On Serlio's constructions of ovals
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On Serlio’s constructions of ovals Paul L. Rosin Department of Computer Science Cardiff University UK [email protected] Abstract In his celebrated Tutte l’Opere d’Architettura published over the period 1537–1575 Sebastiano Serlio introduced four techniques for constructing ovals which have thereafter been applied by many architects across Europe. Using various geometric forms (i.e. the triangle, square, and circle) as a basis they produced ovals made up from four circular arcs. This paper analyses both Serlio’s constructions and some of the many possible alternatives and evaluates their accuracy in terms of the ovals’ approximations to an ellipse. 1 Introduction After the dearth of the Middle Ages, when architectural principles were kept secret by the guilds, the Re- naissance was witness to an explosion of architectural treatises and handbooks [11]. These were mostly written by architects, in manuscript form until the end of the fifteenth century, and provided a combination of theories, rules and patterns concerning all aspects of architecture. Contents range over the suitability and preparation of building materials, the design of plans, facades, and ornamentation, theories of beauty, and primers on geometry, to details of suitable inscriptions for tombs, and methods for repelling and extermi- nating insects. The main prior work was Vitruvius’s De Architectura which was (and remains) the only extant archi- tectural treatise from Roman times. Written before AD 27 it has, despite its failings, enjoyed substantial popularity; numerous manuscript copies can be traced through Europe up until the Renaissance [9] and more than twenty copies made in the fifteenth century alone are known [5]. According to Vitruvius the design of a harmonious, pleasing building requires careful attention to the proportions of its parts. Their sizes are simple multiples of a unit length called the module. The first treatise of the Renaissance (written around 1450 and published in 1485, thirteen years after his death) was Alberti’s De Re Aedificatoria and it followed (or at least was inspired by) classical buildings, and of course Vitruvius. Alberti recommended nine basic geometric shapes. First there are the regular polygons: the square, hexagon, octagon, decagon, and dodecagon, all of which can be constructed from a circle (which he considered the ideal form [32]). In addition, there are three rectangles based on extensions of the square. Buildings could then be planned by taking these forms as the underlying components, and combining them to form a pleasing plan. Following Alberti came a host of other authors elaborating, recodifying, and extending the previous writings. During this period the geometry of the circle predominated, holding “an almost magical power” over the architects [32]. In contrast, the ellipse “was as emphatically rejected by High Renaissance art as it was cherished in Mannerism” [22]. Not only was the circle considered to have the most perfect and beautiful form, but its preeminence over the ellipse was bolstered by arguments based on the circular movements of the human body [22]. Eventually the fashion shifted from Alberti’s restrictive circle geometry, and the ellipse came to provide a “convenient form for the Baroque spirit for dynamic expression” [19], not only for aesthetics, but also astronomy, mechanics, etc. For instance, somewhat later William Hogarth, in The Analysis of Beauty, For example, in the fifteen century we have amongst others Filarete, Georgio, and Grapaldi, while those of the next century include Vignola, Palladio, and Scamozzi. As can be seen, the majority of activity took place within Italy. Actually, the same anthropomorphic argument can be applied to ellipses [12]. As seen from above, the rotation of the two arms sweeps out a pair of circular arcs reminiscent of the circles used in one of Serlio’s schemes to approximate ellipses. 1
Transcript of On Serlio's constructions of ovals
OnSerlio’sconstructionsof ovals
Paul L. RosinDepartmentof ComputerScience
Cardiff UniversityUK
Abstract
In hiscelebratedTuttel’Opere d’Architettura publishedover theperiod1537–1575SebastianoSerliointroducedfour techniquesfor constructingovalswhich have thereafterbeenappliedby many architectsacrossEurope. Using variousgeometricforms (i.e. the triangle, square,and circle) as a basistheyproducedovals madeup from four circular arcs. This paperanalysesboth Serlio’s constructionsandsomeof themany possiblealternativesandevaluatestheiraccuracy in termsof theovals’ approximationsto anellipse.
1 Introduction
After thedearthof theMiddle Ages,whenarchitecturalprincipleswerekeptsecretby theguilds, theRe-naissancewaswitnessto an explosionof architecturaltreatisesandhandbooks[11]. Theseweremostlywrittenby architects,in manuscriptform until theendof thefifteenthcentury, andprovideda combinationof theories,rulesandpatternsconcerningall aspectsof architecture.Contentsrangeoverthesuitabilityandpreparationof building materials,thedesignof plans,facades,andornamentation,theoriesof beauty, andprimerson geometry, to detailsof suitableinscriptionsfor tombs,andmethodsfor repellingandextermi-natinginsects.
Themainprior work wasVitruvius’s De Architectura which was(andremains)theonly extantarchi-tecturaltreatisefrom Romantimes. Written beforeAD 27 it has,despiteits failings,enjoyedsubstantialpopularity; numerousmanuscriptcopiescanbe tracedthroughEuropeup until the Renaissance[9] andmorethantwenty copiesmadein the fifteenthcenturyaloneareknown [5]. According to Vitruvius thedesignof a harmonious,pleasingbuilding requirescarefulattentionto the proportionsof its parts. Theirsizesaresimplemultiplesof a unit lengthcalledthemodule.Thefirst treatiseof theRenaissance(writtenaround1450andpublishedin 1485,thirteenyearsafter his death)wasAlberti’s De ReAedificatoriaandit followed(or at leastwasinspiredby) classicalbuildings,andof courseVitruvius. Alberti recommendedninebasicgeometricshapes.First therearetheregularpolygons:thesquare,hexagon,octagon,decagon,anddodecagon,all of whichcanbeconstructedfrom a circle (which heconsideredtheidealform [32]). Inaddition,therearethreerectanglesbasedon extensionsof thesquare.Buildingscouldthenbeplannedbytakingtheseformsastheunderlyingcomponents,andcombiningthemto form a pleasingplan.
Following Alberti camea hostof otherauthorselaborating,recodifying,andextendingthe previouswritings.
�During this periodthegeometryof thecircle predominated,holding“an almostmagicalpower”
over thearchitects[32]. In contrast,theellipse“wasasemphaticallyrejectedby High Renaissanceart asitwascherishedin Mannerism”[22]. Not only wasthecircleconsideredto havethemostperfectandbeautifulform, but its preeminenceover theellipsewasbolsteredby argumentsbasedon thecircularmovementsofthehumanbody[22].
�Eventuallythefashionshiftedfrom Alberti’srestrictivecirclegeometry, andtheellipsecameto provide
a “convenientform for the Baroquespirit for dynamicexpression”[19], not only for aesthetics,but alsoastronomy, mechanics,etc. For instance,somewhat later William Hogarth, in The Analysisof Beauty,
�For example, in the fifteen centurywe have amongstothersFilarete,Georgio, andGrapaldi,while thoseof the next century
includeVignola,Palladio,andScamozzi.As canbeseen,themajority of activity took placewithin Italy.�Actually, thesameanthropomorphicargumentcanbeappliedto ellipses[12]. As seenfrom above, therotationof thetwo arms
sweepsout apair of circulararcsreminiscentof thecirclesusedin oneof Serlio’s schemesto approximateellipses.
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states“...the oval hasa noblesimplicity in it, moreequalin its variety thanany otherobject in nature�
...[and] is asmuch to be preferredto the circle, as the triangle to the square�” An early proponentwas
Michelangelo,who consideredtheellipseduringhis first projectfor thetombof JuliusII�, andlaterused
it in his designfor thePiazzadel Campidoglio.�
Anotherpioneerin theuseof theellipsewasBaldassarePeruzzi,whohadapredilectionfor difficult or eccentricproblems.However, heneitherbuilt norpublishedany relatedmaterial,possiblydueto his early death. It was left to his pupil, SebastianoSerlio, to writethe celebratedTutte l’Opere d’Architettura which waspublishedover the period1537–1575.The workincorporatedandacknowledgedmaterialleft by Peruzzi,althoughtheaccusationsof wholesaleplagiarismby many (e.g.VasariandCellini) appearunfair. Serlio’sapproachto thewriting of thearchitecturaltreatisewassignificantlynew on severalcounts:[5, 11]
� For thefirst time acoordinatedschemeof architecturaleducationwasdevised.
� It containstheearliestdetailedwork on thefiveOrders.Serlioattemptedto sortout thediscrepanciesbetweenVitruviusandactualRomanmonuments,andestablishclearconsistentrulesfor theOrders.
� It providedpracticaldesignpatterns,andwaswritten for theuseof architectsratherthanperusalbywealthypatronsandnoblemen.
� It waswritten in Italianwhereasmostof thepreviouswritings hadbeenmadein Latin.
� The illustrationswerea substantialcomponentof the work. In contrastto Alberti’s original whichcontainednone,while Vitruvius’ hadbeenlong lost,out of the155pagesof Serlio’s third book118hadoneof moreillustrations.
This combinationof practicalityandeaseof usemadeit oneof themostimportantarchitecturaltreatises,andnumerouseditionsandtranslationsweremade. As Summersonsaid: “The booksbecamethe archi-tecturalbible of the civilised world. The Italiansusedthem,the Frenchowednearlyeverythingto Serlioandhis books,the GermansandFlemingsbasedtheir own bookson his, the Elizabethanscribbedfromhim...”. Of particularrelevanceto this paper, Serlioagreedwith Alberti on theascendanceof thecircularform, but extendedtherangeto includeamongstthe polygonsthe GreekCrossandtheoval, thusherald-ing a new approachto ecclesiasticalarchitecture[18, 32]. Amongstthe churchpatternshe includedonewith an elliptical plan, althoughhe himself did not build any elliptical churches.Consequently, in largepartencouragedby Serlio’sbook,many elliptical churcheswerebuilt in Italy andSpainfrom thesixteenthcenturyonwards.
Thesearecoveredin greatdetail by WolfgangLotz [18] andVincenzoFasolo[6]; see
also[14, 19, 31]. Moreover, elliptical theatredesignbecamepopularfrom theendof theseventeenthcen-tury, first in Italy andthenin France,while theellipsealsofeaturedprominentlyin art [24]. Over theyearsthe ellipsehascontinuedto figure in architecturaldesign. For example,instancesin recentbuildings arethestriking “Lipstick Building” by Philip JohnsonandJohnBurgeewhich consistsof a stackof ellipticalcylinders,theTokyo InternationalForumdesignedby RafaelVinoly whichcontainsthevastelliptical GlassHall, andtheadditionsto theBritish Library’sReadingRoomby FosterandPartners(seefigure1). Wealsonotethattheellipseoccursovera wide rangeof scales.Figure2 showsexamples,startingwith smallscaledecorative detailssuchasthe patera,continuingto the scaleof a singleroom (RobertAdam’s hallway atCulzeanCastle),andcontinuingbeyondthescaleof a singlehouseto JohnWood’s conglomerationof theRoyal Crescentat Bath.
In orderto build elliptical structuresa meansof constructingthe shapeof an ellipseis required. Theinstructionsfoundin booksonbuilding constructionor technicaldrawing fall into threeclasses– geometricrulesfor drawing trueellipses,mechanicaldevicesfor drawing trueellipses,andgeometricrulesfor draw-ing approximationsto ellipsesusingfour or morecircular arcs. The methodsaregenerallybasedon thegeometricpropertiesof ellipseswhich have beenavailablefrom variousclassicaltreatisesof conics. Themostcompleteonethathassurvived,by Apollonius,wasrepublishedin Venicein 1541. However, this ispredatedby workssuchasWerner’s conics(1522)andDurer’sgeometry(1525). In fact,Sinisgalliassertsthat Leonardoda Vinci wasthe first amongthe modernsto draw an ellipse, in around1510. Thusthere
In chapter12 - “Of Light andShade”.�In chapter4 - “Of Simplicity, or Distinctness”.�This view is disputedon lackof evidence[30]. We notethat therearethreemajorarchitecturalworks in Romein which theellipsefiguresprominently: thePiazzadel Campi-
doglio, theColosseum,andBernini’s colonnadeoutsideSt. Peter’s cathedral.�For instance,someof thesixteenthandseventeenthcenturyarchitectswhodesignedelliptical churchesare:Vignola(in 1550this
wasthefirst built onanoval plan),Mascherino,Volterra,Vitozzi, Bernini,Rainaldi,andBorromini.
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Figure1: Examplesof modernelliptical buildings
Figure2: Theellipsein architectureat differentscales
wasplentyof theoryavailableduring theRenaissance.However, translatingthis into the practicalitiesofarchitecturecouldbeproblematic.
Fromthefirst classof ellipseconstructionsthebestknown is the“gardener’smethod”in whichtwo pinsor pegsareplacedat the foci anda lengthof string is attachedto the pegs. Thepenis pulledout againstthepegsagainstthestringandaround,therebydrawing out theellipse.While adequatefor drawing rough,smallellipsesthereareseveraldifficultiesin scalingtheapproachupto tackleaccurate,largeellipses.First,in orderto draw thefull
�������properlythestringmustbe loopedratherthantied to thepegs. Evenso,the
knot in thestringcaninterferewith thedrawing process.Second,particularlywith long sectionsof stringit is difficult to maintaina constanttension,andevenvariationsin humidity andtemperaturecanhave aneffect. And finally, if the penor marker is not kept perfectlyperpendicularto the drawing surfacethenadditionalinaccuracieswill occur.
From the next classof ellipseconstructions,drawing devicessuchastrammelswereusedin the Re-naissance,and thereweremany contemporarydevelopersof ellipse compasses– including well knownartistssuchasLeonardoda Vinci, Durer, andpossiblyMichelangelo[15, 25]. Theseenabledreasonablypreciseellipsesto bedrawn in plans,but againcouldnot bereadilyappliedto the largescalemarkingoutof buildings.
This leaves the third classof ellipse constructionswhich, at the expenseof only approximatingtheshapeof theellipse,providesa morepracticaltool for the architectandbuilder. In fact, thereis a doublebenefitsincethecontinuallyvaryingcurvatureof theellipseotherwisecomplicatesmatters.For instance,measuringtheperimeteralongasectionof anellipsecanonly beapproximated;thereis nosimpleanalyticalsolution. Also, for small ellipsessuchas door or window arches,precisebuilding constructionwouldrequirealargerangeof brick shapes[20]. As in indicationof thedifficulty in dealingwith theellipse,BattyLangley [16] in aneighteenthcenturybuilder’s manualreckonson anextra 50%expensein workmanshipandmaterialsfor constructingelliptical walls ascomparedto straightwalls. Although therearevariouspossibleapproachesto approximatingtheellipse(e.g.Blackwell describesa polygonalapproximation[2])themostpopularapproachhasbeento usea combinationof circulararcs.Theadvantageof working withcircles is they provide a powerful geometrictool: with circlesaloneonecancopy angles,bisectanglesandlines,constructperpendicularsto lines,constructequilateraltriangles,etc.[4]. Moreover, they keeptheconstructionprocesssimpleandreliable,andprovideareasonablyaccurateovalwith apleasingappearance.In his unpublishedpapersPeruzziroughlysketchedout two suchconstructions(typesIII andIV describedbelow); � seefigure3. In hisTuttel’Opered’ArchitetturaSerlioprovides,in additionto aplanof anelliptical
�It is perhapsnot coincidentalthat thesescribblesareadjacentto his designfor PalazzoMassimo,in which heusesa curvedwall
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Figure3: Peruzzi’ssketchcontainingtwo oval constructions
churchanda methodfor constructingtrue ellipses,four constructionmethodsfor drawing ovals,namelypiecewisecircularapproximationsto theellipse.
2 Serlio’s Four Oval Constructions
Serlio’s four alternative constructionsfor approximatingtheellipsefrom his First Book areshown in fig-ure 4. Eachgeneratesan oval consistingof four circular arcswith centres������� ��� and � � � ��! � andradii"$# � and %'&(! respectively. ConstructionsII-IV have beencalledby Giulio Troili in the seventeenthcenturyandotherssinceby themoredescriptive termsovatolongo, ovato, andovatotondo. More recently,the generalterm quadrarc hasbeenusedin their descriptionandanalysis[8, 27]. Kitao [13] incorrectlystatesthatconstructionsII–IV areall specialcasesof constructionI. In fact,themoregeneralconstructionwould beonein which anarbitrarytypeof triangleis usedwith its sidesextendedby anarbitrarylength.This would give many possiblealternative constructionsfor any desiredellipse(i.e. any semi-majorandsemi-minoraxes " and % ). Theimportantfactoris thatdueto theunderlyingtriangleall theconstructionsexhibit circulararcswith tangentcontinuity. ) This geometricconstraintcanbealgebraicallyexpressedas
�+* ! #-,/.10�2,/.10 #4365Serlio’s constructionsI andIV containanunderlyingequilateraltriangle.Both constructionsII andIII useanisoscelestrianglemadeupfrom half asquare,but extendedby differentamounts, 78 � and 9 respectively,where9 is thelengthof thesquare’sside.In addition,thethreesquareconstructiondescribedlateris madeupof a similar triangle,but only extendedby 7 � .
Thepresenceof circles,triangles,squares,etc. underlyingtheconstructionsis not surprisingsincethecombinationof geometricprimitiveswasa favouritedevice of artistsandarchitectsin theRenaissance(aswell asin earlierandlater times). In Italy the normalproportionswerebasedon the equilateraltriangle,leadingto the geometricsystemad triangulum [9]. Many examplesof a preoccupationwith underlyinggeometricformsarefoundin numerology(“sacredgeometry”),leadingon to architecture,andassuchareparticularlyevidentin importantbuildingssuchaschurches.
to cleverly combinetwo adjacentexisting sitesinto asinglebuilding.:DespiteArnheim’s assertionto the contrary[1], circulararcscanbe joined easilyandpleasinglyasdemonstratedin theseoval
constructions.
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Figure4: Serlio’s four constructionsfor drawing ovals
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Figure5: Aspectratioof Serlio’sconstructionI
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Figure6: Thetrinity of rootscontainedin theVesicaPiscis
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2.1 Construction I
This is the only oneof Serlio’s constructionsthatenablesovalswith variedeccentricitiesto be drawn. Itconsistsof two equilateraltriangleswhosebasesarecentredon theorigin. Their intersectionswith theaxesdetermine� and ! , which canbeexpressedas
�$* ";# %< � #43�= !>*< � � "?# % �< � #@3 5
Thelengthsof thecirculararcsarespecifiedby extendingthetriangles’diagonalsidesby a length A . WhenA is increasedboth " and % areincreased,andsoit is not straightforwardto (geometrically)choosecorrectvaluesof � and A soasto achievespecifiedvaluesof " and % . In fact,theratio is givenby
"% * �B&CA
�D�FE # < ��� &4A 5Figure5 shows how, for a trianglewith unit lengthsides,increasingA from zeroinitially rapidly increasesthecircularity of theoval, while as A increasesbeyondonetheaspectratiovery slowly decreases,reachinga perfectcircle in thelimit. Thusnot all aspectratioscanbeachievedwith this methodsincetwo arcsperquadrantareonly drawn when A;G �
suchthat , 0>H �� . 8 �BI � 5KJ � E . Theratioof theradii of thetwo circulararcsalsovaries,andcanbeshown to be L� M/N L .
2.2 Construction II
Thesecondconstructionusesthreecircles,eachpassingthroughtheir neighbouringcircle(s)centres.Thecentresof thecirculararcsareat
�$*O!>* "E
andproduceanapproximateellipsewith thefixedaspectratio of , 0 * < E I 3 5 P 3 P . Theratio of theradiiof thearcsis
< E #43 . Serlioconsideredthis oval to bevery similar to theshapeof a naturalegg. To verifythis a roughtestwasmadein which thetypical aspectratiosof theeggsof onehundredrandomlyselectedspeciesof Europeanbird wereobtained[23]. Themeanaspectratio wasfound to be 3 5 ��Q (with standarddeviation 5 ��R ) which is indeedcloserto constructionII thanthe constructionsIII or IV. We alsonotethatsuchanarrangementof circleswasoftenusedby artistsin theirdesigns.An extensiveexampleof its useisgivenby Pinturicchio’s frescosin theLibrary of SienaCathedral[3].
2.3 Construction III
Thethird constructionusesyetanothergeometricprimitiveasits basis:thesquare.Thisyields
�$*S!T* "< E'& 3with a fixedaspectratioof , 0 * 8 � NU�� 8 � . � I 3 5 � E � , andtheratioof theradii of thearcsis
�� 52.4 Construction IV
Thefinal constructionwasrecommendedby Serlio for its beauty, andis thesimplestandquickestto con-struct.Not surprisingly, thiswasthestandardellipseapproximationusedin architecturalpractise[13]. Theconfigurationof two intersectingcirclesis commonin sacredgeometry[17] whereit is calledthe VesicaPiscis, its symbolismoftenappearingin Egyptian,Christian,etciconography. As anexampleof its numer-ical significance,its proportionscontainthe threebasicroots,asshown in figure 6. Kitao alsonotesthesimpleratiospresentin therelationshipof its parts:
1. thelengthsof thearcsare1:1,
2. theradii of thearcsare1:2,
3. themajoraxisof theoval to thesmallerarcradiusis 1:3.
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Like constructionI thereis anunderlyingequilateraltriangle,althoughthis is not requiredto beexplicitlydrawn to locatethearcs.Thecentresof thesearcsarelocatedat
�V* "� = !T* "< � �andits fixedaspectratio is , 0 * �� . 8 � I 3 5 � E � . Thusthis constructionprovidesa verysimilar approxima-tion to thepreviousconstruction.
3 Other Oval Constructions
Having run throughSerlio’s four oval constructionswe now consideralternative approaches.Somearevariationson themethodsalreadydescribedwhile othersareextensionsor improvements.
3.1 Three square construction
In the sameway that Serlio’s constructionII increasedthe two circlesin constructionIV to threecircleswe considerincreasingthe two squaresin constructionIII to threesquares(figure 7a). The sameshapedunderlyingisoscelestriangleremains,but is extendedby a shortersection,giving
�$*S!T* E "< E'&CE 5Thefixedaspectratio is , 0 * �WN 8 �� . 8 � I 3 5 X E�E , andtheratioof theradii of thearcsis
�� . 8 � I � 5 ��� 3 .3.2 Four circle construction
ContinuingSerlio’s constructionsof two andthreeintersectingcircleswe show four intersectingcirclesinfigure7b,giving
�$*�X" = !>*
< �X
" 5with thefixedaspectratio is , 0 * �8 �YN�� I 3 5 � P � andtheratioof theradii of thearcsis
�8 �YN6� I � 5 ����� .Increasingfrom two to threecirclesproducedanincreasein aspectratio,but moving from threeto four
hasdecreasedit almostbackto thetwo circle aspectratio. This is dueto thealternationof thepositionof! at thetop of thecentralcircle or at theintersectionof themiddletwo circlesdependingon whetherthereareanoddor evennumberof circlesused.
3.3 Half-square triangle
JustasSerlio’s constructionI extendsanequilateraltriangleby variableamountsto generateovalsover arangeof eccentricitiesweapplythesametechniqueto theisoscelestriangleunderlyinghis constructionsIIandIII, yielding
�$*S!T* "B# %E # < E
with anaspectratio of , 0 * M/N LZ 8 � . �\[]M/N L andtheratio of theradii of thearcs L8 � M^N L . A similar behaviour of
aspectratio to extensionlengthis observed: for smallvaluesof A minor changescauselargedifferencesinaspectratio.
3.4 Vignola’s construction
Kitao [13] describesa constructionby GiacomoVignolabasedaroundthePythagoreantrianglewith sidelengths
�, P , and X asshown in figure7c,
�V* "E = !>*_%`* E� "
giving an fixed aspectratio , 0 * �� with the ratio of the radii of the arcsas�� . Golvin alsoclaims this
approachwasusedby theRomansin their building of ampitheatres[7].
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Figure7: Alternativeoval constructions
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Figure9: Two ovalswith identicalaspectratiosgeneratedusingKitao’sconstruction
Kitao [13] generalisesSerlio’s constructionIV by allowing the circlesto move closeror furtherapart(figure7f), resultingin
�+*� "?# E�%a� <
P " % # � " �P = !T*cb " � ";# E�� � 5
Thesolutionbecomescomplex when , 0 G � � . On examinationwe seethecausefor thebreakdown at thispoint in theunusualbehaviour of theoval’saspectratio astheunderlyingcirclesaremovedapart.Startingin thelimit from concentriccircleswhich producea circle, theaspectratio reachesa peakof , 0 * � � when�d* ���e (where e is the radiusof theunderlyingcircles)andthereafterdecreasesto unity againwhenthecirclesno longeroverlap– seefigure8. Thus,for any aspectratio achievableby Kitao’s methodtherearetwo possibleconstructions.Curiouslyenough,theratio of theradii of thearcsis constant,namely
�� – theonly suchcaseof a fixedratio of arc radii over all thevariableaspectratio oval constructionsanalysedinthis paper. Despitehaving identicalaspectratiostheovalsdiffer, asdemonstratedin figure9. Comparedtotheoval in (a), theoval in (b) is madeupwith arcsof largerradii, andthereforehasa squarerappearance.
3.6 Bianchi’s construction
Although Serlio’s constructionI doesallow a rangeof oval eccentricitiesto be approximatedit doesnoteasilyenableany specificeccentricityto be determinedsincethe two parameters(trianglelengthandex-tension)interact. Later constructionsovercamethis difficulty. For instance,Paolo Bianchi’s InstituzionePracticadell’ Architettura Civile in 1766providesthefollowing scheme(seefigure7d):
1. markpoint f suchthatthelength f " *Sgh%2. point � is markedsuchthat gh�$* �� g�f3. anequilateraltriangleis constructed,its verticesdefining � and ! :
�+* P� � "?# % � = !T* P< � � ";# % � 5
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Thisapproachoperatesup until thebreakdown aspectratioof , 0 G �8 � #@3 I � 5 � 3 R .It is interestingto notethat Bianchi’s approachprovidesidenticalovals,but usingan alternative con-
struction,to thoseof the Slantzmethod[33] describedin currentengineeringdrawing books. Anothervariationproducingthesameoval wasgivenby Langley [16] in which theequilateraltriangleis drawn suchthatit touchesthebottomof theellipse(figure7e).
3.7 Equilateral triangle constructions
We notethatmany oval constructionsinvolve theuseof equilateraltriangles.Serlio’s constructionsI andIV andBianchi’sconstructionhavealreadybeendescribed.Two furthermoremodernexamplesareshown.Mott’s construction(his methodnumber7.8) [21] alsoinvolvestwo circles like Serlio’s constructionIV.However, thecirclestouchratherthanintersect,andanequilateraltriangleis drawn with its bottomcornersin thecirclecentres(figure7g),yielding
�$* "E = !>*
< �E " 5
Although it is similar to Vignola’s constructionwith the Pythagoreantrianglereplacedby the equilateraltrianglethereis a furtherdifference:theradiusof thearccentredat ! is setto !?&i% ; this enablesa rangeof aspectratios(althoughonly j 3 & �8 �k� 8 N 8 �� l I j 3 5mX�J�JnP � E 5 3 Q�Q�R l ) to beconstructed,but theovalsarenolongertangentcontinuous.
Hewitt’smethod3 [10] issomewhatdifferentin thatunlikethepreviousexamplestheequilateraltriangleis centredon thesemimajoraxisratherthantheoval centre(figure7h). Theprocedureis:
1. thearccentredat theorigin with radius % is drawn out; it cutsthetriangleat o2. thestraightline through % and o is drawn, andcutsthetriangleagainat p3. theline through p parallelto thetriangleside g�q intersectstheaxesat � and ! , yielding
�$* � "?# % �< � #@3 = !>*< � � "?# % �< � #@3 5
Thusit providespreciselythe sameoval asSerlio’s constructionI. Its advantageis thatgiventhe desiredaspectratioof theoval thentheconstructionfollowseasily. Serlio’smethodprovidesnomeansof achievingaspecificaspectratioexceptby trial anderrorvariationof � and A .3.8 Robert Simpson’s construction
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Figure10: Simpson’soval construction
In around1744,motivatedby his interestin astronomyand the orbits of the planets,JamesStirlingdevisedanapproximationin whichnotonly dothearcjoinshavetangentcontinuity, but they alsolie on thetrueellipse[29]:
�$* � "?# % �ar " &4%a& < " � & � " %a&C% �^s"B# %a& < " � & � " %a&C% � = !T* � "?# % �ar " & � %t& < " � & � " %a&4% �^sP % 5
10
Stirling proposedto RobertSimpsonthathefind a geometricconstructionfor his algebraicdescription.InamanuscriptSimpsonprovidedthefollowing solutionin 1745,shown in figure10
1. thesemicircle% " g is drawn andbisectedat point u2. thecircle
�wvwith centreu andradius uh% intersectstheellipseat point x
3. theisoscelestriangleis drawn asshown; its intersectionwith theaxesprovides � and !Stirling’sapproximationwasrecentlyshown to beextremelyaccurate[27]. Unfortunatelyhowever, duetothe shallow intersectionof the circle andellipsein stage2 it is difficult to determinepoint x , makingitsgeometricconstructionratherimpractical. In addition,it requiresthe trueellipseto be drawn prior to theoval!
3.9 Squaring of the circle
φ1
(xt,yt)
(a)
1
φ
(b)
Figure11: Squaringof thecircle
In theratherdifferentcontext of sacredgeometryLawlor [17] describesaconstructionthatcouldleadtoanoval. Figure11 illustratesanattemptat squaringthecircle, i.e. usingonly a compassandstraight-edgeto constructa squarewith perimeterequalto the circumferenceof a givencircle. First a circle with unitradiusis drawn with two half-sizedcirclesinside.Takingthetopof theoutercircleascentre,acirculararcis drawn suchthatit is tangentto boththeinnercirclesasshown in figure11a.Thepoint of contactis
�zyk{|�\}~{ � * 3<X �F���U� #h3 �
where �O* 8 � NU�� I 3 5 � 3 Q is the GoldenRatio which often appearsin sacredgeometryandtheoriesofproportionsand aesthetics.The radiusof the arc is thus � , and the intersectionof the arc with the Xaxis is at y�* < � . Drawing anothercircle centredat the origin andpassingthroughthe intersection(i.e.with radius
< � ) producesa circle whosecircumferenceis En� < � I J�5 R�R . The perimeterof the squarecircumscribingthe initial circle equalseight, andthusprovidesa very closeestimateto the final circle’scircumference.Figure11aalsoprovidesa meansof constructinga circular approximationto an ellipse,usingtheinnercircles(radius
�� ) andthearcsof radius< � joining at �zy { �\} { � . Thecentresareat
�$* "E = !>* " 5
thefixedaspectratioof theoval is � , andtheratioof theradii of thearcsis E�� .
11
1.0� 1.5� 2.0� 2.5�a / b�
0
2
4
6
8
max
imum
err
or
�
Serlio I�Serlio II�Serlio III�Serlio IV�3 squares�4 circles�half-square triangleVignola�KitaoMottBianchiSimpson�golden ratio�optimal C1�
Figure12: Approximationerrorsof theovalswith respectto thedesiredellipse
4 Fidelity to the Ellipse
Having definedasubstantialrangeof oval constructionstheobviousquestionis: how well do they approx-imatethedesiredellipse?Theanalyticsolutionsarecomplicatedby the involvementof theelliptic curveandsoa numericalapproximationto theapproximationerrorsis calculatedinstead.Thecirculararcsaresampledat approximatelyequallyspacedpoints,andat eachpoint the distancealong the normal to theellipseis approximated.See[26, 27] for moredetails. Fixing %>* 3 ��� andusing1000samplepointsintotal thegraphin figure12wasgenerated.Althoughonly themaximumerroris displayedtheaverageerrorexhibitsa similarpattern.
Theerrorincurredby theoptimaltangentcontinuous( u �) oval (whichwasnumericallyestimatedby a
1D search)[27] is includedfor reference.WecanseethatSerlio’sconstructionsdoreasonablywell, but arecertainlynot theclosestto theellipse(althoughof coursethis maynot reflecttheir aestheticqualities).Forinstance,Simpson’sconstructiondoesuniformly well andis generallysuperior(it waspreviously foundtooutperformmostof themoremodernmethodstoo[27]). In addition,Vignola’sconstructiondoesespeciallywell. Nevertheless,the extensionsof all of Serlio’s constructions(i.e. half-squaretriangle,3 squares,4circles,andKitao’s generalisation)mostly performpoorly (the exceptionis the half-squaretrianglecon-structionat low eccentricities).This shows how apparentlyplausibleconstructionsdo badly, andsuggeststhatsomecarewastakenin developingSerlio’s original constructions.This argumentis supportedby thefactthatSerlio’s constructionI is alwaysbetterthanBianchi’s,andmostlybetterthanMott’s construction.It is interestingto notethatfor ovalswith aspectratiosjust lessthantwo Serlio’sconstructionI approachestheoptimalapproximation.
The oval approximationsof ellipseswith low eccentricityaremostly good,andthe errorsarebarelynoticeable.For moreelongatedellipseswe canseethediscrepanciesmoreclearly, asshown in figure13.The first two show four-arc ovals discussedin this paper, andthe superiorityof Stirling’s approximationoverSerlio’sconstructionis obvious.Thefollowing two ovalsareconstructedusingeight-arcs– detailsaregivenin [28]. Naturally, usingmorearcsit is possibleto improvetheaccuracy of theapproximationto theellipse,asis achievedby Walker’smethod.Perhapssurprisingly, anumberof eightarcovalswerefoundtobeinferior to four arcovals,asdemonstratedby Lockwood’soval; seealsoRosin[27].
�]�This circle is in factthelocusof arcjointssatisfyingthe � �
constraint[27].
12
(a)Serlio (b) Stirling
(c) Lockwood (d) Walker
Figure13: Variousoval constructionsshowing their discrepanciesagainsttheellipse
5 An Application
We finish by showing that Serlio himself madegooduseof his methodsof oval construction.Figure14shows his plan for anoval churchfrom theTutte l’Opere d’Architettura. Thebestfit ellipseis overlaidinfigure14aandthediscrepanciesatthediagonalsareevident.Thebestfit ovalwasdeterminedby performinganoptimisationover all parametersusingPowell’s method,andascanbeseenin figure14bnot only doesit provideabetterfit to thechurch’sperimeterbut is clearlySerlio’sconstructionIV.
Acknowledgements
I would like to thankIan Tweddlefor providing a copy of his translationof Simpson’s constructionfromtheoriginalLatin.
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