[ Pobierz całość w formacie PDF ]
TMME,vol.6,no.3,p.297
TwoApplicationsofArttoGeometry
ViktorBlåsjö
Geometryandartexploitthesamesourceofhumanpleasure:theexerciseofourspatialin-
tuition.Itisnotsurprising,then,thatinterconnectionsbetweenthemabound.Applications
ofgeometrytoart,ofwhichweshallindicateafew,gobackatleasttoAlberti’sDePictura
(1435).Butalthoughgeometrystartedout,asitsooftendoes,asamostcourteoussuitorin
itsrelationshipwithart,itwassoontobeaffectionatelyrewarded.Weshallstudytwoofthese
rewards.
Geometryappliedtoart
Letusindicatebrieflyhowgeometrymaybeappliedtoart.Aperspectivepaintingdistortssizes
andshapes.Abuildinginthedistancemaybesmallerthanaman’shead,thecircularrimofa
cupbecomesanellipse,etc.Lines,however,alwaysremainlines.Thissimplefactisthekeyto
drawingtiledfloors(figure1),asAlbertiexplainedinDePictura,becauseitguaranteesthatthe
diagonalofthefirsttileisalsothediagonalofsuccessivetiles.Furthermore,alllinesparallelto
theviewer’slineofsightwillmeetatonepointinthepicture,namelythepointperpendicularly
infrontoftheviewer’seye(theso-called“centricpoint”).Thehorizonisthehorizontalline
throughthispoint,becauseiftheobserverlooksdownwardsfromthere,nomatterhowlittle,
thentherayfromhiseyewillhittheground,whereasishelooksupwardsitwillnot,sothisis
indeedtheboundarybetweengroundandsky(hereweareassuming,ofcourse,thattheearth
isflat).Thus,forexample,placingthecentricpointclosetothegroundgivestheviewerthe
impressionthatheislyingdown.ThistrickisusedtogreateffectbyMantegnainSt.Jamesled
toExecution(figure2).Italsofollows,inthewordsofAlberti,thatthehorizonis“alimitor
boundary,whichnoquantityexceedsthatisnothigherthantheeyeofthespectator...Thisis
whymendepictedstandingintheparallel[tothehorizon]furthestawayareagreatdealsmaller
thanthoseinthenearerones—aphenomenonwhichisclearlydemonstratedbynatureherself,
forinchurchesweseetheheadsofmenwalkingabout,movingatmoreorlessthesameheight,
whilethefeetofthosefurtherawaymaycorrespondtotheknee-levelofthoseinfront.”(De
Pictura,BookI,§20,quotedfromthePenguinedition,Alberti(1991,p.58).)Formoreonthe
roleofgeometryinRenaissanceartsee,e.g.,Kline(1985,ch.10)andIvins(1973).
E-mail:viktor.blasjo@gmail.com.
TheMontanaMathematicsEnthusiast,ISSN1551–3440,Vol.6,No.3,pp.297–304.
2009©TheMontanaCouncilofTeachersofMathematics&InformationAgePublishing
Blåsjö
Figure1:Drawingatiledfloor.
Figure2:Mantegna’sSt.JamesledtoExecution.
Newton’sclassificationofcubiccurves
Letusnowturntotheapplicationsofarttogeometry.OurfirstexampleisNewton’sclassi-
ficationofcubiccurves.Theclassificationofcurvesisthezoologyofmathematics—indeed,
Newtonspokeofdividingcurvesintodifferent“species.”Artprovidesapicturesquecriteriafor
whethertwocurvesshouldbeconsideredtobeofthesamespeciesornot:twocurvesareof
thesamespeciesifoneisaprojectiveviewoftheother,i.e.,ifwhenpaintingthepictureofone
curveyouobtaintheother.Newton(1695),§5,usedthisideatoclassifycubics“byshadows,”
ashesaid,intothefiveequivalenceclassesillustratedinfigure3(formoredetailsseeNewton
(1981),vol.VII,pp.410–433,Newton(1860),Ball(1890),BrieskornandKnörrer(1986),
Stillwell(2002)).Weshallshowwherey
=
x
3
fitsintothisclassificationbyshowingthatitis
equivalenttoy
2
=
x
3
(themirrorimageofthemiddlecurveinfigure3).Theclassificationof
cubicsisanaturalsettingfortheuseofprojectiveideasbecausecubicsarethenextstepbeyond
conics,whicharethemselvestooeasy:projectively,theyareallthesame;anysectionofadou-
bleconeprojectedfromthevertexofthecone(theeyepoint)ontoaplaneperpendiculartothe
axis(thecanvas)comesoutasacircle.
Weimagineourselvesstandingontopoftheflatpartofy
=
x
3
andpaintingitsimageona
canvasstandingperpendiculartotheplaneofthecurve(figure5a).Isaythatthepaintingcomes
outlookinglikefigure5b.Firstofall,thedashedlinerepresentsthehorizon.Letusfocusfirst
TMME,vol.6,no.3,p.299
Figure3:Thefiveprojectiveequivalenceclassesofcubiccurves.(FromNewton(1860).)
y
=
x
3
y
2
=
x
3
Figure4:Twoequivalentcubiccurves.
onthepartoffigure5bbelowthehorizon,whichissupposedtobetheimageofeverything
infrontofus.Apparently,eventhoughthecurvey
=
x
3
goesoftoourright,wewillseeit
meetingthehorizonstraightaheadofus.Weunderstandwhybylookingatthesupportlines
drawninthefigures.Thedottedlineandthebrushstrokelineonourrightareparallelsointhe
picturetheyshouldmeetatthehorizon(likerailroadtracks,ifyouwill).Sincethecurvey
=
x
3
essentiallystaysbetweenthesetwolines(almostallofit,anyway),itmuststaybetweenthem
inthepictureaswell,soitisindeedforcedtomeetthehorizonstraightaheadofus.Thepart
abovethehorizonissimilar,butwemustallowforamathematicaleyethatcanseethroughthe
neck,sotospeak.Todrawtheimageofanypointinfrontofusweconnectittooureyewith
alineandmarkwherethislineintersectsthecanvas.Todrawtheimageofanypointbehindus
weusethesameprocedure,ignoringthefactthatthecanvasisnolongerbetweentheeyeand
thepoint.
Blåsjö
(a)
(b)
Figure5:Projectiveequivalenceofy
=
x
3
andy
2
=
x
3
.
Desargues’theorem
WeshallnowseehowDesargues’theorememergesbeautifullyfromnaturalideasofperspective
painting,namelythe“visualrayconstruction”of’sGravesande(1711)(seeAndersen(2006)for
amoderncommentary).Desargues’theoremisoneofthegreatresultsofprojectivegeometry.
Letusfirstlookbrieflyatwhatitsaysandhowwecanthinkaboutit.Thetheoremsays:iftwo
triangles(ABCandA
0
B
0
C
0
)areinperspective(i.e.,AA
0
,BB
0
,CC
0
allgothroughthesame
point,O)thentheextensionsofcorrespondingsides(ABandA
0
B
0
;BCandB
0
C
0
;ACand
A
0
C
0
)meetonaline.Desargues’theoremisespeciallyeasytothinkaboutinthreedimensions,
asindeedDesargueshimselfdid(asconveyedtousbyBosse(1648);seeFieldandGray(1987,
chapterVIII)).Consideratriangularpyramid.Cutitwithtwoplanestogettwotriangles.The
threepointsofintersectionoftheextensionsofcorrespondingsideswillorcoursebeona
line(theintersectionofthetwoplanes).Byprojectingthetrianglesontooneofthewallsof
thepyramidwegettwoplanetrianglesinperspectiveandthetheoremholdsforthemalso.So
Desargues’theoremholdsforanytwotrianglesinperspectivethatcanbeobtainedbyprojection
fromatriangularpyramid.Wefeelthatanytrianglesinperspectivecanbeobtainedinthisway
soDesargues’theoremisproved.Nowletusseewhatithastodowithart.
Visualrayconstructionoftheimageofaline.Weshalldrawtheperspectiveimageofa
groundplane.Todothiswerotateboththeeyepointandthegroundplaneintothepicture
plane:thegroundplaneisrotateddownaboutitsintersectionwiththepictureplane(the“ground
line”)andtheeyeisrotatedupaboutthehorizon.ConsideralineABinthegroundplane.The
intersectionofABwiththegroundlineisofcourseknown.TheimageofABintersectsthe
horizonwheretheparalleltoABthroughtheeyepointmeetsthepictureplane,andparallelityis
clearlypreservedbytheturning-inprocess.SotoconstructtheimageofABweturnitintothe
TMME,vol.6,no.3,p.301
O
A'
C'
B'
A
C
B
Figure6:Desargues’theorem.
pictureplaneandmarkitsintersectionwiththegroundlineandthendrawtheparallelthrough
theeyepointandmarkitsintersectionwiththehorizon;theimageofABisthelineconnecti
ng
thesetwopoints.
Collinearitypropertyofthevisualrayconstruction.Drawthelineconnectingaturned-in
pointAandtheturned-ineyepoint.TheimageofAisonthislinebecauseifweturnthingsback
outtheeye-point–to–horizonpartofthelinewillbeparalleltotheA–to–groundlinepartoft
he
line,sothattheimagepartofthislineisindeedtheimageoftheA–to–ground-lineline.
turned-in eye plane
turned-in eye point
horizon
horizon
e
y
e
p
l
a
n
e
picture plane
ground line
B
g
r
o
u
n
d
p
l
a
n
e
A
turned-in ground plane
ground line
Figure7:Thevisualrayconstruction.
Desargues’theorembythevisualrayconstruction.ConstructtheperspectiveimageA
0
B
0
C
0
ofatriangleABC.Bytheimage-of-a-lineconstruction,intersectionsofextensionsofcorre-
spondingsidesareallonaline,namelythegroundline,andbythecollinearitypropertyA
0
B
0
C
0
andABCareinperspectivefromtheeyepoint,sowehaveDesargues’theorem.
  [ Pobierz całość w formacie PDF ]