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Alberto Lastra
Parametric Geometry of Curves and Surfaces Architectural Form-Finding
MathematicsandtheBuiltEnvironment Volume5
SeriesEditors
MichaelOstwald ,BuiltEnvironment,UniversityofNewSouthWales,Sydney, NSW,Australia
KimWilliams,KimWilliamsBooks,Torino,Italy
EditedbyKimWilliamsandMichaelOstwald.
Throughouthistoryarichandcomplexrelationshiphasdevelopedbetween mathematicsandthevariousdisciplinesthatdesign,analyse,constructandmaintain thebuiltenvironment.
Thisbookseriesseekstohighlightthemultifacetedconnectionsbetweenthe disciplinesofmathematicsandarchitecture,throughthepublicationofmonographs thatdevelopclassicalandcontemporarymathematicalthemes–geometry,algebra, calculation,modelling.Thesethemesmaybeexpandedinarchitectureofanyera, cultureorstyle,fromAncientGreekandRome,throughtheRenaissanceand Baroque,toModernismandcomputationalandparametricdesign.Selectedaspects ofurbandesign,architecturalconservationandengineeringdesignthatarerelevant forarchitecturemayalso beincludedintheseries.
Regardlessofwhetherbooksinthisseriesarefocusedonspecificarchitecturalor mathematicalthemes,theintentionistosupportdetailedandrigorousexplorations ofthehistory,theoryanddesignofthemathematicalaspectsofbuiltenvironment.
Moreinformationaboutthisseriesat http://www.springer.com/series/15181
AlbertoLastra ParametricGeometry ofCurvesandSurfaces ArchitecturalForm-Finding AlbertoLastra DepartamentodeFísicayMatemáticas UniversidaddeAlcalá AlcaládeHenares,Spain
ISSN2512-157XISSN2512-1561(electronic) MathematicsandtheBuiltEnvironment
ISBN978-3-030-81316-1ISBN978-3-030-81317-8(eBook) https://doi.org/10.1007/978-3-030-81317-8
MathematicsSubjectClassification:53A04,53A05,00A67
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2021
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations.
ThisbookispublishedundertheimprintBirkhäuser, www.birkhauser-science.com,bytheregistered companySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Amisabuelos
Preface Theparametricaspectsofcurvesand surfaceshavebeenstudiedfromthepoint ofviewofdifferentialgeometrythroughhistory.Indeed,manydifferentstudies havebeendevelopedsincethenineteenthcenturyonthisdiscipline,whichcanbe foundindetailintextssuchasdoCarmo(1976),Tapp(2016),Umeharaetal. (2017).Apartfromthetheoreticalrelationofacurve(orsurface)toanyofits parametrizations,onecangoastepfurtheranddescribeitfromapracticalpoint ofview.Thegeometricschemeofacurveorasurfacehasprovidedinspiration fornumerousworksofartandarchitecture.Thesecreationsnotonlyrespondto physicalneedssuchascertainacousticproperties,lighting,etc.,butalsotoahuman desiretocreatestructureswithsimplegeometricshapes.Thisbooksdescribesthe classicaltheoryofparametrictoolsinthegeometryofcurvesandsurfaceswithan emphasisonapplicationstoarchitecture.
Thisbookisbasedonadecadeofteachinga classongeometryinarchitectural studiesatUniversidaddeAlcalá(Spain).Thisclass,“DrawingWorkshopII”, combinedarchitecturaldesignandmathematics.Nevertheless,itcanalsobeused asatextondifferentialgeometryformathematicsstudents,orabasicreference onmathematicsforarchitectsanddesigners(especiallythoseworkingwithCAD). Ihopethatthelatterwillfindthistextusefulandinteresting,sheddinglighton thetheoreticalaspectsoftheirwork,aswellasontheapplicabilitytoarchitecture. Thetechniquesusedintheexamplesprovidedinthetextserveasthemathematical realizationofmanygeometrictoolsusedinCADprogramssuchastheconstruction ofanhelix,extrusions,revolutionorruled surfaces,projections,andmanyothers. Ialsoprovidealgorithmsrelatedtosomeofthegeometricobjectsandshowhow differentactionsontheparametrizationschangethenatureofthegeometricobject itself.Thesemathematicaltoolsareimportanttounderstandthestructureofa geometricobjectandtoknowhowtomodifyitconsciously.
Thestructureofthebookisasfollows.
Thefirstchapterisdevotedtothestudyofparametrizationofplanecurves, withspecialfocusonconics.Inthischapter,theimplicitationandapproximation ofcurvesisalsoillustratedwithgeometricexamples.
Fig.1 Structureofthebook
Thesecondchapterdescribesaparalleltheoryonspacecurves,andtheappearanceofsuchcurvesinarchitecture.Geometrictransformationsareperformedona spacecurve,makingexplicitthemathematicsbehindusualactionsincurvedesign.
Thethirdandfourthchaptersconsider,respectively,generalsurfacesandother particularclassesofsurfaceswhichare ofwidespreaduse.Theexamplesrange fromclassicsurfacesinarchitecturetotheparametrizationofsuchfamiliesorthe constructionofotherwhichremainofparticularinterestregardingtheirproperties. Moreprecisely,wefocusonsomecurveslyingonsurfacesandontheintersection ofcurves.Surfacessuchasquadrics,ruledsurfaces,surfacesofrevolution,minimal anddevelopablesurfacesarealsostudied andappliedtoarchitecturalelements.
ThestructureofthebookisillustratedinFig. 1
Themathematicalprerequisitesforthisbookarefirstcoursesintopology,linear algebraandcalculus(bothsingleandmulti-variable),asamplycoveredinthe booksSalasetal.(2003),MarsdenandTromba(2012),andLang(1986);Strang (1993).Forcompleteness,wehaveincludedtwoappendicescoveringknowledge thatwillbeusefulforunderstandingthematerial.
ThefiguresofgeometricobjectshavebeencreatedwithGeogebrasoftware. Iwanttoexpressmygratitudetoeveryonewhowasinvolvedinthisclass, speciallytoProf.ManueldeMiguel,whointroducedmetotheworldofarchitecture. IalsowanttoexpressmygratitudetoRemiLodh,whohasguidedmeonits publicationwithhighprofessionalismandalsotoKimWilliamsforherenthusiasm, professionalismandeffortintherevisionofthemanuscript,andalsogivingrelevant andinterestingdetails.
SuggestedFurtherReading Thefollowingsourcesaresuggestedtoin terestedreadersseekingadditional material(AAG 2008, 2010, 2013, 2014, 2016, 2018;Bridges 2003, 2004, 2008, 2011, 2012, 2014, 2016, 2018).
AlcaládeHenares,SpainAlbertoLastra 2021
1ParametrizationsandPlaneCurves
1.1PlaneCurvesandParametrizations
1.2SomeClassicCurvesinArchitecture
1.3SomeElementsofRegularPlaneCurves
1.5SomeConicsinArchitecture
1.6OntheImplicitationandParametrizationofCurves
1.7ApproximationandInterpolationofCurves
1.8SuggestedExercises
2ParametrizationsandSpaceCurves
2.2SomeElementsofRegularSpaceCurves
2.3SomeClassicSpaceCurvesinArchitecture
2.4RigidTransformationsin
2.5SomeTransformationsonaHelix
2.6SuggestedExercises
3ParametrizationsandRegularSurfaces
3.2SomeClassicSurfacesinArchitecture
3.3ProjectionsofSurfacesontoPlanes
3.4CurvesinSurfacesandIntersectionofSurfaces
3.5SuggestedExercises ....................................................136
4SpecialFamiliesofSurfaces
4.1RuledSurfaces
4.2SomeSubfamiliesofRuledSurfaces
4.3ParametrizationofSomeRuledSurfaces
4.4SurfacesofRevolution
4.6QuadricsRevisited:SomeExamplesinArchitecture
4.7Curvature:MinimalandDevelopableSurfaces .......................190
4.7.1FinalComments ...............................................200
4.8SuggestedExercises ....................................................201
ACoordinateSystems ..........................................................205
BMathematicalToolKit .......................................................213
B.1IntroductiontoLinearAlgebra .........................................213
B.1.1SystemsofLinearEquations ..................................213
B.1.2VectorSpaces ..................................................214
B.1.3EuclideanVectorSpaces .......................................215
B.1.4Diagonalization:EigenvaluesandEigenvectors ..............215
B.2RealFunctionsofOneVariable ........................................216
B.3FunctionsofSeveralRealVariables ...................................219
B.4DifferentialEquationsandSystemsofDifferentialEquations .......221
CSolutiontotheSuggestedExercises ........................................223
C.1Chapter1 ................................................................223
C.2Chapter2 ................................................................235
C.3Chapter3 ................................................................245
C.4Chapter4 ................................................................249
AbouttheAuthor AlbertoLastra hasaPh.D.inMathematicsbytheUniversityofValladolid.He isassociateprofessorattheUniversity ofAlcalá(Spain).Hehasbeenteaching mathematicsinthedegreeofArchitectureandFundamentalsinArchitectureand UrbanismattheUniversityofAlcalásince2011,insubjectsunderthepointof viewofinnovationandinterdisciplinarythinkinginArchitecture.Hisresearch interestsdonotonlygointhepreviousdirection,butalsointhestudyofasymptotic analysisoffunctionalequationsinthecomplexdomainandrelatedtopics,symbolic computation,ororthogonalpolynomials.Heisamemberoftheresearchgroups ECSING-AFAoftheUniversityofValladolidandASYNACS(CT-CE2019/683) oftheUniversityofAlcalá.Hehasalsobeenavisitoratforeignresearchcenters duringthelastdecade,suchastheUniversityofLille(France),theUniversityof Warsaw(Poland),theUniversityofLaRochelle(France),UniversidadeFederalde MinasGerais(Brazil),amongothers.
ListofSymbols AT Transposematrixofthematrix A
C ∞ (U) Setofscalarorvectorfunctionswhicharedifferentiableforevery degreeofdifferentiationintheopenset U
d(P,Q) Euclideandistancefromapoint P ∈ Rn to Q ∈ Rn
dX(u0 ,v0 ) (v) differentialof X : U ⊆ R2 → R3 at (u0 ,v0 ) ∈ U ,evaluatedat v ∈ R2
D(P,r) Disccenteredatthepoint P andradius r> 0
d dx derivativewithrespecttothevariable
∂
∂x , ∂ ∂y , ∂ ∂z ,...Partialderivativewithrespectto x , y , z ,...
Mm×n (K) Setof m × n matriceswithcoefficientsinafield K
I(ω1 ,ω2 ) Firstfundamentalform
II(ω) Secondfundamentalform
∇ f(P) Gradientofthefunction f ,evaluatedatthepoint P
⊥ Orthogonal
rank(A) Rankofamatrix A
∼ Asymptoticequivalence
· , · or · Innerproductin Rn
× Crossproductin R3
[·, · , ·]
Scalartripleproduct
· Euclideannormin Rn
C Setofcomplexnumbers
Q Setofrationalnumbers
R Setofrealnumbers
R Setofrealnumbers,exceptfromtheorigin. R \{0}
Z Setofintegernumbers
NZ \{0, 1, 2,...}
PQ vectorfromthepoint P tothepoint Q ofaEuclideanspace
v vectorofanEuclideanspace
Im(f) Rangeofafunction,i.e. {f(x) : x ∈ X },whenever f : X → Y
Ker(f) Kernelofafunction f : X → Rn ,i.e. {x ∈ X : f(x) = 0}
v ||w Thevectors v and w areparallel
xiii
ListofFigures Fig.1Structureofthebook .................................................viii
Fig.1.1Circlecenteredat (0, 0) andradius R = 3(left), lemniscateofBernoulli(right) ......................................2
Fig.1.2Cardioid(left)andepicycloid(right) ...............................3
Fig.1.3Regularcurve .........................................................5
Fig.1.4CounterexampleofregularcurveinExample1.1.10 ..............8
Fig.1.5KimbellArtMuseum ................................................9
Fig.1.6Rollingcircleproducingacycloid.QRCode1 ....................10
Fig.1.7Catenary, a = 1.QRCode2 ........................................11
Fig.1.8Catenaryarchs .......................................................12
Fig.1.9Lemniscate ...........................................................13
Fig.1.10Thehigh-speedtrainstationReggioEmiliaAV Mediopadana,Italy ..................................................14
Fig.1.11Spirals:CasaBatló,byAntoniGaudí(left)andStaircase (right) .................................................................14
Fig.1.12Logarithmicspiral. a = 1,b = 0.3 .................................15
Fig.1.13Secant ................................................................18
Fig.1.14Normalline ..........................................................19
Fig.1.15Successiveapproximationsofthelengthofacurveby segments .............................................................22
Fig.1.16Curvatureatapoint,I ...............................................24
Fig.1.17Curvatureatapoint,II.QRCode3 .................................25
Fig.1.18Conicsassectionsofaconebyaplane,I ..........................27
Fig.1.19Conicsassectionsofaconebyaplane,II .........................28
Fig.1.20Orthogonaltransformationofacoordinatesystem ................29
Fig.1.21Campidogliosquare,Rome.Engraving:ÉtienneDupérac, 1568 ..................................................................39
Fig.1.22AerialviewofSt.Peter’ssquare,inRome .........................39
Fig.1.23CathedralofBrasilia,byOscarNiemeyer ..........................40
Fig.1.24OceanogràficbyFélixCandela .....................................41
Fig.1.25QRCode4 ...........................................................44
Fig.1.26Graphof y = x 2 rollingaround y =−x 2 (symmetric parabolaof y =−x 2 withrespecttothetangentlines) ............44
Fig.1.27Rouletteofapointdrawingacissoid ...............................44
Fig.1.28Constructionofthecissoid ..........................................45
Fig.1.29ConchoidofNicomedes .............................................47
Fig.1.30QRCode5(left)Deltoidwith R/r = 3;QRCode6 (right)Astroidwith R/r = 4 .......................................48
Fig.1.31QRCode7(left)Cardioidwith R/r = 1;QRCode8 (center)Nephroidwith R/r = 2;QRCode9(right) Epicycloidwith R/r = 3 ...........................................48
Fig.1.32Anephroidasanepicycloid .........................................49
Fig.1.33Deltoid(left)andastroid(right)ashypocycloids ..................49
Fig.1.34StainedglasswindowfromSaint-ChapelleinParis
Fig.1.35Boor–deCasteljaualgorithm ........................................52
Fig.1.36QRCode10 ..........................................................52
Fig.1.37TeatropopularinNiterói,Brazil,byOscarNiemeyer .............54
Fig.1.38TheTeatropopularwithauthor’soverlay ..........................55
Fig.2.1Circlecenteredat P = (0, 0, 1) andradius R = 3,at height z = 1(left);Helix(right) ....................................60
Fig.2.2CurvesinFig.2.1determinedbytheintersectionof surfaces ...............................................................60
Fig.2.3Viviani’scurve(left),andsolenoidtoric(right) ....................61
Fig.2.4 α(t) = (t 2 ,t 3 ,t 4 ), t ∈ ( 2, 2) ......................................62
Fig.2.5CurveinExample2.1.5 .............................................63
Fig.2.6Parametrizations (I1 ,α1 ) and (I2 ,α2 ) associatedto Viviani’scurve .......................................................63
Fig.2.7Regularcurve ........................................................64
Fig.2.8Secantlines ..........................................................68
Fig.2.9Approximationofacurvewithpolygonalchains ..................71
Fig.2.10Frenettrihedroninaspacecurve ...................................75
Fig.2.11LinesandplanesassociatedtotheFrenettrihedron ...............76
Fig.2.12Positivetorsion(left)andnegativetorsion(right)in Example2.2.27 ......................................................78
Fig.2.13Localshapeofacurveatapoint ....................................82
Fig.2.14Exampleoflocalshapeofacurveatapoint .......................83
Fig.2.15Someosculatingcirclesassociatedtoacurve ......................84
Fig.2.16QRCode11 ..........................................................84
Fig.2.17Circularhelix ........................................................87
Fig.2.18Projectionsofthecircularhelix .....................................88
Fig.2.19QRCode12 ..........................................................88
Fig.2.20Examplesofhelicesinarchitecture .................................89
Fig.2.21Twistedcubic(2.26),for a = b = c = 1 ...........................90
Fig.2.22Differentprojectionsofthetwistedcubic ..........................90
Fig.2.23CapitalGateTowerinAbuDhabi ...................................91
Fig.2.24Translationofvector v = (1, 2, 3) ofthecurvein Example2.4.3.QRCode13 ........................................93
Fig.2.25Rotationofangle β = π/4around {y = z = 0} ofthe curve (I, α) inExample2.4.3.QRCode14 ........................94
Fig.2.26Reflectionwithrespecttotheplane x = 0ofthecurve α inExample2.4.3.QRCode15 .....................................95
Fig.2.27Circularhelix(black)vs. (R,α0 ) (green) ..........................97
Fig.2.28Helix (R,α1 ) for ρ = 1(black)vs. ρ = 3(green) .................97
Fig.2.29Helix (R,α2 ) for ρ = 1, h = 1(black)vs. h = 1/2 (green) ................................................................98
Fig.2.30Helix (R,α3 ) for ρ = 1, h(t) = t (black)vs. h(t) = t + π 2 (green) ................................................................99
Fig.2.31Helix (R,α3 ) for ρ = 1, h(t) = t (black)vs. h(t) = t 3 (green) ................................................................100
Fig.2.32Helix (R,α3 ) for ρ = 1, h(t) = t (black)vs. h(t) = exp(t) (green) ................................................100
Fig.2.33Helix (R,α3 ) for ρ = 1, h(t) = t (black)vs. h(t) = sin(t/3) (green) ..............................................101
Fig.3.1Spherecenteredat (1, 0, 0) andradius R = 2(left); hyperbolicparaboloid(right) ........................................104
Fig.3.2 { ∂x ∂u (u0 ,v0 ), ∂y ∂u (u0 ,v0 ), ∂z ∂u (u0 ,v0 ) , ∂x ∂v (u0 ,v0 ), ∂y ∂v (u0 ,v0 ), ∂z ∂v (u0 ,v0 ) } ...............................105
Fig.3.3Autointersection .....................................................106
Fig.3.4Regularsurface ......................................................106
Fig.3.5Localcoveringsoftheunitsphere ..................................107
Fig.3.6Localcoveringsoftheconeminusthevertex ......................108
Fig.3.7HelixcontainedinacylinderinExample3.1.13 ...................112
Fig.3.8Schemeoftheconstructionoftheregularcurvecontained inaregularsurface ..................................................113
Fig.3.9Schemeoftheconstructionofaregularplanecurvefrom aregularcurvecontainedinasurface ..............................113
Fig.3.10Tangentplaneto z exp((x 1)2 + y) = 0at P = (1, 0, 1) ....114
Fig.3.11Normalvectorto X(u,v) = (u,v,e (u 1)2 +v ) at P = (1, 0, 1) .........................................................116
Fig.3.12Torus .................................................................117
Fig.3.13Geometricconstructionofthetorus ................................118
Fig.3.14Dubai’sMuseumoftheFuturebyKillaDesign ...................119
Fig.3.15QRCode16 ..........................................................120
Fig.3.16DetailoftheconstructionofaMöbiusband .......................121
Fig.3.17Möbiusband .........................................................121
Fig.3.18PhoenixInternationalMediabyShauWeipingofBIAD ..........122
Fig.3.19Kleinbottle. r = 3 ...................................................123
Fig.3.20Orthogonalprojection π onaplane ................................124
Fig.4.23Surfaceofrevolution ................................................167
Fig.4.24Torus; r = 1, R = 3 .................................................168
Fig.4.25ApproximationoftheWatertowerinFedalaasasurface ofrevolution .........................................................169
Fig.4.26Orthogonaltransformationofacoordinatesystem ................170
Fig.4.27Ellipsoid ..............................................................172
Fig.4.28Sectionsofanellipsoidincanonicalformbythe coordinateplanes ....................................................173
Fig.4.29Hyperboloidofonesheet ............................................173
Fig.4.30Sectionsofahyperboloidofonesheetincanonicalform bythecoordinateplanes .............................................174
Fig.4.31Hyperboloidoftwosheets ..........................................175
Fig.4.32Sectionsofahyperboloidoftwosheetsincanonicalform bythecoordinateplanes .............................................175
Fig.4.33Cone ..................................................................176
Fig.4.34Ellipticparaboloid ...................................................177
Fig.4.35Sectionsofanelliptic paraboloidincanonicalformat positiveheight(left)andwith y = 0(right) ........................178
Fig.4.36Hyperbolicparaboloid ...............................................178
Fig.4.37Sectionsofahyperbolicparaboloidincanonicalformat positive(left),negative(center)andnull(right)height ............178
Fig.4.38Sectionsofahyperbolicparaboloidincanonicalform withtheplanes x = 0and y = 0 ....................................179
Fig.4.39QuadricdefinedinEq.(4.12)andparametrizedby Eq.(4.13) ............................................................185
Fig.4.40Sheratonhotel ........................................................188
Fig.4.41Apple ..................................................................188
Fig.4.42JamesS.McDonnellPlanetariumbyGyoObata ..................189
Fig.4.43ScotiabankSaddledomebyGECArchitecture .....................190
Fig.4.44CatenoidofExample4.7.5,with a = 1 ............................195
Fig.4.45OlympiastadioninMunichbyFreiOtto ............................198
Fig.4.46AnEnnepersurface ..................................................199
Fig.4.47QRCode20 ..........................................................202
Fig.4.48Exampleofcurveshifting.QRCode21 ............................203
Fig.A.1ACartesiancoordinatesystem ......................................206
Fig.A.2Polarcoordinatesystem .............................................207
Fig.A.3Cartesiancoordinatesofthepoint (1, 2, 3) .........................208
Fig.A.4LinesintheGranViaCapitalHotel,SpainbyLaHoz Arquitectura ..........................................................209
Fig.A.5Cylindricalcoordinatesofthepoint (1, 2, 3) .......................210
Fig.A.6Sphericalcoordinatesofthepoint (1, 2, 3) .........................210
Fig.A.7CloudGateinChicagobyAnishKapoor ..........................211
Fig.B.1 f(x) = sin(x) andsomeTaylorpolynomialsat x = 0 ............217
Fig.B.2Areabetween f(x) = 2xe x 2 4x and OX from x = 0 and x = 1 ............................................................219
Fig.C.1Cusp.Exercise1.5 ...................................................224
Fig.C.2Tangentlineinapointofalemniscate.QRCode22. Exercise1.6 ..........................................................225
Fig.C.3Orthogonallinestothesymmetryaxisoftheparabola. Exercise1.14 .........................................................228
Fig.C.4Twosecantlines.Exercise1.16 .....................................229
Fig.C.5Hyperbola.Exercise1.17 ............................................231
Fig.C.6Ellipse.Exercise1.18 ................................................232
Fig.C.7CatenaryarcandthreeLagrangeapproximationsin [0, 1], a = 1.Exercise1.24 ..........................................235
Fig.C.8Spacecurve.Exercise2.1 ...........................................236
Fig.C.9TheconicalspiralofPappus.Exercise2.1 .........................237
Fig.C.10Exampleofspacecurve.Exercise2.2 ..............................237
Fig.C.11Exampleofaloxodromecontainedinatorus.Exercise2.3 .......239
Fig.C.12Constructionofacurveofconstantcurvatureandtorsion.
Exercise2.7 ..........................................................241
Fig.C.13Lamésurfacefor p = 1/2.Exercise3.7
Fig.C.14Intersectionofsurfaces.Exercise3.8
Fig.C.15Intersectionoftwoparabolas.Exercise3.9
Fig.C.16Cylindricalsurface.Exercise4.1
Fig.C.17Conicalsurface.Exercise4.2
Fig.C.18Tangentdevelopablesurface.Exercise4.3
Fig.C.19Quadric.Exercise4.4
Fig.C.20Geometricscheme.Exercise4.5
Fig.C.21Geometricscheme.Exercise4.6
Fig.C.22Geometricscheme.Exercise4.7
Fig.C.23Elliptictorus.Exercise4.11
Fig.C.24Generalizedelliptictorus.Exercise4.12
Fig.C.25Conoidstructure.Exercise4.13
Fig.C.26Secondconoidstructure.Exercise4.13
Fig.C.27Pseudosphere.Exercise4.14
Fig.C.28QRCode23.Exercise4.15
Fig.C.29Sweepingcurve.Exercise4.15
PhotographCredits Figure 1.5 Source:ByPhoto:AndreasPraefcke—Self-photographed,Public Domain,
Link: https://commons.wikimedia.org/w/index.php?curid=8382419
Figure 1.8 (left)Source:Mattancherrykoonankurish,Kochi,Kerala,India. KoonanKurishPalli,Flickr.com.
Link: https://flic.kr/p/V4vMDm
Figure 1.8 (right)Source:PhotobyJohnsonLiuonUnsplash. Link: https://unsplash.com/photos/C3SEO9ORkMg
Figure 1.10 Source:PhotobyLucaBravoonUnsplash
Link: https://unsplash.com/photos/alS7ewQ41M8
Figure 1.11 (left)Source:PhotobyAndreaJunqueiraonUnsplash Link: https://unsplash.com/photos/mNoMLlDDJbg
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Chapter1 ParametrizationsandPlaneCurves Thisfirstchapterisdevotedtothestudyofplanecurvesfromthepointof viewofdifferentialgeometry.Asmentionedinthepreface,thistheoryisquite classicalandcanbefoundindifferentundergraduateandgraduatetextbookssuch asdoCarmo(1976),andmorerecentbooks(Tapp 2016;Umeharaetal. 2017). Thetheoryisillustratedwithexamplesinspecificarchitecturalelements,together withmathematicaltechniquesappliedonplanecurves.Amongthem,wefocuson implicitation,approximationandinterpolationtechniques.
Differentapproachesmightbefollowedinordertoreachanadequateand consistentdefinitionofcurve:ontheonehand,thepointofdeparturecanbelocated atthedefinitionofaparametrizedcurve,arrivingattheconceptofthearcofacurve inimplicitform,whilstotherauthorsprefertoapproachtheconceptofcurveviathe levelcurvesofcertainsurfacesandconcludewithlocalparametrizations.Wewill followthissecondroute,relatingbothapproachesbymeansoftheimplicitfunction theorem.Certaintopologicaldiscussionsareimportantwhengoingfromonetothe otherpointofview.Forourpurpose,wewillonlydealwithcurveswhicharelocally homeomorphictoanopensegment,i.e., suchthatateachofitspointsthecurveis “essentially”abentopenintervalnearthatpoint.
Boththeimplicitandparametricformsturnouttobeimportantwhenhandling acurve.Theuseofaparametrizationofacurveismoremanageablewhentrying toconstructacurvebygivingdifferentvaluestotheparameter,andtheimplicit formallowsustodetectdirectlywhetherapointbelongstoagivencurveornot. Anotheradvantageofhavingtheimplicitrepresentationofacurveisthenumerical applicationofthefalsepositionmethod(seeBurdenandFaires 2000,p.72).Also, continuityofthefunctiondeterminingthecurveinimplicitformmakesitpossible toobtainslightlyperturbedcurveswhichassembleaplanecurvedeterminedby f(x,y) = 0andthecurveassociatedtothefunction f(x,y) + c ,byconsidering smallenough c ∈ R.
Moreover,someelementsrelatedtoacurvesuchasthecurvature,tangentand normalvectors,etc.provideawiderknowledgeofthecurve.Inadditiontothis,
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2021 A.Lastra, ParametricGeometryofCurvesandSurfaces,MathematicsandtheBuilt Environment5, https://doi.org/10.1007/978-3-030-81317-8_1
thedefinitionofacurveitselfmayincorporatedifferentphysicalor/andaesthetic propertiessuchasbuildingacoustics,theconfigurationofloads,architectural lighting,etc.
Wehavedecidedtoconsiderthestudyofspacecurvesseparatelytoemphasize theimportanceofsuchcurves,inparticularhelices,inarchitecture.Asamatterof fact,thetwopreviousapproachescanbe rephrasedafternaturaladjustments.
1.1PlaneCurvesandParametrizations Wefirststatetheconceptofaplanecurve,mainlyconsistingintheso-calledlevel curveofaregularsurface.Intuitively,aplanecurverepresentsthepathfollowedby amovingpointinsideaplane.
Acircleandalemniscate(seeFig. 1.1)areexamplesoftheintuitiveconceptofa planecurve.
Thecircleconsistsofallpointsintheaffineplane, (x,y) ∈ R2 ,whose componentssatisfytheequation x 2 + y 2 = 9:
Thelemniscatecorrespondstotheset
Wedefineaplanecurvefollowingthisapproach,bymeansofalevelcurveofa scalarfunction f intworealvariables.
Definition1.1.1 Let ∅ = U ⊆ R2 beanopensetandlet f : U → R bea C ∞ functionin U ,i.e.,afunctionthatisdifferentiableforeverydegreeofdifferentiation in U .Wewrite f ∈ C ∞ (U)
Fig.1.1 Circlecenteredat (0, 0) andradius R = 3(left),lemniscateofBernoulli(right)
Incasetheset C ={(x,y) ∈ U : f(x,y) = 0}
isnonempty,wesaythat C isa planecurve
Acardioidconsistsofthepoints (x,y) ∈ R2 suchthat (x,y) = (2cos(t) cos(2t), 2sin(t) sin(2t)), (1.2)
forsome t ∈ R.Anepicycloidisaplanecurvedeterminedbythepoints (x,y) ∈ R2 satisfying (x,y) = (r1 + r2 ) cos(t) r2 cos r1 + r2 r2 t ,(r1 + r2 ) sin(t) r2 sin r1 + r2 r2 t ,
forsome t ∈ R,andwhere r1 ≥ r2 arefixedpositiverealnumbers.
Anotherapproachtodefineaplanecurve(orpartofaplanecurve)isbymeans ofaparametrizationofsuchcurve,aswehavealreadyobserved(seeFig. 1.2).
Definition1.1.2 A parametrization isapair (I,α),where I ⊆ R isanopen intervaland α : I → R2 isa C ∞ functionon I ,i.e. α ∈ C ∞ (I).
Itisworthpointingoutthatthehypothesisof f belongingtotheset C ∞ (I) istoo restrictivefortheunderlyingtheory.Asamatteroffact,thepreviousdefinitioncan bestatedonothermoregeneralsubsets I ⊂ R.Wehavedecidedtomaintainsuch restrictionsforthesakeofsimplicityinourreasoning,andplaceanemphasisonthe applicationsinarchitectureratherthantheaccuracyonthehypothesesmadeonthe results.
Wewillmainlyworkwithregularparametrizations.
Cardioid(left)andepicycloid(right)
Fig.1.2
Definition1.1.3 Aparametrization (I,α) isa regularparametrization ifit satisfiesthefollowingconditions:
• α (t) = (0, 0) forall t ∈ I ;
• α : I → R2 isaone-to-onefunction.
ThefirststatementinDefinition 1.1.3 allowsustodefinethetangentlineateach pointofthecurveassociatedtotheregularparametrization.
Intheexampleofthecardioid,wehavethat
α(t) = (2cos(t) cos(2t), 2sin(t) sin(2t)),
t ∈ ( π,π) coversthewholecurveexceptfromthepoint P1 = ( 3, 0).Observe that
α (t) = ( 2sin(t) + 2sin(2t), 2cos(t) 2cos(2t)),
therefore α (0) = (0, 0)
Thepropertyofinjectivityassociatedto aregularparametrizationisneededin ordernottoobserveself-intersectionpointsinthecurve,ashappenedintheexample ofthelemniscate.
InordertofulfillthetwoconditionsinDefinition 1.1.3,theuseof local parametrizations ofcurvesisuseful.Sofar,wehavedefinedplanecurvesbymeans ofthelevelcurvesofsomefunctionintwovariables.Givenaparametrization,itis naturaltothinkthattheimageofaparametrizationdeterminesacurve(orpartof it).
Definition1.1.4 Givenaplanecurve C ,wesaythattheparametrization (I,α) isa parametrizationofthecurve C if C = α(I)
Theimagesetofaparametrization (I,α), α(I),isknownasan arc.Anarcis saidtobe regular ifitisdescribedbyaregularparametrization.
Thepreviousdefinitionstatesthatgivenacurve C parametrizedby (I,α),its associatedarcis C (Fig. 1.3).
Regardingthecardioid,thearcdeterminedbytheparametrization
α(t) = (2cos(t) cos(2t), 2sin(t) sin(2t)), for t ∈ (0, 2π),coversthewholecurve,exceptforthepoint P2 = (1, 0).With respecttothelemniscate,onecanconsidertheparametrizations
whichtogetherdrawthewholecurve,exceptfortheorigin.
Fig.1.3 Regularcurve
Thepreviousexamplesmotivatethefollowingdefinitionofaregularcurve.
Definition1.1.5 Aset ∅ = C ⊆ R2 isa regular(plane)curve ifforevery (x0 ,y0 ) ∈ C thereexistsadisc D((x0 ,y0 ),r) ⊆ R2 ,suchthat D((x0 ,y0 ),r) ∩ C is aregulararc.
Thefollowingresultisadirectapplicationoftheimplicitfunctiontheorem.
Theorem1.1.6 Let U ⊆ R2 beanonemptyopensetof R2 .Let f : U → R with f ∈ C ∞ (U) and (x0 ,y0 ) ∈ U suchthat ∇ f(x0 ,y0 ) = ∂f ∂x (x0 ,y0 ), ∂f ∂y (x0 ,y0 ) = (0, 0).
Weconsidertheset
C ={(x,y) ∈ U : f(x,y) f(x0 ,y0 ) = 0} =∅
Then,thereexist r> 0,anopeninterval I ⊆ R andafunction α : I → R2 , α ∈ C ∞ (I) suchthat
• α (t) = (0, 0) forall t ∈ I ,
• α : I → R2 isaone-to-onefunction,and α(I) = D((x0 ,y0 ),r) ∩ C
Intermsoftheconceptsintroducedabove,Theorem 1.1.6 canbestatedas follows:
Theorem1.1.7 Let U ⊆ R2 beanonemptyopensetof R2 .Let f : U → R with f ∈ C ∞ (U).Considertheset
C ={(x,y) ∈ U : f(x,y) = 0}
If C =∅ andforevery P ∈ C itholdsthat
∇ f(P) = ∂f ∂x (P), ∂f ∂y (P) = (0, 0), then C isaregularcurve.
Proof Let P = (x0 ,y0 ) ∈ C .Weassumethat ∂f ∂x (P) = 0,withoutlossof generality.Theimplicitfunctiontheoremguaranteestheexistenceofanopen interval x0 ∈ I ⊆ R and α : I → R2 suchthat α(x0 ) = y0 and f(t,α(t)) = 0for every t ∈ I
Fromthecontinuityofthefunction t ∈ I → ∂f ∂x (t,α(t)) andthefactthat ∂f ∂x (x0 ,y0 ) = ∂f ∂x (x0 ,α(x0 )) = 0,
thereexists x0 ∈ I1 ⊆ I suchthat ∂f ∂x (t,α(t)) = 0forall t ∈ I1 .
•Fromtheconstructionof α weobtainthat (t,α(t)) ∈ C forall t ∈ I1 .
• α (t) = 0forall t ∈ I1 .Otherwise,assumetheexistenceofsome t0 ∈ I1 with α (t0 ) = 0.Takingderivativesin f(t,α(t)) = 0weobtainthat
∂x (t,α(t)) + ∂f ∂y (t,α(t))α (t) ≡ 0,t ∈ I1 .
Thisyieldsthat ∂f ∂x (t0 ,α(t0 )) = 0,whichcontradictsthechoiceof I1 .Therefore, α (t) = 0forall t ∈ I1 .
• α : I1 → R2 isaone-to-onefunction.Ifthereexist t1 ,t2 ∈ I1 suchthat α(t1 ) = α(t2 ),then,Rolletheoremguaranteestheexistenceof t3 ∈ I1 with α (t3 ) = 0 whichcontradictsthepreviousstatement.
Weobservefromtheproofthatanadequateparametrizationcanbeconsidered foreachpointinthecurve,theresultbeingofalocalnature.Moreover,theexistence ofalocalparametrizationisguaranteedby theimplicitfunctiontheorem,depending onthecomponentofthegradientwhichdoesnotvanish.
Areciprocallocalresultisalsovalid,describingthereciprocalrelationship betweenlocalregularparametrizationsandlevelcurvesofscalarfunctionsintwo variables.
Theorem1.1.8 Let C bearegularcurve.Forevery (x0 ,y0 ) ∈ C thereexist r> 0 andascalarfunction f : D((x0 ,y0 ),r) → R, f ∈ C ∞ (D((x0 ,y0 ),r)),suchthat
C ∩ D((x0 ,y0 ),r) ={(x,y) ∈ D((x0 ,y0 ),r) : f(x,y) = 0},
and ∇ f(Q) = ∂f ∂x (Q), ∂f ∂y (Q) = (0, 0),
forall Q ∈ D((x0 ,y0 ),r).
Proof Let (x0 ,y0 ) ∈ C .Weconsiderthedisc D((x0 ,y0 ),r1 ) ⊆ R2 suchthat D((x0 ,y0 ),r1 ) ∩ C isanarcofregularcurve.Thisentailstheexistenceofa regularparametrization (I,α) with α(I) = D((x0 ,y0 ),r1 ) ∩ C .Letuswrite α(t) = (α1 (t),α2 (t)).Let t0 ∈ I with α(t0 ) = (x0 ,y0 ) andassume,withoutlossof generality,that α1 (t0 ) = 0.
Fromthecontinuityof α1 in I ,wecanguaranteetheexistenceofanopeninterval ∅ = I1 ⊆ I inwhich α1 (t) = 0forall t ∈ I1 .Thisentailsthatthefunction α1 is invertiblein I1 ,with α1 (I1 ) = I2 forsomeopeninterval I2 .Itisnotdifficultto verify,reducing I1 ifnecessary,theexistenceof0 <r ≤ r1 suchthat (x0 ,y0 ) ∈ α(I1 ) = D((x0 ,y0 ),r) ∩ C .Thepair (I1 ,α) isaregularparametrizationwhich parametrizesanarccontainedin C .
Let f : D((x0 ,y0 ),r) → R bedefinedby
f(x,y) = y α2 (α 1 1 (x)).
Observethat f iswelldefinedforall (x,y) ∈ D((x0 ,y0 ),r1 ) duetoinjectivity of α1 .Given (x,y) ∈ D((x0 ,y0 ),r) ∩ C, wehave x = α1 (t) and y = α2 (t),for some t ∈ I1 .Then, t = α 1 1 (x) and y = α2 (α 1 1 (x)).Thefunction f isinfinitely differentiablein D((x0 ,y0 ),r).Inadditiontothis, ∂ ∂y f(x,y) = 1 = 0.
Thepreviousresultsgiverisetotheconceptofregularcurve,whenthepointof departureisanimplicitexpression.
Definition1.1.9 Let U ⊆ R2 beanonemptyopensetof R2 .Let f : U → R with f ∈ C ∞ (U).Wesaythattheset
C ={(x,y) ∈ U : f(x,y) = 0}
isa regularcurve if C =∅ andforall P ∈ C
∇ f(P) = ∂f ∂x (P), ∂f ∂y (P) = (0, 0).
Theexistenceofadiscforeachpoint (x0 ,y0 ) ∈ C suchthattheintersection ofthatdiscandthecurvecoincideswiththeimageofaregularparametrization isessentialinthehypothesesofthedefinitionofaregularcurvegiveninterms ofregularparametrizations(seeDefinition 1.1.5).Theusualtopologyon R2 determinescounterexamplesinthisdirection.
Example1.1.10 Letusconsiderthearcdeterminedbythefollowingparametrization.Thatarciscontainedinthelemniscatedescribedabove inthischapter:
α turnsouttobea C ∞ ( 1, ∞) function.Thepair (( 1, ∞),α) isaregular parametrization:
•Itisdirecttoverifythat α (t) = (0, 0) for t ∈ ( 1, ∞) • α isaone-to-onefunction.
However,foreverydisccenteredat α(0) = (0, 0),thesetobtainedbyintersection ofthecurveandthedisccannotbeparametrizedbyanyregularparametrization. Thatsubsetisnothomeomorphictoasegment,i.e.,itcannotbetransformedby continuoustransformationsintoasegment(thissethastwoconnectedcomponents whereasasegmenthasonlyone(Fig. 1.4)).
Fig.1.4 CounterexampleofregularcurveinExample 1.1.10
1.2SomeClassicCurvesinArchitecture Inthissection,weshowhowclassicplanecurvesareusedtodescribeandinspire architecturalelements.Thisisnotaccidental,andismorelikelyduetosome necessityinthestructure,foraestheticreasons,etc.InthebookHanh(2012),several particularstudiesaremadeonplanecurvesappliedinarchitecturaldesign.
Cycloid,andtheKimbellArtMuseum TheKimbellArtMuseum(ForthWorth,Texas,1972)byLouisKhanisabuilding ofknowngeometriccomplexity.Itsroofconsistsintheconcatenationofseveral vaultsbuiltupfromaplanecurveandparallellinespassingthroughthatcurve(see Fig. 1.5).
Eachofthevaultsisbasedontheplanecurveknownasthe cycloid.Thecycloid isaplanecurvedefinedbyaphysicalphenomena.Letacirclerollonaline.The trailleftbyanyfixedpointinthecircleafterthismovementdrawsacycloid(see Fig. 1.5).
Theequationsdefiningaparametrizationofthecycloidcanbederivedfromthe physicaldefinition,makinguseofelementarytrigonometryandfundamentalsof physics(Fig. 1.6).
Wewrite r> 0fortheradiusoftherollingcircle.Assumetheinitialpositionof therollingcircleisgivenbytheequation x 2 + (y r)2 = r 2 ,andthedistinguished
Fig.1.5 KimbellArtMuseum
