Approaching developable surfaces through shadow and penumbra

Vol.:(0123456789) Nexus Network Journal https://doi.org/10.1007/s00004-023-00647-x Published: 21 January 2023 RESEARCH Approaching Developable Surfaces Through Shadow andPenumbra AndrésMartín‑Pastor1 · FranciscoGonzález‑Quintial2 Accepted: 2 January 2023 © The Author(s) 2023 Abstract This paper enquires into the relationship between the rigorous graphic construction of penumbra and shadow and the idea of a developable surface circumscribed to two cores (whether these are curves or surfaces). The first rigorously constructed graphic representations of shadow and penumbra appearing in different historical treatises are contextualised and contrasted with the first works addressing the mathematical description of such surfaces. We describe how artisans and artists have employed physical and mechanical models to construct circumscribed developable surfaces, examining both their formal exploration and constructive formalisation. Next we extrapolate the mechanisms for generating circumscribed surfaces to digital and parametric models. The two procedures presented here evince the existing geometric limits of this way of generating a surface, thus influencing its constructive applications. Lastly, several examples of ephemeral architecture based on such procedures are offered, in which the digital design and manufacturing of experimental pavilions with circumscribed developable surfaces are addressed. Keywords Descriptive geometry· Shadow· Penumbra· Developable surfaces· Circumscribed surfaces· Digital manufacturing· Experimental pavilions· Augmented graphic thinking * Andrés Martín‑Pastor archiam[email protected] 1 Department ofGraphic Engineering, Escuela Técnica Superior de Ingeniería de la Edificación (ETSIE), Universidad de Sevilla (US), Seville, Spain 2 Department ofArchitecture, Escuela Técnica Superior de Arquitectura (ETSA), Universidad del País Vasco (UPV‑EHU), Donostia, Spain A.Martín-Pastor, F.González-Quintial The Rigorous Graphic Control ofPenumbra andShadow Shadow has been studied since Antiquity. According to Pliny, outlining the shadow of a man with lines (umbra hominis lineis circumducta) is associated with the mythical origin of painting.1 Since then, the importance of shadow, as one of the major problems of painting, has been evidenced by the works of painters and theoreticians of painting, as found in Leonardo’s notebooks or in Alberti’s De pictura. However, it is in treatises on perspective that the evolution of the geometric mechanism for controlling and representing shadow can be fully appreciated. Dürer’s Underweysung (1525) offers the first example of controlling shadow in perspective by means of double orthogonal projection. This development was contextualised in the sixteenth century, above all in the treatise by Vignola‑Danti (1583), in which shadow was represented with indefinite edges. In the seventeenth century there was a renewed interest in the study of shadow and the rigorous definition of its edges.2 This change should be attributed to the work of Guidobaldo Del Monte (1600) and the observations made by Galileo with his telescope, which had a notable impact on Florentine academic and artistic circles at the beginning of the century.3 We know that penumbra is a natural phenomenon addressed in optics and studied in astronomy. The issue of indefinite edges, or half‑light, is commented on in the works of Alhacen, Maurolico and Kepler, to whom we owe the current term “penumbra” (Raynauld 2019). Leonardo also speculated on half‑light, or penumbra, as reflected in the drawings appearing in many different codices, although in all these cases, his study was restricted to spherical lumens and their manifestation in a dark room. Curiously, and despite being a well‑known phenomenon, the graphic control of penumbra is a topic that has received a lot less attention than shadows (Fig.1). It was not addressed by the most prominent sixteenth‑ and seventeenth‑century perspective authors who had indeed defined shadow projection with precision, such as François d’Aguillon, Samuel Marolois, Jean Dubreuil, Bosse and Desargues, among others. The pioneers in the rigorous representation of penumbra were a small group of Florentine authors very closely related to Galileo’s circle, such as Ludovico Cardi and Pietro Accolti. It was in this cultural context in which special attention was paid not only to shadow, but also to penumbra, which ceased to be an astronomical problem to become a graphic one, as can be seen precisely in the work of Cardi and Accolti. The first case of rigorous representation of penumbra is to be found in the treatise by the painter Ludovico Cardi(c. 1613). In his manuscript, he rigorously 1 As regards the huge implications of shadows in Western culture and art, see Gombrich 1995; Ronchi 1983. A first geometric study is found in Kaufmann (1975). 2 The shadow cast by a point of light (a candle or torch) was the first case study, while the projection of a shadow produced by the sun came about after half a century of controversies. For a comprehensive study of this subject, see Martín‑Pastor etal. (2017). 3 Ludovico Cardi, a painter and disciple of Galileo, represented his Virgin of the Immaculate Conception (Roma, Sta. Maria Maggiore) standing on a moon defined by craters and shadows. His treatises have been comprehensively studied in Camerota (2010). Approaching Developable Surfaces Through Shadow andPenumbra defines the outline of penumbra and shadow cast by a prismatic body when illuminated by diffuse light coming through a window. The graphic construction used by Cardi (Fig.2 Left) allows for defining the projection using points on the perimeter of the window, in a sort of minimal envelope of luminous rays. This same problem was also addressed in a treatise by Accolti (1625) (Fig.2 Centre) and subsequently by Níçeron (1646) (Fig.2 Right). The graphic model proposed by Cardi, Accolti and Níçeron might seem evident in the current context. It warrants noting, however, that the proposed solution was a milestone in the history of graphic expression and the seed of the synthetic conception of the problem of developable surfaces which, as will be seen further on, consists in finding a surface circumscribed to both bodies. The rectangular form of a window and the prismatic body reveal their limitations when the intention is to resolve more complex cases of penumbra and shadow. For this reason, the geometric approach seems to have been designed for multifaceted bodies. However, the assumption that these first illustrations served as a source of inspiration for mathematicians at the end of the eighteenth century for linking penumbra and shadow to the problem of developable surfaces is not entirely outlandish. Fig. 1 Penumbra and shadow as developable surfaces. Image: authors A.Martín-Pastor, F.González-Quintial Developable Surfaces asPenumbra andShadow: Mathematics andRepresentation The history of developable surfaces is well known since its beginnings (Lawrence 2011). According to Hachette (1818 VII), the first geometrician to address the theory of these surfaces was Euler, in his De solidus quorum superficem in planum explicare licet (Of solids whose [entire] surface can be unfolded onto a plane) (1772). In addition to its being a text revolving around mathematical analysis with original results in the field of algebra, it should be stressed that it also proposes the first approach to the problem from the perspective of “the question of light and shadow”: third solution of the principal problem derived from the theory of light and shadow . §39 But what especially extends to our undertaking from this theory, is that the shapes of the shadows are always of such a nature that their surface can be unfolded onto a plane; ... (Euler 1772: 27; trans. Alexander Aycock). The next mathematical treatise was published by Gaspard Monge a few years afterwards. In the prologue to the Mémoire sur les propriétés de plusieurs genres de surfaces courbes, particulièrtement sur celles des surfaces d’eveloppables (Dissertation on the properties of several kinds of curved surfaces, particularly on those of developable surfaces) (1780), Monge stated: Having addressed this issue, on the occasion of a dissertation that M. Euler delivered on developable surfaces in his book of 1771, of the Academy of St Petersburg, in which the illustrious geometrician provides formulas for recognising whether a given curved surface has the attribute of being able to be unfolded on a plane, I have obtained results that seem to be much simpler and much easier to use for the same purpose. From this I deduced the resolution of the penumbra and shadow of a body of any shape, illuminated by a luminous body in any way (1780: 382; our trans.). It is important to stress that Monge proposed the generalisation of the problem to all cases, as can be seen in the “Problem V. Find the equations of the surfaces that Fig. 2 Left Ludovico Cardi, Prospettiva pratica, “Libro Secondo. Quinta Parte/Terza Regola/Degli sbattimenti del sole” (ca. 1613: fol. 83V); Centre Pietro Accolti, Lo inganno de gl’occhi: prospettiua pratica di Pietro Accolti… (1625: 137); Right Jean‑François Níçeron, Thaumaturgus Opticus (1646: Table42, Pl. 87) Approaching Developable Surfaces Through Shadow andPenumbra envelop the penumbra and shadow of any opaque body, illuminated by any luminous body, giving the two bodies figures and positions in space” (1780:407). The essential idea underlying Monge’s text is that of a mobile plane rolling over two bodies, while remaining tangential to both. In this way, the problem of the relationship between developable surfaces and penumbra and shadow is perfectly expressed from a synthetic point of view: From the foregoing it can be inferred that the required surfaces should be generated by the continuous intersection, with itself, of a plane that revolves around the two bodies while always remaining tangential to them, with the sole difference that the plane should touch the two bodies on the same side to generate the surface of pure shadow. Therefore, let us first find the equation of any plane tangential to the surfaces of both bodies (Monge 1780: 407; our trans.). The next important milestone is found in Jules de la Gournerie’s Traité de Géométrie descriptive (Treatise on descriptive geometry) (1860). La Gournerie represents, for the first time and in a rigorous fashion, a developable surface in an orthographic projection system (Figs.3 and 4). As with Monge, La Gournerie defines developable surfaces “as those that, supposedly flexible and inextensible, can be unfolded onto a plane without breaks or duplications” (La Gournerie 1860: 50; our trans.). In his treatise, he also expounds on other procedures for determining developable surfaces on the basis of the envelope of the motion of a cone and as a surface of constant slope (La Gournerie 1860: 53, 104), which still appear in descriptive geometry texts (Izquierdo‑Asensi 1985). The same synthetic formulation of the problem of developable surfaces, associated with the movement of a mobile plane put forward by Monge is also to be found in the treatise by Leroy (1842), which ended up becoming one of the most important and most read in the field of descriptive geometry. Leroy has the following to say in this respect: So, for example, when rolling, a mobile plane can be affixed to fixed surfaces, constantly remaining tangential to these two surfaces, provided that none of them are developable” (Leroy 1842: Chap. 184). Since then and to this day, there have been extraordinary mathematical advances in the state of the knowledge of developable surfaces. A compilation of the results of the geometrical design of nondegenerate developable surfaces can be found in Krivoshapko and Shambina (2012). Circumscribed Developable Surfaces inTrades andHandicrafts The control of developable surfaces is directly related to trades and handicrafts employing materials that are manufactured flat and which can be curved without breaks or distortions, whether of leather, fabric, metal, wood or any other flat material. These traditional trades include tailoring, shoemaking, shipbuilding and A.Martín-Pastor, F.González-Quintial so forth. Treatises summarising geometric knowledge have been written on nearly all of them, providing exceedingly valuable information for their pursuit.4 The Spanish translation of Leroy’s treatise by Andrés Antonio de Gorbea y Gancedo (1845) offers a subtle example of the relationship between the theory of developable surfaces and handicrafts: Fig. 3 Determining the shadow of an ellipse illuminated by a circle. Plan, elevation, section and flat development of a developable surface. Image: La Gournerie (1860: Pl. XXIV–XXVII) 4 In Spain alone there are the treatises on tailoring by Juan de Alcega (1580), Diego Freyle (1588), and Albayzeta (1720), among others. Information retrieved from Gentil‑Baldrich (2021). Approaching Developable Surfaces Through Shadow andPenumbra It is also with the help of the envelopes that tinsmiths make developable surfaces; because they make use of a cylindrical or conical anvil to double, bit by bit, the sheet of tin along a series of lines drawn on their plan (Gorbea y Gancedo 1845: 117; our trans.). It is interesting to see how in the different pattern papers appearing in sixteenth‑ century treatises on tailoring (Fig. 5 Left) and (Fig. 6 Right), the problems that arose are the same as those faced now in digital manufacturing. Specifically, they have to do with the optimisation of material usage, on the one hand, and that of the alphanumeric coding of the cuts or machined parts on flat pieces, with a view to placing or assembling them in their ultimate spatial form, on the other. Circumscribed Developable Surfaces intheArts andArchitecture It is well known that the twentieth‑century artistic avant‑gardes discovered an inexhaustible source of inspiration in geometry and mathematics. In the field of sculpture, Antoine Pevsner, one of the fathers of Russian Constructivism, was the pioneer in the formal exploration of developable surfaces (Farinella and Baglioni 2017). Since then, there has been an endless stream of new contributions, including the work of Ilhan Koman (Tevfik etal. 2006). Fig. 4 3D representation of Fig.3 appearing in the treatise by La Gournerie. Image: authors Fig. 5 Left Juan de Alcega, Libro de geometria, practica y traça… (1580: fol. 31r); Right Diego Freyle, Geometria y traça para el oficio de los sastres… (1588: Fol. 16v) A.Martín-Pastor, F.González-Quintial The sculpture series of the minimalist artist Richard Serra, Torqued ellipses, has a special relationship with the object of study.5 It involves the generation of a surface circumscribed to two rotated ellipses (Fig.6 Bottom left). As is apparent from the many interviews that the artist has given, both the process of generating the form and the curving of the huge steel sheets with heavy machinery are directly related to the geometric processes analysed above. In one of his interviews, he has the following to say in this respect: They [the artists] constantly come up with ways of informing themselves by inventing tools or techniques or processes that allow them to see into a material manifestation in the way that you would not if you dealt with standardized or academic ways of thinking. These ellipses only came about because we had invented a wheel to make these things in order for us to understand what we Fig. 6 Top left Rotary mechanism by Richard Serra. Image retrieved from the video Charlie Rose Conversation (2001), Time 33:32, reproduced by permission; Top right Calculating the developable surface on a plane with the rotary mechanism. Source: author; Bottom left Torqued Ellipse in Brancusi- Serra Exhibition at the Guggenheim Museum Bilbao between 2011 and 2012. Photo: authors; Bottom right Movement of the sheet as it is unrolled, going from its flat to spatial form. Images: authors 5 On this topic and the geometric errors detected in the explanatory posters of the Guggenheim Museum in Bilbao, see González‑Quintial etal. (2022). Approaching Developable Surfaces Through Shadow andPenumbra were doing. And albeit it’s a small invention and in another sense, it has never come up in the history of form‑making before (Art21 2013). The wheel formed by two interconnected ellipses that are rotating on the same horizontal plane (Fig. 6 Top left) uses gravity to find a geometric solution to an analytically complex problem.6 As the ellipses rotate, the horizontal plane on which the set rests acts as the tangent plane common to both. On completing a full rotation, the wheel draws the outline of the developable surface circumscribed to the two ellipses (Fig.6 Top right). In a way, Serra’s “invention” is the mechanical extrapolation of the synthetic procedure set out by Monge, represented by La Gournerie and exemplified by Leroy and Gorbea y Gancedo in the case of tinsmiths (Fig.6 Bottom right). The application of developable surfaces in the field of architecture is currently being comprehensively revised and updated (Glaeser and Gruber 2007). The team of engineers of Frank Gehry paved the way for the formal control of these surfaces for their architectural use.7 Since then, the list of significant contributions from a mathematical and computational perspective has grown far too long to address here.8 From a more modest standpoint, the widespread use of digital and parametric graphic tools is also favouring the geometric control of these surfaces on the basis of graphic thinking (Martín‑Pastor 2019). Penumbra, Shadow andtheLimits oftheRotary Mechanism It is important to understand that parallel or conical illumination, in which there is only shadow and no penumbra, are simpler—and more exceptional—cases of a physical problem. The presence of penumbra is a natural phenomenon that manifests itself when the light is not a point but covers a certain surface or volume (Fig.7). The model of the window through which diffuse light enters clearly exemplifies this phenomenon. Always provided that there are two closed curves, one can act as a window and the other as an opaque object. At this point, it is should be noted that the rotary mechanism does not allow for the complete construction of a surface circumscribed to two cores (curves or surfaces) in all cases. In Fig.8 Left, the mechanism would only roll on the outer edge of the two curves, leaving an occluded inner area. The model based on the shadow cast by a window (Fig. 8 Centre) provides a complete solution (Fig. 8 Right). The field of application of the rotary mechanism as a procedure for finding the circumscribed surface is limited to two surfaces (or curves) with a positive curvature, as can be seen in Fig.9. 6 The description of the mechanism of the double ellipse is found Cooke etal. (1997) and commented on in Mattsson (2000). This idea is also described in the video Art21 (2013): https:// www. youtu be. com/ watch?v= G‑ mBR26 bAzA and Charlie Rose Conversation (2001), https:// charl ierose. com/ videos/ 18060. 7 Gehry’s approach to developable surfaces has been studied in Shelden (2002). 8 Special mention should go to Yang etal. (2006) and the works of Pottmann etal. (2007, 2008, 2010), Jung etal. (2015), Tang etal. (2016), Bo etal. (2019) and Hoffmann etal. (2022). A.Martín-Pastor, F.González-Quintial Fig. 17 Left Discontinuities in the rectifying strip on anticlastic surfaces; Centre Discontinuities in the strip circumscribed to the two equidistant curves; Right Amplified area of discontinuity. Images: authors Fig. 18 Dieste Pavilion. Image: authors It can be observed how similar results can be obtained by resolving this same problem with the placement of a developable strip circumscribed to two equidistant curves of the geodesic (Fig.17 Centre) and (Fig.17 Right). Applications As has been seen, not all surfaces circumscribed to two cores can be used for construction purposes. The parametric tools that we have generated allow us to trace in a rigorous fashion the penumbra and shadow of all types of bodies, as well as to explore and define developable surfaces on the basis of two curves or surfaces. We will now offer several examples in which we have considered and put into practice the issues addressed above. Dieste Pavilion The proposal is an installation formed by three surfaces of plywood with a thickness of between 3 and 4mm which, after being assembled on the ground, acquires its spatial position by folding and tying together the three surfaces. The shape was inspired by the work of the Uruguayan engineer Eladio Dieste (Fig.18). Approaching Developable Surfaces Through Shadow andPenumbra Starting from any two circumferences located in the space, the surface circumscribed to both is a generic developable surface.12 Based on this idea, it is possible to generate two types of surfaces depending on whether the top plane intersects the bottom circumference (Type 1, Fig.19 left) or not (Type 2, Fig.19 right). The proposal involves the alternate concatenation of these two types of surfaces on a sole sheet. As has been seen, the condition for guaranteeing the continuity and constructability of the surface is that there are common tangent planes in each pair of inflection points of both curves (Fig.20). Dieste Pavilion is an experimental pavilion made in collaboration with different universities, in which a large work team was involved.13 Cactus Pavilion This pavilion was designed as an itinerant interpretation centre in the framework of the project ‘Architectural & Natural Heritage Rescue Project of Santiago de Anaya’. It is a biomimetic architectural proposal in the shape of a hollow wooden cactus, made from developable strips formed by plywood panels of a thickness of between Fig. 19 Developable surface between two circumferences. Left Type 1; Right Type 2. Image: authors Fig. 20 Developable surfaces tangentially linked by a common tangent plane. Image: authors 12 In the event that the two circles were co‑spherical sections, it would be a cone. 13 Department of Graphic Engineering ‑ETSIE‑ University of Seville; DepInfo and Fablab MVD, FADU, University of the Republic, UdelaR. A.Martín-Pastor, F.González-Quintial 3 and 4mm. The shape as a whole serves structurally as a sole self‑supporting skin (Fig.21). The geometric structure of the cactus is formed by segments comprising three types of developable surfaces. Each one of these three surfaces has been generated as a surface circumscribed to two open flat curves. The exploration of the shape of the pavilion was possible thanks to the development of the method of geometric places, which made it possible to manipulate those developable surfaces on the basis of the flat curves on which they rested, the crests and the valleys of the segments of the cactus (Fig.22). Cactus Pavilion is a multidisciplinary project with the participation of many entities.14 Fig. 21 Cactus Pavilion. Image: authors Fig. 22 Segments of the pavilion based on surfaces circumscribed to two flat curves. Image: authors 14 The Department of Graphic Engineering ‑ETSIE‑ University of Seville; Fablab Donostia ETSA‑ AGET UPV‑EHU; The Master’s Program and Doctorate in Architecture of The National Autonomous University of Mexico and the Laboratory of Conservation of the Natural and Cultural Heritage, UNAM. The project received an honourable mention in Laka Competition 2018. Credits and project at https:// lakar eacts. com/ winne rs/ cactus‑ pavil ion/. Approaching Developable Surfaces Through Shadow andPenumbra Acknowledgements We would like to thank Charlie Rose Inc. for granting us permission to use their image in Fig.6 top left. All other images are by the authors. This research has been possible thanks to the internationalisation grants awarded to the Instituto Universitario de Arquitectura y Ciencias de la Construcción (IUACC) by virtue of the XI Plan Propio de Investigación y Transferencia of the University of Seville, with the support of the research group “HUM‑976. Expregrafica. Lugar Arquitectura y Dibujo”. It has also been partly funded by the ETSA AGET of the University of the Basque Country and the Basque government. Special thanks are also due to the DepInfo and Fablab MVD, FADU of the University of the Republic (UdelaR) and the Master’s and PhD Programme in Architecture of the National Autonomous University of Mexico (UNAM) and its Natural and Cultural Heritage Conservation Laboratory. Funding Funding for open access publishing: Universidad de Sevilla/CBUA. Data availability All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Andrés Martín‑Pastor and Francisco González‑Quintial. The first draft of the manuscript was written by Andrés Martín‑Pastor and Francisco González‑Quintial and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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Yang, Liu, Helmut Pottmann, Johannes Wallner, Yong‑Liang Yang and Wenping Wang. 2006. Geometric Modeling with Conical Meshes and Developable Surfaces. ACM Transactions on Graphics 25(3): 681‑689. https:// doi. org/ 10. 1145/ 11419 11. 11419 41 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Andrés Martín‑Pastor holds degree in Architecture with a PhD from Universidad de Sevilla (Spain) since 2009. He works as a lecturer in the Department of Graphic Engineering in the same University. His publications focus on the study of Perspective and Geometry in Architecture, making a thorough review from the graphic tradition inherited to new digital tools. He also has been visiting lecturer at several Universities where he has given workshops on Architectural Geometry. His work of developable surfaces in pavilions has been recognized with some awards in Ephemeral Architecture. Francisco González‑Quintial earned his Ph.D. in Architecture at the University of the Basque Country in 2012, where he currently teaches in the Department of Architecture at the School of Architecture in the UPV‐EHU (University of the Basque Country). He is specialized in Structures, Geometry, Parametric Architecture and Digital Fabrication. Since 2016 he has been in charge of the FabLab Donostia, a digital fabrication laboratory that belongs to ETSA‐AGET.