from bézier to nurbs: integrating material and digital...

156 ACADIA05: Smart Architecture

M. Cabrinha / From Bézier to NURBS

From Bézier to NURBS:Integrating Material and Digital Techniquesthrough a Plywood Shell

Mark Cabrinha1

1 Rensselaer Polytechnic Institute

Abstract

The development of digital fabrication has reintroduced material processes with digital processes. There hasbeen much discussion about the tool and the objects of the tool, but little discussion of the implication of thematerial process on the digital process. A brief historical review on the development of computer numericalcontrol and the origins of the Bézier curve reveals an instrumental fact: computer numerical controlled toolsnecessitated advancements in computational surfaces which eventually led to NURBS (Non-Uniform RationalB-Splines) surfaces. In other words, the origins of NURBS surfaces resides in its relation to material processes,rather than many current approaches that develop free form surfaces and then force the tool onto the materialwithout regard to the material properties.From this historical and mathematical review, this project develops toward more intelligent constructionmethods based on the integration of NURBS differential geometry paired with material qualities and processes.Specifically, a digital technique of developing conceptual NURBS geometry into piecewise surface patchesare then flattened based on the material thickness and density. From these flattened patches, a materialtechnique is developed to intelligently remove material to allow the rigid flat material to re-develop intophysical surface patches. The goal of this research is to develop digital and material techniques towardintelligent construction based on the correspondence between digitally driven surface and digitally drivenmaterial processes. The application of this technique as a rational and flexible system is to support thedynamic response of form and material toward such performative aspects as structure, daylight, ventilation,and thermal properties.

“Each idea must be related to the principle of a material system, simple and primitive though it may look, onwhich a variable solution could be based.”

Gerald Farin, Curves and Surfaces for Computer Aided Geometric Design

ACADIA05: Smart Architecture 157


Introduction

The recent focus of digital fabrication inarchitecture has reintroduced material processeswith digital processes. Since at least the earlynineties, the accessibility of both computersoftware and hardware motivated a formalrenaissance in architecture propelled by glossyrenderings. This formal emphasis has only beencompounded with the ease of formal manipulationenabled by non-uniform rational b-splines(NURBS). With the recent introduction ofcomputer numerical control (CNC) in architecture,the ability to build this formal manipulation is nolonger novel. Moreover, the approach to formaldevelopment and material fabrication is met witha certain schizophrenia: formal development upto a point, and then the form is handed over to besliced up, often in orthographic sections, to thenbe milled. Although NURBS were introduced toarchitecture before CNC technologies, the originsof NURBS are tied directly to a material system.

This research project was motivated by the factthat Pierre Bézier, arguably the forefather ofNURBS, wrote a book on numerical control nearlyten years before publishing his slightly more wellknown book on the Unisurf CAD System. (Bézier)Further reading reveals a startling fact: Bézier andothers at the time developed the foundation of whatis now NURBS as a result of the development ofcomputer numerical control in the 1960’s. Theirony is that early CNC technologies inspired thefoundation for NURBS - not the other way around.In other words, the tool necessitated more robusttechniques to exploit the tool. While CNCtechnology gives the possibility to slice up anyshape and fabricate it, is there not a more robustapproach to integrating form and material throughthe relation between NURBS and computernumerical control?

The motivation behind this question is two-fold.First it is presumed, and all too frequently the case,that the software leads the development of formthrough its encoded emphasis with the designer’sdirection and intentions filtered through thatmedium. For example, in one case study “FormFollows Software,” it is argued that each softwareinterface either enhances or hinders thedevelopment or 3-d alternatives beyond the defaultprimitives. (Serriano 187) Furthermore, it isargued that the formal flexibility of NURBS-based

programs, which is more akin to digital clay, is anaccomplice in obscuring the tectonics of theseartifacts. (Serriano188-9) Although there is nodoubt evidence to support these two theses, is thisnecessarily so? First, even if one onlyconceptually understands the mathematicalprinciples behind NURBS, learning the softwareis a question of preferable interface, not of encodedbias or proprietary tricks.1 Secondly,understanding the origins of the material processesthat inspired NURBS development implies anobvious link to material systems and processes,bringing into the digital process a question ofmaterial making. The argument presented here isthat there is a great deal more informationembedded within the NURBS surface than merelyrepresenting a shape, but indeed in directlyfabricating it.

Briefly three built examples are given which formthe parameters of this particular research project.Then, a brief overview of b-spline principles areintroduced that lead to both a working vocabularyand certain techniques that are a more materiallyinformed approach to working with NURBS.Specifically, a digital technique of developingconceptual NURBS geometry into piecewisesurface patches are then flattened based on thematerial thickness and density. From theseflattened patches, a material technique isdeveloped to intelligently remove material to allowthe rigid flat material to re-develop into compoundcurved surface patches. This technique isdemonstrated through a rigorous and replicablecomputational process, and fabricated as a full-size installation. The goal of this research is todevelop a technique toward intelligentconstruction based on the correspondence betweendigitally driven surface and digitally drivenmaterial processes. The application of thistechnique as a rational and flexible system is tosupport the dynamic response of form and materialtoward such performative aspects as structure,daylight, ventilation, and thermal properties.

Precedent

Examples of digitally fabricated structures are nowubiquitous, at least in the architectural media.Despite their formal eloquence, these projectsgenerally rely on similar techniques of descriptionand development to fabricate these forms. Theevolution of materially informed processes ondigital development occurs at a much slower



timeframe than the speed of which formalrepresentation can be published through imagesand publications. For example, a close inspectionof Frank Gehry’s work from the Guggenheim inBilbao, to the Experience Music Project in Seattle,to the Sosnoff Theater at Bard College, and theDisney Concert Hall in Los Angeles reveals a slowbut significant impact of material processes anddigital development revealed in the actualexperience of the work through smaller details thatare often glossed over in publications. Althoughthe Disney Concert Hall technically predates theGuggenheim project, the long delay andconsequent redesign and redevelopment of theDisney Concert Hall reveals the development andknowledge based experience of both FrankGehry’s office and the fabricators that his officeworks with, with the most obvious example,among many, being the move from a curvedlimestone skin to a titanium skin. Three moresuccinct examples that illustrate an evolution oftechnique are Bernard Franken’s BMW pavilions“The Bubble,” “The Wave,” and “TheBrandscape” developed between 1999-2000.These pavilions are simple programmatically andillustrate an explicit focus on developing material/digital processes through these projects.2

The Bubble’s main structure was developedthrough orthogonal slicing yielding a rigid egg-crate representation of the form. The precise formis then enclosed with acrylic sheets thermoformedfrom CNC milled polyurethane molds. Despite theegg-crate structure, the pavilion’s novelty residesin its use of compound curved surface patches asopposed to geometrically reduced ruled surfaces.3

While these acrylic patches directly correspondto the digital model, they are clearly not structural.

The following year, Franken’s pavilion “TheWave” and similarly developed “Brandscape”focused on evolving the primary structure awayfrom the egg-crate approach. Instead ofrepresenting the form through the approximationof orthogonal slicing, the evolution in theseprojects is using the iso-parametric informationembedded into the digital model to develop thedoubly curved steel pipe structure. However, theCNC pipe-bending machine only curved pipe inone direction, and therefore the doubly curvedstructural section was segmented intoapproximately 100 singly curved pieces. Thesecondary structure similarly consisted of a latticeof double curved aluminum tubes, but due to the

flexibility of the aluminum section, and the gentlerdegree of curvature in one direction, eachaluminum lattice line could be CNC bent in onedirection and curved into the second direction asthey are fixed in place. The skin in this pavilionis simply fabric stretched between the latticeopenings.

Figure 1. Orthogonal Slicing of Form

Figure 2. Thermoformed Acrylic Patches

Figure 3. Iso-Parametric Structure



Although these projects are now five years old,the evolution these projects present appears tohave met an impasse, as there has been little furtherdevelopment in this direction. Specifically, despitethe Bubble’s egg-crate structure, the thermoformedsurface patches are developed from a directtranslation of the digital form to material form,albeit with the less than desirable intermediarymold. In the Wave, the primary structure developsdirectly from the iso-parametric informationembedded in the surface. The limitation of thisproject is the compounded cost in assembly of thesingly curved steel segments to build the doublycurved iso-paramteric lines of structure, yet anopportunity is also revealed through the naturalcharacteristic of certain materials to bend in at leastone direction, such as the aluminum tubes.Combining the assets of both of these projects, isthere not a way to combine the embeddedinformation of the iso-parametric lines with astructurally rigid surface patch to develop a truefree-form self-supporting structurally rigid shell?

From Bézier to NURBS: Principles of B-SplineDifferential Geometry

It is not the intention of this historical review tofollow the derivations of the differential calculusof these mathematical curves, but to uncover theimplications of this mathematical basis on materialform. It cannot be overstated that these advancedmathematical techniques were motivated by theburgeoning methods of computer numericalcontrol in the automotive industries at Citroen andRenault in Paris, and Ford and GM in Detroit.(Farin 363)

Through the understanding of a couple principlesbehind b-spline geometry, a more intelligentcorrespondence between digital and materialprocess is possible. Of principle concern here isfirst the idea of piecewise construction exemplifiedthrough the evolution of the Bézier curve into theb-spline. Second, exemplified through thedevelopment of Coons surfaces, which ismathematically speaking a patch, outlines thedifferences, and potential, of distinguishingbetween a patch and a surface. The combinationof these two principles leads to a materiallyinformed technique of rebuilding conceptualNURBS surfaces into piecewise bi-cubic patches.

Although many mathematicians were working onthe same problem, the French mathematicianPierre Bézier was the first to publish themathematical basis of what is now known as theBézier curve. (Farin XV) Generally speaking,the Bézier curve is the basic building block of b-splines and NURBS surfaces. A Bézier curve iscomprised of a series of control points, whereinthe curve only passes through the first and lastcontrol points, or end points. The lines thatconnect these control points are known as thecontrol polygon. Despite their free form, thesecurves are the graphical expression of theparametric equation: C(u)= ni=0 Ni,p(u)Pi , wherein{Pi} are the control points, and the {Ni,p(u)}represents the pth degree of the Bézier curve.(Piegl81) In the simple Bézier form, the degree of thecurve follows the number of line segments in thecontrol polygon, or one less than the number ofcontrol points. For example, the illustration showsa cubic Bézier curve (3rd degree), which is the most

Figure 4. Cubic Bézier Curve and B-Spline Curve as Piecewise Cubic Bézier Curves



common type in computer graphic applications.Alternatively, a Bézier curve of degree one (linear)is a line segment despite being derived from thesame parametric expression above. A b-spline,however, can be described as a piecewiseconstruction of Bézier curves where theintermediary control points with the line passingthrough them are defined as knots.

From these simple elements, a surface can begenerated by continuously moving either Bézieror b-spline curves through space, connectingsimilar curves in the opposing direction to thecontrol points of the generating curve, indicatedas u and v directions. Mathematically speaking,this is known as the tensor product approach.(Farin 271) In lieu of modifying a given curveover time, a given surface can be modified throughits control net, or lattice. These types of surfacemanipulation are so familiar to architects workingwith NURBS that their mathematical origins aretypically unknown. It is not the particular maththat is significant but the rationality in buildingup the system that is fundamental. Where the b-spline curve is a piecewise construction of Béziercurves with the knot marking the beginning/endpoint of subsequent piecewise Bézier curves,these knots become iso-parametric lines in thesurface. Note that a true Bézier surface wouldtherefore have no intermediary iso-parametriclines, which is defined as a patch. Whereas theBézier curve was seen as the mathematicalbuilding block of the b-spline, the Coons patchcan be seen as the mathematical basis of b-splinesurfaces. (Farin XV)

The mathematician S. Coons4 is attributed for hisderivation of a surface from four boundary curves.Most architects are familiar with lofting, forminga surface from a series of curves or line segments.

Lofting between only two curves results in a linearinterpolation between them or a surface generatedby straight lines or rulings, known as a ruledsurface. Coons developed his method of a surfacedefined by four boundary curves through atechnique of bi-linear blending. Each opposingset of curves are lofted with the resultant gridblended into a continuous surface. While thesolution of bi-lineal blending this ruled mesh iselegant, it is also problematic as a result of thelinear interpolation from one curve to the next.For variable surfaces which rely on more than fourboundary curves, as most do, the Coons patch isnot informed by neighboring patches therebycreating creases at the boundary conditions.Mathematically speaking, “cross boundarytangents along one boundary depend on data notpertaining to that boundary.”(Farin 369) Althoughthere are blending functions to smooth out thesedifferences, they are frequently undesirable as theyalter the defining boundary curves.

This brief knowledge of Coons patches is usefulfor two reasons. First, although patches may notbe as advantageous in conceiving form, thereprimary benefit is to rationally reconstruct asurface from a given network of curvesmaintaining the logic of that network. Secondly,with the knowledge that these patches are bendedfrom two ruled surfaces, it is possible to degreereduce these patches in one direction resultingback into a ruled surface.

Digital Technique: Conceptual NURBSGeometry into Piecewise Bi-Cubic Patches

While the subject of this paper is not formalconception, it does imply the introduction ofmaterial systems and processes at much earlier

Figure 5. Coons Patch Development



stages of formal development. To this end, thetechnique presented here relies on simpleconceptual geometry, yet the ease of which thesetwo can be integrated suggests an increase inintegration between form and material at theearliest stages of formal development. The verylimitation of the patch, that “cross boundarytangents along one boundary depend on data notpertaining to that boundary,” becomes theopportunity for this technique of rebuilding theconceptual NURBS surfaces into piecewise bi-cubic patches. If developed from an appropriatelyformed network of NURBS, the boundary curvesof the patch can be developed as a seam betweenadjacent patches where the patch boundary edgesseparated by that seam do share the same tangent.The result is the same continuous curvature of theconceptual NURBS surface but separated aspatches.

Figure 6. Conceptual NURBS Geometry from Torus

Figure 7. Rebuilt Materially Correlated Surface

More significantly, the piecewise construction isnot merely a different representation of the sameNURBS surface, but the spacing between the iso-parametric b-splines (isoparms) can be correlatedto the particular material unit sizes. In thisexample, the conceptual NURBS geometry wasrebuilt based on a standard 4x8 sheet size ofplywood. These dimensions can be verifiedthrough flattening a particular patch (see below)or simply verifying the appropriate spline length.

In conceptual development, a minimum of iso-parametric lines will likely be desired to maintainglobal control. However, once the form is set, itis suggested here that adjusting iso-parametric linedensity is directly related to product scale anddetail when the surface is connected to a materialprocess. For example, the isoparm density wouldbe different for say a stapler, than for a building.It should be made clear that for representationalpurposes this is nearly irrelevant. However, whentied to material parameters and processes theseisoparms exhibit a potential much greater thansimply representation. The isoparms can be seenas seams in a material process whereby adjustingthe isoparm density creates a correlation with amaterial dimension and/or a material process. Inthis way, the process of slicing is not necessary,as the object or building can be broken apart atthe seams. For example, in an injection-moldedproduct, I might choose to split the surface moreor less down the center into two equal surfaces.However, for much larger products, such asarchitecture composed of many large pieces,distinct techniques are necessary.

After the given conceptual NURBS surface isrebuilt based precisely on particular materialparameters5, the surface curves can be extractedand reconstructed as bi-cubic patches in piecewiseform6 . These are termed “bi-cubic” as the b-splinecurves in opposing direction are both of cubicdegree. Although this is a basic andstraightforward approach, when NURBS are onlyseen as digital representations of form the materialimplications of this technique would not beapparent. Furthermore, other approaches such astriangulation have dealt with the roughapproximation of continuous curvature, but aredeveloped from the reconstruction of the entiresurface, globally as a mesh, thus dropping theparametric expression of the generating curves.Triangulation also yields a field of complex jointangles that must be dealt with. By rationally



constructing NURBS geometry, continuouscurvature can be achieved while simultaneouslysimplifying the assembly of joints as a result oftheir shared tangents, and creating a digital/material synthesis through the correlation ofmaterial parameters and process through definediso-parametric seams. What was the specific iso-parametric line of the NURBS surface, becomesthe defining edge of the patch.

Figure 8. Piecewise Bi-Cubic Patches

While the technique of piecewise bi-cubic patchessimplifies the joint, the challenge resides ineconomically fabricating these bi-cubic(compound curve) patches. Of course thesesurfaces could be milled from a solid chunk ofmaterial, though this is obviously time and materialconsuming. Similar to the acrylic patches inFranken’s Bubble, intermediary molds could bemilled and thin sheets of material could belaminated to the mold.7 However, the mosteconomically and materially desirable solution isto flatten these patches utilizing flat stock materialand basic two and three axis CNC technology asthese are the most common in the industry.(Kolarevich 34)

At this point in the process, there are two optionsin flattening the patches to work with flat stockmaterial. The first option is to degree reduce thepatch in one direction into a developable surface,and the second option is to use specific softwarethat can flatten compound curve material basedon the material thickness and density. Adevelopable surface is a sub-set of ruled surfaceswhose rulings are parallel (a cylinder) orconcentric (a conic section). As a result,developable surfaces can be iso-metricallymapped, or unrolled onto a plane. A compoundcurve is simply a curve in both directions, or

doubly curved, as in the bi-cubic patches.Although both options require specific softwaresolutions, option one rebuilds the surface into adevelopable surface such that it can be unrolledregardless of material properties. The challengeof flattening compound curve material is that thematerial is actually stretching, creasing, and/ortearing and therefore the density and thickness ofthe particular material must be taken intoconsideration by the software.

Option One: Degree Reduction of Bi-CubicPatches to Developable Surface

Although ultimately this project proceeds with thesecond option, this option results from the reviewof the mathematical principles of NURBS and willlikely be invaluable in different applications.While by definition, the bi-cubic patches (curvedin both directions) are not developable; a degreereduction in one direction will yield a ruledsurface, or technically a dual degree patch (cubicin one direction and linear in the other direction).Furthermore, as a result of the piecewiseconstruction of bi-cubic patches, each boundarycurve shares the same tangent with the adjacentpatch, and therefore through reducing the degreein one direction this technique will yield adevelopable surface. Pause must be given here tostress that degree reduction in one direction of anyNURBS surface will not necessarily yield adevelopable surface, but will yield a ruled surface.However, in this technique, as the conceptualNURBS geometry is rebuilt through a piecewiseconstruction of patches, this approach to degreereduction should yield a developable surface ateach patch.

Although it would be easy to proceed with thisproject in developable surfaces by unrolling themthrough Rhino8, this formal approximation is notsatisfactory compared to the intended compoundcurve model. Care needs to be taken in whichdirection to reduce in degree as it not only willeffect the appearance of the surface, but dependingon the geometry can also split the seams.Furthermore, projects which successfully use ruledsurfaces approach ruled surfaces at the conceptualstage of design as a material/geometric restraint,such as the physical paper strips in Frank Gehry’sdesign process, and not as an approximation of adesired compound curve surface. (Killian 78)Scale is also an issue, as larger projects utilizingsmaller ruled surfaces may appear as compound



curved as Gehry’s work exemplifies. The attemptto develop this project through ruled surfacessuggests that the successful use of ruled surfacesis more a part of conceptual development,accepting the material constraint of curving in onlyone direction.

Figure 9. Option One: Developable Surfaces

Figure 10. Option Two: Bi-Cubic Surfaces

Option Two: Flattening Compound Curvaturethrough a Software Approach

The software approach to flattening compoundcurvature was previously the purview of advancedaerospace, automotive, and aeronautical softwareand therefore presented a daunting learning curveto mention nothing of expense. With the recentintroduction of Lamina Design software, this haschanged9. This stand alone, single functionsoftware is both inexpensive and easy to use.Unique to this software, is its ability to flattencompound curvature based on material density andthickness. This is intended neither as a productendorsement nor software tutorial, but thisdirection would not have been pursued without

Lamina. The illustrated examples on the Laminawebsite are of smaller sculptural objects andfurthermore are objects formed by only thesurface. This is problematic as architectureobviously deals with much larger pieces and isalways a composite of multiple systems. ForLamina to work, surfaces must be rebuilt as a meshthereby dropping the iso-parametric information.However, through the piecewise technique of bi-cubic patches presented here, the boundarycondition is all that is necessary so long as it isproportionate to the material process employed(e.g. in this example a 4x8 sheet of plywood witha 4x8 CNC router). Although this may appear asmerely procedural, all too often I have encounteredan indifference to surface construction stemmingfrom the digital model as only a representationalmodel of form expecting the software to do it foryou. In this way, this is not a critique of thisparticular software, but supports the need for thedigital process to be informed by the materialprocess – an intelligence from the designer, notthe software.

Material Technique: Routing Uniform B-Splines at Lines of Stress

For clarity, this paper has presented a linear processof moving from digital to material, with theknowledge of the material parameters, such as aunit size, informing the digital process. However,as both the digital process and material processwere being tested at the same time, it cannot bestressed enough that the material developmentproceeded in tandem with the digital development.While a great deal of research was geared towardthe digital technique of intelligently rebuilding theNURBS geometry, an equal amount of physicalresearch was based on the ability of the materialto form a compound curve as well as simply testingthe file translation process from digital to materialprocesses. This project proceeds with plywoodas the material of choice for a number of factors,but particularly for the ease of use of working withwood, its relative inexpensive cost, and most ofall that I had unlimited access to a woodshop witha 3-axis CNC router. Certainly other materialscould be used and their specific properties wouldmodify this process.

Test Patch

A small test patch was fabricated as an initialdemonstration of this process. Curvature analysis



was used in Rhino to locate the patch with themost extreme curvature. From this analysis, asection of this patch and frame was flattened withLamina. The frame was simple to fabricate andassemble with dados at the corners to register thepieces. For the 3/4” structural skin, a techniquewas necessary to selectively remove material atregular intervals relative to the double curvaturedesired, which was a particular challenge as allfour edges were of unique curvature due tounrolling the compound curve patch. Although itadded for a complex workflow, the solution waseasy: use the blending function in AdobeIllustrator. This function rebuilds these lines as ablending of uniform b-splines, whereas thedifferential geometry of NURBS is formed by thenon-uniform basis of isoparms. These paths werethen milled using a 3/4" ball endmill to a depth ofabout 5/8" of a 3/4" piece of plywood. A ballendmill was used instead of a square endmill toavoid stress failure at the internal corners. Thispatch was then easily screwed to the frame. Thisquick sample identified three issues for furtherdevelopment. First, the blending lines were moredense than necessary in one direction, but moreimportantly were not dense enough in the seconddirection creating a noticeable segmentation at theedges. Second, the material does not naturallywant to bend in both directions, and thereforeformed a flat spot at the center of theapproximately 1' by 2' sample. Third, althoughthe skin is inherently stronger as a result of thecompound curvature, because of the routedgrooves it easily supported about 30 pounds, theweight of a child, but began to crack upon my fullweight.

The structural integrity of the skin was of utmostconcern and two options were considered. Thefirst was to infill the cavity with polyurethane foamwhich expands creating an insulated andpotentially rigid structure.10 The second was tolaminate a thin sheet of material to the groovedplywood such that the grooves could not flex. Asheet of 1/8" masonite was milled to the sameflattened path profile, glued to the plywood andthen screwed to the frame to dry overnight. Afterremoving the screws, the laminated skinmaintained the exact shape of the frame, and thenew shell would support my full weight withoutcracking and minimal deflection.This introduceda more robust approach toward a true structuralskin independent of an intermediary frame, andtherefore this option was pursued in this project.

Through this move toward lamination, an obviousoption of using several thinner sheets wouldeliminate the need for the grooves. Although thisis a viable option, it was avoided in the attempt tominimize the number of pieces and the morecomplex apparatus required to laminate multiplesheets together. Moreover, as thinner material wasmore flexible, it also meant an increase in densityof support structure for fabrication possibly evennecessitating a continuous mold, which was to beavoided in the first place. Finally, the thinnerlaminations resulted in nearly double the materialcost as thinner sheets are not priced proportionallyto their thickness. Therefore, it was desirable todevelop a technique mating two skins each withan exposed exterior face, unlike the laminatedsample above with 1/8" masonite. It was decidedto laminate two skins of 1/2" wood consideringparticle board, plywood sheathing, and veneerplywood.

Figure 11. Curvature Analysis

Figure 12. Test Patch Exterior



Figure 13. Test Patch Interior

Figure 14. Laminated Skin

Frame and Skin

If the frame was not to be used in the finalinstallation, could it be avoided all together?Theoretically, if there was a way to register thetwo offset shells through pins or dowels, the framecould be avoided. It must be made explicit thatthis would only work if the two skins were offsetin the digital model and precisely flattened takinginto account there thickness resulting in theslightest dimensional variation. Yet as a result ofthe surfaces being offset from each other, theyshare the same surface normals - that is, any linedrawn perpendicular to the surface would likewisebe perpendicular to the offset surface. Therefore,when these sheets are flattened, a hole drilledperpendicular to the surface would then becomenormal to the surface when the pieces take theirshape. More significantly, if the holes are preciselyplaced, the holes should align only when thesurface takes its shape and could be doweled fixingtheir position. Holes were proportionately located

Figure 15. Blend Pattern

Figure 16. Sample Fabrication on Particle Board

based on the same blending lines developed fromAdobe Illustrator, only now the blend was mademore dense to create a pattern of grooves andalternating holes. The ability to register the twolaminations through dowels proved indispensablein laminating the skins, however it provedimpossible to wrestle the wood into compoundcurvature without the frame. The frame then reallybecomes a stretcher and also a physical check thatthe skin is at precisely the correct curvature. Ifthe previously described option one, of using ruledsurfaces was used, this same technique of holealignment and peg registration could be usedwithout a supporting frame or mold, as the woodhas no problem curving in one direction, so longas there is minute movement between dowel andhole, including material flexure.

In addition to the need for lamination, the test patchrevealed two other issues, one regarding visualsegmentation as a result of the grooves, and



second, a flattening of the material at itsunsupported center. Numerous material studieswere made on 1/2” 3 layer cabinet grade birchveneer plywood and 1/2” particle board adjustinggroove spacing and depth, testing for flexibility.As plywood is not an isotropic material, the graindirection of the laminations has a significantimpact on its ability to bend, and therefore deepergroove depths were tested perpendicular to thestrength of direction. While particle board is moreiso-tropic than plywood, it has almost no flexuralstrength and therefore segmented or slightlycracked at the grooves. Exterior plywoodsheathing was briefly tested, but is rarely perfectlyflat and therefore difficult to precisely mill thegrooves. Furthermore, plywood sheathing hasnumerous knots, plugs, and internal voids thatresulted in failure. For the project developed here,it was decided to proceed with 1/2” cabinet gradeveneer plywood with grooves spaced at maximumof 3” on center, and 3/8” depth, leaving 1/8" ofmaterial, or about one of the three laminations.Further structural analysis may suggest alternatingthe grain direction between layers. However, asplywood generally comes with the grain strengthin the long direction, to alternate the graindirections the piecewise construction of bi-cubicpatches would have to be based on a 4’x4' maxmodule and not the 4’x8' max module used here,doubling the number of pieces and thereforecompounding the number of joints.11 Theflattening of the material at its unsupported centerwas simply overcome by adding an intermediaryvertical b-spline support in the final frame.

Frame Final Assembly

The frame itself demonstrates the direct translationof iso-parametric information to material structure.While orthographic slicing yields a series of flatsections, utilizing the iso-parametric lines yieldsruled surfaces. These lines were offset 4” forstructural depth, and digitally modeled as amodular series of nine frame patches. Theseframes were then unrolled, and tested in a lasercut scaled physical model. This physical modelindicated a sequence of construction, as well asthe need for curved corner bucks to give theboundary of the frames their correct curvature, andwas instrumental for assembling the full sizeframe, suggesting that the model is not merely forrepresentation, but a prototype of actualconstruction. The precise location of eachplywood spline was marked with dados.

Modular Skin: Design for Assembly

The full size frame was left assembled as skin andjoint options were tested on it. Through thesenumerous material and lamination tests, greaterattention was placed on its surface quality creatinga screen-like object at a scale between furnitureand architecture. As a demonstration project, thegrooves became articulated at the surfaces inconjunction with their structural necessity.Routing all the way through the material in onedirection, the material took the curvature as wellas expressing its construction. Similar to thedowels normal to the surface, the grooves onlyaligned as the panels took their shape.

As a full size installation/demonstration project,an additional constraint was to create an objectthat would easily be assembled, demounted andre-assembled elsewhere. This placed emphasis ona removable joint between the nine prefabricatedpanels. Several options were considered,including doweling, puzzle pieces, and a fingeredspline joint. In the end, the simplest joint, a 1/2”lap joint was used. However, due to the size andgeometry of these panels, a great deal of stress isplaced on the joint, and therefore the simple 1/2”lap joint only became feasible when a system ofpost-tensioning was used in conjunction with thelap joint. By including vertical ball end groovesback to back, an approximately 3/4” round voidwas created, making it possible to run a continuousdowel in that void for additional support.However, a wooden dowel created too muchfriction, and was therefore impossible to remove.This introduced a new material to the project: highdensity poly propylene (HDPE). HDPE isrelatively inexpensive, is very flexible, has a verylow co-efficient of friction, with a high tensilestrength, and is easy to fabricate. These HDPErods were tapped at the ends and used as tenons topost-tension the panels which aided greatly in theassembly of the structure.

Skin Final Fabrication

The final fabrication of this project was no smallaffair, with the frame fabricated over spring break,and the laminated skins fabricated at the end ofthe academic year when the wood shop was moreor less vacant. Laminating each patch over theframe was a blend between choreography andchaos. Each patch required at least ten clampsthat were sequentially tightened to distribute the



Conclusion

The motivation of this research project is basedon the relation of the material process on the digitalprocess. In particular, through the mathematicaland historical review of NURBS surfaces, it isfound that there is a strong correlation betweenNURBS and material processes movingarchitecture beyond representations of formtoward physical processes informing form andenabling the design process. The emphasis of thispaper has been on the digital technique ofdeveloping conceptual NURBS geometry intopiecewise bi-cubic patches informed by material

stress as uniformly as possible. However, despitethe rigorous digital process and numerous smallertest pieces, the full size patches built-up enoughsurface tension along the larger surface that eachpiece buckled at one fairly consistent location.Despite the frustration of this physical failure,pushing the material to its limits is also indicativeof the need for further material research in tandemwith the digital techniques presented here. Oneof the three bays was able to be assembled as ademonstration of the potential of this installation.

Figure 17. Full-Size Frame Installation

Figure 19. Post-TensioningFigure 18. Tenons in Assembly



parameters and processes into a materially relateddigital model. From this, two options arepresented: one in working with ruled surfaces andone in working with compound surfaces. Amaterial technique of routing uniform b-splinegrooves was used to develop the directionalmaterial properties toward compound curvature.

As a result of the material failures in finalfabrication, one of the conclusions of this researchis that working with ruled surfaces and compoundcurved surfaces is more than simply a question ofdegree reduction, mathematically speaking, but afundamental question of material resistance – aparameter not included in the “materialparameters” of the digital model presented here.While the digital techniques presented here arean important development of integrating digitaland material techniques, the material techniquesrequire future work with emphasis placed on the

Figure 20. Final Fabrication

Figure 21. Clamping Laminated Skin

Figure 22. Final Assembly

Figure 23. Detail



material resistance inherent in the material as aresult of the curvature. Material research needsto focus on material composition: iso-tropic vs.non-iso-tropic materials, material density, andthickness of laminations in compound curvature.In addition to wood and metal, this would includenew materials such as the HDPE introduced in thisproject. For prefabricated assemblies, structuraltesting is necessary through computationalprocesses such as finite element analysis (FEA)as well as physical destructive testing on entireassemblies including the joints betweenassemblies. Building applications in polyurethaneinjected compound curved stressed skin panels aswell as compound curved concrete formwork aretwo clear structural applications utilizing thetechniques presented here.

Notes

1. The ease of interface is the question of bias, asSerriano argues, to say nothing that NURBS couldbe said to be biased from the first.2. To the degree that “Form Follows Software,”at issue here is the development from form tofabrication and not the formal morphogenesis ofthese projects. See New Technologies inArchitecture and Digital Design andManufacturing, for the morphogenesis of theseprojects.3. Frank Gehry’s Conde Nast project uses a similarapproach with its glass surfaces while using ruledsurfaces for the titanium interior. See NewTechnologies in Architecture and Digital Designand Manufacturing for more on this project aswell.4. Ironically, I was first introduced to Coonssurfaces through MasterCAM.5. As a result of the scale of architecture, thistypcially means an increase in isoparm density, ifa reduction of isoparm density is required, this willmost likely alter the surface.6. Alternatively, it may be possible to split thesurface at the isoparms as in Rhino. However,the accuracy here is questionable. Trimming thesurface is not an option, as trimmed surfacesmerely mask the entire mathematical surface fromview.7. See also Bechtold, Martin. “Surface Structures:digital design and fabrication” in Fabrication:Examining the Digital Practice of Architecture.8. see www.rhinocerous3d.com, Rhino is wellknown for its unique “unroll” command to unrolldevelopable surfaces.

9. see www.laminadesign.com.10. This remains a viable option for future workas well as product application, but this projectmoved toward the more demanding applicationof a structural skin.11. In hindsight, the largest pieces that approached8’ were also the most prone to failure both fromtheir own weight, as well as surface tensiondeveloped over the longer length of the piece, andtherefore, alternating laminations on a 4’x4’ maxmodule may be more successful.

References

Bechtold, Martin et al, ed. (2000) NewTechnologies in Achitecture: Digital Design andManufacturing Techniques. Cambridge, MA:Harvard Design Books.

Bechtold, Martin. (2004) Digital Design andFabrication of Surface Structures. P. Beesley (Ed.)ACADIA 2004: Fabrication: Examining theDigital Practice of Architecture. (pp 88-99)Toronto, Canada.

Bézier, Pierre. (1972) Numerical Control:Mathematics and Applications. New York, NY:Wiley.

Farin, Gerald. (1993) Curves and Surfaces forComputer Aided Geometric Design, 3rd ed. SanDiego, California: Academic Press.

Killian, Axel. (2003) Fabrication of PartiallyDouble-Curved Surfaces out of Flat Sheet MaterialThrough a 3D Puzzle Approach. K. Klinger (Ed.)ACADIA 2003: Connecting-Crosscroads ofDigital Discourse (pp 75-83) Indianapolis,Indiana.

Kolarevic, Branko ed. (2003) Architecture in theDigital Age: Design and Manufacturing. London,England: Spon Press.

Piegl, Les & Tiller, Wayne. (1995). The NURBSBook. Berlin, Germany: Springer-Verlag.

Schodek, Daniel et al., eds. (2005) Digital Designand Manufacturing. New York, NY: Wiley.

Serriano, Pierluigi. (2003) Form FollowsSoftware. K. Klinger (Ed.) ACADIA 2003:Connecting-Crosscroads of Digital Discourse (pp185-205) Indianapolis, Indiana.

from bézier to nurbs: integrating material and digital...

Documents