Quadrics Theorems as an Introduction to Geometry,
                                         ParametricDesignandDigitalFabrication
                                                                                           1                                 2
                                                           Roberto Narvaez-Rodriguez , Jose Antonio Barrera-Vera
                                                          1,2
                                                             University of Seville
                                                          1                                  2
                                                           http://personal.us.es/roberto http://personal.us.es/barrera
                                                          1,2
                                                             {roberto|barrera}@us.es
                                                          TheCaterpillar gallery was a teaching innovation project intended to integrate
                                                          geometry, parametric design and digital fabrication at the earliest stage of the
                                                          undergraduate training period. This paper shows the contents, based on this
                                                          project, of the workshop carried out within the 33rd eCAADe conference, Vienna
                                                          2015. The geometrical principles -stemming from certain quadrics theorems-, the
                                                          parametric definition and the digital fabrication of the reduced-scale model
                                                          executed in the workshop are outlined and illustrated.
                                                          Keywords:Architectural geometry, Quadrics, Parametric Design, Digital
                                                          Fabrication, Education
                                         INTRODUCTION                                              forms. Compared with the conventional paradigm,
                                         The use of computation in architectural design has        this means major changes in the way in which ge-
                                         definitely opened a new paradigm in architecture           ometry is applied, represented and even managed
                                         (Terzidis, 2003). Thefocusofthenewdesignstrategy          along the different stages of a project.       In fact,
                                         hasmovedfromtheobjecttotheprocessitself. Algo-            the new relationships between applied geometries
                                         rithmsacquiretheroleofthenewmeansofrepresen-              -descriptive, projective, algebraic...- and related dis-
                                         tationasthelanguagewhichtranslateshumanthink-             ciplines in architecture -CAAD, programming, digital
                                         ing for the combination power of computer-based           fabrication...- foster the emergence of a new disci-
                                         processes. Indeed scripting languages integrated in       plinewhichintegratesallofthemtogether,Architec-
                                         CADsystemsgobeyondvisualandmouse-basedop-                 tural Geometry(Pottmann,2007).
                                         erations establishing a new way of interacting with           The ways of thinking, conceiving, developing
                                         the geometry involved in the project. On the other        and materialising an architectural work are affected
                                         hand,theparallel developmentofdigitalfabrication          by the new paradigm. In most architecture schools,
                                         perfectly matches with digital design tools, allowing     training matching this new concept is provided
                                         a totally digital architectural process, from concep-     withinpostgraduateprogrammes. Nevertheless,just
                                         tion to materialisation.                                  because the fundamentals are affected, the authors
                                              Thisnewparadigmplacesgeometryinanewpo-               of this paper strongly believe -after years of tested
                                         sition with new roles. The explicit use of geometry is    teachingexperience-thattheimplementationofthe
                                         almosttheonlylinkbetweenprogramming/scripting             digital realm must take place at the beginning of the
                                         languagesandarchitecturalspatialrelationshipsand          undergraduate training period, which is exactly the
                                                                                                         WorkshopContributions-Volume2-eCAADe33|29
            time when students shape their design methodol-       concepts for the course, including an introduction
            ogy.                                                  to algorithmic thinking applied to architectural de-
                The Caterpillar   gallery  (Narvaez-Rodriguez,    sign, human scale control, vector geometry, basic
            Martin-Pastor, & Aguilar-Alejandre, 2015) was a       curves, developable surfaces, intersections, true size
            project intended to produce this implementation       andshapeforfabrication...
            under a carefully-planned pedagogical structure to        The result was not a closed-ended design but
            avoidgapsofknowledgeandcompetences. Itwasa            a generative law based on the use of developable
            ten-meter-longbuiltfull-scaleprototypewhichfacili-    surfaces and simple geometrical operations, which
            tatedtheintegrationofdigitaltoolsinacourseabout       allowed students to propose a variety of solutions.
            the Fundamentals of Architectural Geometry for un-    Theparameterisationofaquadraticsurfacetheorem
            dergraduate first-year students. This paper shows      by Gaspard Monge (Taibo, 1983), regularly used in
            thecontents, based on this project, of the workshop   teaching to give explanation to classic vaults, is the
            carriedoutwithinthe33rdeCAADeconference,held          starting point to address exploration. The statement
            in the Vienna University of Technology (Austria) in   basically says that if two quadratic surfaces are cir-
            September2015. The geometrical principles -based      cumscribedaboutathird,theintersectioncurvede-
            onquadricstheorems-,theparametricdefinitionand         composesintotwoplanarcurves.
            thedigitalfabricationofthe1:6scalemodelexecuted           In order to work with simple and developable
            in the workshopareoutlinedandillustrated.             surfaces, the quadrics chosen are rotational cones or
                                                                  cylinders circumscribed about spheres to force the
                                                                  intersectionstobeplanarcurves,particularlyellipses.
            GEOMETRICAL PRINCIPLES AND PARA-
                                                                  Thespheresareplacedattheverticesofapolygonal
            METRICDEFINITION
                                                                  path with an elevation by way of zig-zag lines and a
            Asanexercisetobeaddressedbybeginnerstudents,
                                                                  plan defining the route to follow, as the idea was to
            thedesignofthepavilionwasintendedtorespondto
                                                                  createalongitudinalspace. Moreover,thediameters
            a series of requirements; simplicity, constructive ef-
                                                                  of the spheres control and define the spaciousness
            ficiency to be built and assembled by the students,
                                                                  of the project at every point of the path at two dif-
            and spatial interest both from the installation itself
                                                                  ferent levels; at the ground level and at the average
            andtheenvironmentrelationship. Atthesametime,
                                                                  humansize. Finally, the shell is defined by keeping
            the piece must contain the intended geometrical
                                                                                                                            Figure1
                                                                                                                            Pictures of the built
                                                                                                                            full-scale prototype,
                                                                                                                            TheCaterpillar
                                                                                                                            gallery, University
                                                                                                                            of Seville (Spain),
                                                                                                                            2014.
        30|eCAADe33 -WorkshopContributions-Volume2
      Figure2
      A-left. Humanscale
      andspacecontrol
      withthesetof
      spheresatthetwo
      levels. A-top-right.
      Setofcones
      circumscribed
      aboutthespheres
      todefinetheshell.
      A-bottom-right.
      Secondquadrics
      theoremshowing
      thetwo-
      dimensional
      constructionto
      simplify the
      parametric
      definition. B.
      Differentstagesof
      theassembly
      process;theconical
      fractions are linked
      totheground
      throughtheuseof
      woodenwedges
      andtotheother
      fractions with the
      useofnyloncable
      ties. Excepting the
      rigid position of the
      wedgesalongthe
      intersection of the
      coneswiththe
      ground,therestof
      thelinkagescanbe
      directly executed,
      asthedrillingsfor
      thetiesarelaidout
      togeometrically
      constrainthe
      surfacestoacquire
      thedesiredshape.
      C. Pictures of the
      resulting modelat
      1:6 scale.
                                              WorkshopContributions-Volume2-eCAADe33|31
            every conical fraction located between the two pla-    conclusions related with the assembly strategy can
            nar intersection curves with the adjacent cones and    beachieved. Thefiguresillustratedifferentstagesof
            thegroundfloorplane.                                    this process.
                The parametric definition also takes advantage
            of another quadric theorem which states that the
                                                                   CONCLUSION
            two planar curves resulting from the previous the-
                                                                   The quadrics theorems stated in this paper, in the
            orem are projected onto the plane of symmetry of
                                                                   particular case of cones and cylinders circumscribed
            the two quadrics as two straight lines. This simpli-
                                                                   aboutspheres,provideeffectiveandappropriatege-
            fies, even more, the three-dimensional geometrical
                                                                   ometricalprinciplestointegrategeometry,paramet-
            relationships to an easy two-dimensional construc-
                                                                   ric design and digital fabrication at the earliest stage
            tion. The spheres turn into circles and the rest of
                                                                   of the undergraduatetrainingperiod.
            theelements,cones,cylindersandtheirintersections
                                                                       Geometry plays new roles within the digital
            canberepresentedasstraightlines. Inaddition, this
                                                                   realm in contemporary architecture which must be
            is a good exercise for students to get trained in the
                                                                   translated to the educational context.    Although
            useofprojectivepropertiesofthree-dimensionalge-
                                                                   The Caterpillar gallery was a successful innovation
            ometries.
                                                                   projectinthissense,thisisafieldwhichstillneedsre-
                                                                   search,developmentandtestinginmanyotherarchi-
            DIGITALFABRICATIONANDASSEMBLY                          tectureschools. Workshops,suchastheoneshowed
            The impact of digital fabrication laboratories in ar-  in this paper, where lecturers, researchers and stu-
            chitectural education can also be channelled to pro-   dentsfromdifferentcountriesareinvolved,fosterin-
            duce benefits from the earliest stage of the train-     terestingandmotivatingdiscussionstokeepadvanc-
            ingperiod,introducingstudentstotherelationships        ingonthistopic.
            amonggeometry, software, the production process
            and the control over building components and ma-
                                                                   REFERENCES
            terials (Celani, 2012).
                                                                   Celani, G 2012, 'Digital Fabrication Laboratories: Peda-
                In this case, due to the possibility of unrolling
                                                                       gogyandImpactsonArchitecturalEducation',Nexus
            the surfaces defining the project, all building com-        NetworkJournal,14(3),pp. 469-482
                                                                   Narvaez-Rodriguez, R, Martin-Pastor, A and Aguilar-
            ponents can be laid out on a plane. Consequently
                                                                       Alejandre,  M 2015, 'The Caterpillar Gallery:
            theycanbeobtainedfromsheetsorpanelsofdiffer-
                                                                       QuadraticSurfaceTheorems,ParametricDesignand
            entmaterials, according to their functions and thick-
                                                                       Digital Fabrication', in Block, P, Knippers, J, Mitra, NJ
            nessesrequired, andfabricatedwithalasercutteror
                                                                       and Wang, W (eds) 2015, Advances in Architectural
            athree-axisCNCmillingmachine.
                                                                       Geometry2014,Springer
                The structural behaviour differences between        Pottman, H 2007, Architectural Geometry, Bentley Insti-
                                                                       tutePress,Exton
            thebuiltfull-scaleprototypeandthe1:6scalemodel
                                                                   Taibo-Fernandez, A 1983, Geometria Descriptiva y sus
            must be pointed out. At 1:1 scale, the shell was
                                                                       Aplicaciones. TomoII, Tebar Flores, Madrid
            materialisedwithfive-millimetre-thickplywoodpan-
                                                                   Terzidis, K 2003, ExpressiveForm: AConceptualApproach
            els.  The fractions of conical surface are not self-
                                                                       toComputationalDesign,SponPress-Taylor&Fran-
            supporting. Theyareusedascomponentsthat,once
                                                                       cis Group, NewYork
            assembled, bring about the emergence of the de-
            sired structural behaviour without any kind of auxil-
            iarystructure. Thisbehaviourisdifficulttosimulateat
            areducedscale,thereforeconclusionsonthismatter
            are not reliable at 1:6 scale. Nevertheless, essential
        32|eCAADe33 -WorkshopContributions-Volume2