The Caterpillar Gallery: Quadric Surface Theorems, Parametric Design and Digital Fabrication

The Caterpillar Gallery: Quadratic Surface Theorems, Parametric Design and Digital Fabrication Roberto Narváez-Rodríguez University of Seville Andrés Martín-Pastor University of Seville María Aguilar-Alejandre University of Seville Abstract. The use of certain quadratic surface theorems has mainly been associated in architecture with the design of classical vaults, domes and piping. The work presented by the authors is intended to explore the potential of these theorems to be used in the generation law for more complex shapes in contemporary architecture. The paper shows the case study of a built full-scale prototype, The Caterpillar gallery, a project stemming from the combination of geometric research and teaching innovation. Formal and structural experimentation take place in this project where, by starting from geometrical considerations, an efficient way of generating longitudinal spaces is proposed. One of the mentioned theorems applied to rotational cones provides the starting point for the generation of a set of concatenated surfaces that, once assembled, constitute a very stable selfsupporting structure with a variety of possible applications. Figure 1: Photographs of detail and general view of the installation. R. Narvaez-Rodríguez, A. Martín-Pastor and M. Aguilar-Alejandre 1 Origin and purpose The use of parametric design and digital fabrication techniques is becoming widespread in architectural practice and provides the focus for many studies. The capacity of today's and tomorrow's software shifted conventional process into the realm of digital [Bentscheff and Gengnagel 2010], hence this should have direct repercussions within the academic sphere. Indeed, a number of courses on these topics are offered, both in undergraduate and graduate studies. However, these technologies tend to be implemented in the last years of the curriculum and are commonly unfastened from the beginnings of the training period. The Caterpillar is a project to materialise research carried out over several years to blend advances in geometry and technology with those of teaching, and to make them available to students at the earliest stage of their undergraduate training. The idea is that students learn the fundamentals of architectural geometry within a context where technological advances are commonplace. This would allow students to acquire competences in the use of these technologies so that they can be developed and rendered during the rest of their training period in a more motivating and efficient way. On the other hand, attention must be paid in order not to miss essential competences and generate gaps of knowledge. Although these academic matters related to the development of the project were also revealing, they lie outside the scope of this paper, which focuses on the case study of the architectural process itself and the potential of its findings for application or use in other pieces of work or research. Figure 2: Virtual model of the project. Left: East-front view. Right: South-side view. Within this context, the challenge was to investigate the creation of a motivating and attractive project to develop, fabricate and assemble a full-scale prototype. The main components were: simple generation law, covering of the fundamentals of architectural geometry throughout the whole process (with an emphasis on the use of developable surfaces), and the possibility of rendering the results through further research and development. Computational systems, basic parametric design, and digital fabrication provided the tools for the project to come to fruition. The generation of a ten-meter-long gallery was proposed which symbolically connects the two schools (Architecture and Building Engineering of the University of Seville - Spain) in their adjoining gardens. The longitudinal path had The Caterpillar Gallery: Quadratic surface Theorems, Parametric Design and Digital Fabrication predominance over the other dimensions and the generative law was articulated around that path to finally result in a set of conical surfaces. Parameterisation was also useful to respond to the location and environmental requirements during the design process. Figure 3: Photographs of the built full-scale prototype. Left: Top view. Right: Interior view. 2 Quadratic surface theorems and generation of the form Although simple uses of these theorems can be found in architecture in ancient times, no written evidence has been found about their systematisation until the modern day. The geometers who contributed towards the statement of theorems of the intersection of quadrics come from the French tradition in the 18th century; this statement was later developed in the 19th and 20th centuries, and has even been continued by certain authors in recent times [Gentil-Baldrich 1997]. As a result of several studies on classic vaults, where theorems of the intersection of quadrics were involved, it was understood that their implicit geometric relationships were also applicable as a resource for the generation of compositions of a more complex nature through the use of parametric design. For the generative law proposed in this paper, the following statement by Gaspard Monge, has been chosen [Taibo 1983]: “If two quadratic surfaces C1 and C2 are circumscribed about a third C3 along their contact curves c1 and c2, their intersection curve decomposes into two planar curves i1 and i2, which pass through the points A and B, which are common to the contact curves c1 and c2.” By circumscribed is meant that C1 and C2 osculate or touch C3 along curves on which the surfaces share the local tangent plane. In this work, quadrics or quadratic surfaces C1 and C2 are rotational cones and quadric C3 is a sphere. This ensures that the intersection curves of the cones are planar curves, particularly conic sections and, in this case, ellipses. The interest of forcing the curves to be planar lies mainly in the analysis and construction simplicity. Cones are surfaces which offer an easy-to-model geometry and, since they are developable, are also easy to construct. Apart from this, from the design point of view, cones provide the possibility of defining various degrees of spaciousness since their geometry converges at a point. R. Narvaez-Rodríguez, A. Martín-Pastor and M. Aguilar-Alejandre 2.1 Geometry Due to the lineal nature of the required space, the first element of the generation law consists of a 3D polygonal line that travels through a determined path. This polygonal line is composed of vertices whose positions are alternately near the level of the floor or near the average height of a human. Each of the segments of this polygonal line constitutes the axis of the rotational cones that form the gallery. These cones should be built by circumscribing them about spheres so that their intersections result in planar curves (ellipses). From these curves, only those that connect the inner space of each pair of cones should be preserved. Finally, only the fragment of the conical surface between the planar curves that connect the cones and the plane defined by the floor is maintained. Figure 4: Perspective view of the generation law for the first pair of conical fragments. From left to right: polygonal line and spheres at every vertex of the line, cones circumscribed about the spheres, cutting plane containing the common ellipse of connection for the two inner spaces, cutting plane defined by the floor. Figure 5: Front view of the growth law for the first four conical fragments. Left: cones 1 and 2 circumscribed about the upper sphere 2. Middle: cones 2 and 3 circumscribed about the lower sphere 3. Right: cones 3 and 4 circumscribed about the upper sphere 4. The Caterpillar Gallery: Quadratic surface Theorems, Parametric Design and Digital Fabrication Figure 6: Top view of the generation and growth processes. Top: Axes defining the generation path. Middle: Lower spheres (dark) and upper spheres (light) defining the spaciousness at the two levels. Bottom: Intersection ellipses between the conical surfaces and floor defining the boundary at the floor level. At every point of the 3D polygonal line, there is the centre of a sphere which defines the spaciousness required by every point position along the path, thereby providing the possibility of generating a different space at the floor level from the space at the ceiling level. In the end, the parameter to control the volume of the R. Narvaez-Rodríguez, A. Martín-Pastor and M. Aguilar-Alejandre interior space is the radius of the relevant sphere. For this purpose, six spheres at the lower level and five spheres at the upper level were created. Figure 5 shows the polygonal, the spheres and the boundary resulting from the intersection between the set of cones and the plane defined by the floor. This boundary not only defines the limits of the gallery on a ground level, but it also plays a decisive role as an auxiliary element of support and acquisition of the specific curvature of each wooden surface. 2.2 Parametric definition The generation law was defined parametrically in Grasshopper, under Rhinoceros, to facilitate the creation and control of the form within the environment provided for the installation. The geometric principles defined in previous sections and the implicit relationship between the three-dimensional shapes and certain twodimensional constructions were key concepts to define the parameterisation. The plane defined by the two axes of every pair of adjacent cones was the construction plane to carry out the relevant two-dimensional geometric constructions. The definition was thought so that different solutions were easily created by modifying the two groups of input data: (i) the set of points defining the path of the 3D polygonal line and (ii) the spaciousness desired at every point (radius of the relevant sphere), where odd points refer to the lower level and even points refer to the upper level. Figure 7: Parametric definition of the model in Grasshopper for a pair of adjacent cones. Top: 2D construction to start the definition for a pair of cones. Bottom left: different variants of the resulting model after changing input data. Bottom right: Grasshopper screen definition. The Caterpillar Gallery: Quadratic surface Theorems, Parametric Design and Digital Fabrication These two groups of parameters control the position and the space created by the definition and are opened to be modified by the final user without modifying the geometric principles which govern the definition. 3 Structural analysis and experimentation The use of developable surfaces considerably facilitated the experimentation with paper and cardboard as well as the relevant tests to transfer the scale model behaviours to the full-scale model. These are also surfaces which are simple to understand, generate and handle. The scale models turned out to be very useful to physically experiment with structural and constructive behaviour and with the assembly process. Furthermore, finite-element analysis was also implemented before the final step was taken. For this purpose, and before full-scale experimentation, two scale models were built; the first at 1:10 scale, and the second at 1:4 scale. Figure 8: Pictures of the students working on the initial model with an auxiliary structure. Left: interior view. Right: External view during the assembly process. 3.1 Initial model in paper at 1:10 scale This model was mainly used as an introduction to the laser cutter machine within the academic tasks. It was useful to check certain spatial relationships as well as some perceptive matters. However, technical experimentation was very limited due to the small size and the difference in behaviour between the paper and the wood for the final prototype. An auxiliary structure made up of elliptic arches (common planar curves) was laid out as a guide to bend the paper and make it acquire the 3D form of the conical surfaces. It was later demonstrated that this structure was unnecessary. Nonetheless, this model was successful in generating sponsorship from companies. 3.2 Approaching the real behaviour; 1:4 scale model After the first model, a new model was needed on a bigger scale (1:4) for the following purposes: R. Narvaez-Rodríguez, A. Martín-Pastor and M. Aguilar-Alejandre • To approach the real size, and consequently, the real structural behaviour of the set of surfaces. This allowed us to check that surfaces, once linked together, constituted a stable set with considerable rigidity. • To define details and test their behaviour. Two major developments where achieved in this regard: (i) The use of joining strips for panel assembly, which had to satisfy the main target on this matter; to ensure curvature continuity between adjacent panels after bending the planar developments. For this purpose, the necessary width for the strips was established at 150mm. Moreover, the joint had to be executed with four lines of screws; two along the longest boundaries of the strips and another two on both sides of the panel junction line. (ii) The solution of the junctions between the surfaces and the floor; thirty-millimetre-thick wooden wedges were used to hold the surface and cover the angle between the surface and the floor. • To find a procedure to turn the installation into an easy-to-assemble construction. In this regard, and considering the ephemeral nature of the gallery, the use of nylon cable ties proved highly convenient. The key geometrical principle was the fact that joining points from the planar developments of different adjacent surfaces, whose positions in the space coincide, would make the surfaces acquire their 3D shape automatically. Thus the procedure consisted of using the nylon ties by way of threads to make the relevant points meet at their common position in the space. In order to ensure the correct operation of this principle, it was essential to first lay out the surface-floor edge along the rigid ellipses defined by the wedges. Otherwise, there would have been an infinite number of undesired 3D forms different from the original model. Figure 9: Left: Undesired form acquired by the linked surfaces without having been previously set to the rigid shape defined by the wedges on the floor. Right: In contrast, this picture shows the simple way to acquire the form by linking the surfaces first to the rigid shape of the floor. Joining strips between pieces to make up the planar unfolding of the surfaces can be appreciated, as well as the continuity of the surface after bending. The Caterpillar Gallery: Quadratic surface Theorems, Parametric Design and Digital Fabrication 3.3 Experimenting at 1:1 scale Finally, experimentation at 1:1 scale was also carried out, especially to test the behaviour differences as well as to check the feasibility of the assembly and of the set-up procedure. The main points and decisions were: • Approve the material. After considering metal sheets and wooden panels, the final election was the use of only five-millimetre-thick black poplar wood laminate panels to materialise the project. The test carried out with an isolated conical surface revealed considerable deformations in all three dimensions as well as instability, with symptoms of near collapse. Nevertheless, the behaviour of the joining strips was very successful and curvature continuity after bending was no problem at all as can be appreciated in Figure 9. Moreover, as expected, the curvature radius was perfectly supported by the panels without cracking; the minimum existing radius in the project was 793mm while the radius supported by the panels in the tests was under 600mm before starting to crack. • Test the structural behaviour of the linked surfaces. Although the test of an isolated surface was not successful in terms of structural behaviour, the gallery was expected to behave successfully after linking all the surfaces. The emergence of this new structural behaviour was proved with a finiteelement structural analysis. Whereas the deformed shape of the isolated surface showed displacement values over 700mm in the z axis, the analysis of the whole set of linked surfaces showed displacement values under 20mm at the same point. Figure 10: Left: Finite-element analysis. Maximum deformations are represented in blue: 700mm for the isolated surface (top) and 40mm for the set of surfaces (bottom). Photographs of the full-scale experiments. Middle: Unacceptable deformation of the isolated conical surface. Right: Successful behaviour of the joining strips to ensure curvature continuity after bending.