October2020. ^JudithV.Field;RobertoLunardi;ThomasSettle(1989)."TheperspectiveschemeofMasaccio'sTrinityfresco".Nuncius.4(2):31–118.doi:10.1163/182539189X00680.DominiqueRaynaud(1998).L'Hypothèsed'Oxford.Paris:PressesuniversitairesdeFrance.pp. 72–120. ^DominiqueRaynaud(2016).StudiesonBinocularVision.Cham:SpringerInternational.pp. 53–67.;DominiqueRaynaud(2021)."LasfuentesópticasdeLeonardo".LeonardodaVinci.Perspectivayvisión,ed.LuisRamón-Laca.AlcaládeHenares:UAH.pp. 61–62. ^"MesserPaolodalPozzoToscanelli,havingreturnedfromhisstudies,invitedFilippowithotherfriendstosupperinagarden,andthediscoursefallingonmathematicalsubjects,Filippoformedafriendshipwithhimandlearnedgeometryfromhim."Vasarai'sLivesoftheArtists,ChapteronBrunelleschi ^El-Bizri,Nader(2010)."ClassicalOpticsandthePerspectivaTraditionsLeadingtotheRenaissance".InHendrix,JohnShannon;Carman,CharlesH.(eds.).RenaissanceTheoriesofVision(VisualCultureinEarlyModernity).Farnham,Surrey:Ashgate.pp. 11–30.ISBN 978-1-409400-24-0. ^Hans,Belting(2011).FlorenceandBaghdad:RenaissanceartandArabscience(1stEnglish ed.).Cambridge,Massachusetts:BelknapPressofHarvardUniversityPress.pp. 90–92.ISBN 9780674050044.OCLC 701493612. ^Livio,Mario(2003).TheGoldenRatio.NewYork:BroadwayBooks.p. 126.ISBN 0-7679-0816-3. ^O'Connor,J.J.;Robertson,E.F.(July1999)."LucaPacioli".UniversityofStAndrews.Archivedfromtheoriginalon22September2015.Retrieved23September2015. ^Goldstein,AndrewM.(17November2011)."TheMale"MonaLisa"?:ArtHistorianMartinKemponLeonardodaVinci'sMysterious"SalvatorMundi"".BlouinArtinfo. ^MacKinnon,Nick(1993)."ThePortraitofFraLucaPacioli".TheMathematicalGazette.77(479):206.doi:10.2307/3619717.JSTOR 3619717. ^MathographicsbyRobertDixonNewYork:Dover,p.82,1991. ^"...theparadoxispurelyconceptual:itassumesweviewaperspectiverepresentationasaretinalsimulation,wheninfactweviewitasatwodimensionalpainting.Inotherwords,perspectiveconstructionscreatevisualsymbols,notvisualillusions.Thekeyisthatpaintingslackthedepthoffieldcuescreatedbybinocularvision;wearealwaysawareapaintingisflatratherthandeep.Andthatishowourmindinterpretsit,adjustingourunderstandingofthepaintingtocompensateforourposition.""Handprint :Perspectiveintheworld".Archivedfromtheoriginalon6January2007.Retrieved25December2006.Retrievedon25December2006 Sources[edit] Edgerton,SamuelY.(2009).TheMirror,theWindow&theTelescope:HowRenaissanceLinearPerspectiveChangedOurVisionoftheUniverse.Ithaca,NY:CornellUniversityPress.ISBN 978-0-8014-4758-7. Furtherreading[edit] Andersen,Kirsti(2007).TheGeometryofanArt:TheHistoryoftheMathematicalTheoryofPerspectivefromAlbertitoMonge.Springer. Damisch,Hubert(1994).TheOriginofPerspective,TranslatedbyJohnGoodman.Cambridge,Massachusetts:MITPress. Gill,RobertW(1974).PerspectiveFromBasictoCreative.Australia:Thames&Hudson. Hyman,Isabelle,comp(1974).BrunelleschiinPerspective.EnglewoodCliffs,NewJersey:Prentice-Hall. Kemp,Martin(1992).TheScienceofArt:OpticalThemesinWesternArtfromBrunelleschitoSeurat.YaleUniversityPress. Pérez-Gómez,Alberto,andPelletier,Louise(1997).ArchitecturalRepresentationandthePerspectiveHinge.Cambridge,Massachusetts:MITPress. Raynaud,Dominique(2014).OpticsandtheRiseofPerspective.AStudyinNetworkKnowledgeDiffusion.Oxford:BardwellPress. Raynaud,Dominique(2016).StudiesonBinocularVision.Cham:SpringerInternational. Vasari,Giorgio(1568).TheLivesoftheArtists.Florence,Italy. Externallinks[edit] WikimediaCommonshasmediarelatedtoPerspectivedrawings. WikimediaCommonshasmediarelatedtoEvolutionofPerspective. Atutorialcoveringmanyexamplesoflinearperspective TeachingPerspectiveinArtandMathematicsthroughLeonardodaVinci'sWorkatMathematicalAssociationofAmerica MetaphysicalPerspectiveinAncientRoman-WallPainting HowtoDrawaTwoPointPerspectiveGridatCreatingComics vteVisualizationoftechnicalinformationFields Biologicaldatavisualization Chemicalimaging Crimemapping Datavisualization Educationalvisualization Flowvisualization Geovisualization Informationvisualization Mathematicalvisualization Medicalimaging Moleculargraphics Productvisualization Scientificvisualization Softwarevisualization Technicaldrawing Userinterfacedesign Visualculture Volumevisualization Imagetypes Chart Diagram Engineeringdrawing Graphofafunction Ideogram Map Photograph Pictogram Plot Sankeydiagram Schematic Skeletalformula Statisticalgraphics Table Technicaldrawings Technicalillustration People JacquesBertin CynthiaBrewer StuartCard SheelaghCarpendale ThomasA.DeFanti BordenDent MichaelFriendly GeorgeFurnas PatHanrahan 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Infraredphotography Lensflares Lightingeffects Multipleexposure Filtration Rearprojection Reversemotion Schüfftanprocess Shuttereffects Slit-scan Tiltedplanefocus Time-lapse Fastmotion Slowmotion Speedramping Visual Chromakey Compositing (digital) Computer-generatedimagery Gomotion Introvision Matchmoving Opticalprinting Smallgantics Splitscreen Stopmotion vteMathematicsandartConcepts Algorithm Catenary Fractal Goldenratio Hyperboloidstructure Minimalsurface Paraboloid Perspective Cameralucida Cameraobscura Plasticnumber Projectivegeometry Proportion Architecture Human Symmetry Tessellation Wallpapergroup Forms Algorithmicart Anamorphicart Architecture Geodesicdome Islamic Mughal Pyramid Vastushastra Computerart Fiberarts 4Dart Fractalart Islamicgeometricpatterns Girih Jali Muqarnas Zellij Knotting Celticknot Croatianinterlace Interlace Music Origami Sculpture Stringart Tiling Artworks Listofworksdesignedwiththegoldenratio Continuum Octacube Pi PiintheSky Buildings CathedralofSaintMaryoftheAssumption HagiaSophia Pantheon Parthenon PyramidofKhufu SagradaFamília SydneyOperaHouse TajMahal ArtistsRenaissance PaoloUccello PierodellaFrancesca LeonardodaVinci VitruvianMan AlbrechtDürer Parmigianino Self-portraitinaConvexMirror 19th–20thCentury WilliamBlake TheAncientofDays Newton JeanMetzinger Danseuseaucafé L'Oiseaubleu GiorgiodeChirico ManRay M.C.Escher CircleLimitIII PrintGallery Relativity Reptiles Waterfall RenéMagritte Laconditionhumaine SalvadorDalí Crucifixion TheSwallow'sTail CrockettJohnson Contemporary MaxBill MartinandErikDemaine ScottDraves JanDibbets JohnErnest HelamanFerguson PeterForakis SusanGoldstine BathshebaGrossman GeorgeW.Hart DesmondPaulHenry AnthonyHill CharlesJencks GardenofCosmicSpeculation AndyLomas RobertLonghurst JeanetteMcLeod HamidNaderiYeganeh IstvánOrosz HinkeOsinga AntoinePevsner TonyRobbin AlbaRojoCama RezaSarhangi OliverSin HiroshiSugimoto DainaTaimiņa RomanVerostko MargaretWertheim TheoristsAncient Polykleitos Canon Vitruvius Dearchitectura Renaissance FilippoBrunelleschi LeonBattistaAlberti Depictura Dereaedificatoria PierodellaFrancesca Deprospectivapingendi LucaPacioli Dedivinaproportione LeonardodaVinci ATreatiseonPainting AlbrechtDürer VierBüchervonMenschlicherProportion SebastianoSerlio Regolegeneralid'architettura AndreaPalladio Iquattrolibridell'architettura Romantic SamuelColman Nature'sHarmonicUnity FrederikMacodyLund AdQuadratum JayHambidge TheGreekVase Modern OwenJones TheGrammarofOrnament ErnestHanburyHankin TheDrawingofGeometricPatternsinSaracenicArt G.H.Hardy AMathematician'sApology GeorgeDavidBirkhoff AestheticMeasure DouglasHofstadter Gödel,Escher,Bach NikosSalingaros The'Life'ofaCarpet Publications JournalofMathematicsandtheArts MakingMathematicswithNeedlework RhythmofStructure Viewpoints:MathematicalPerspectiveandFractalGeometryinArt Organizations ArsMathematica TheBridgesOrganization EuropeanSocietyforMathematicsandtheArts GoudreauMuseumofMathematicsinArtandScience 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