Migliari, R. Baglioni, L. Fallavollita, F., Fasolo, M., Mancini, M.F. Romor, J., Salvatore, M. Piero della Francesca, Edizione nazionale del *De Prospectiva Pingendi*, Edizione critica del codice 616 Bibliothèque Municipale, Bordeaux. Volume III.B, Disegni, Istituto Poligrafico dello Stato, p. XIII-XXVI.

Migliari, R. Baglioni, L. Fallavollita, F., Fasolo, M., Mancini, M.F. Romor, J., Salvatore, M. Piero della Francesca, Edizione nazionale del *De Prospectiva Pingendi*, Edizione critica del codice 616 Bibliothèque Municipale, Bordeaux. Volume III.B, Disegni, Istituto Poligrafico dello Stato, p. XIII-XXVI.

The geometric certainty of perspective presents the Renaissance artist with a choice: should the work create the illusion representing the concrete, sensible world, or, on the contrary, should it portray the transcendent, speculative, abstract world of thought? This dichotomy would substantially characterise the culture of representation in the West from the Renaissance on. Perspective drawing becomes both the perfect experiment and the tangible testimony of this choice. For this reason, the phenomenon of perspective demands a unique analysis that considers this duality, since the informational intent reflected in the system for its realisation is different.

]]>This contribution identifies the recurring problems in geometric restitution of architectural perspectives and proposes a working method. The inverse problem of perspective proposes the reconstruction of the fundamental elements in space. However, when the observer and spectator are not congruent, a question is raised: should we reinstate illusory space in realistic or surrealistic way? To answer this question, a new approach to restitution is suggested, which is organized into three moments: visual analysis, geometrical analysis and three-dimensional reconstruction, and critical interpretation.

]]>by Federico Fallavollita, Marta Salvatore

This paper aims to highlight the utility of the synthetic approach in solving geometry problems thanks to the contribution of new digital technologies. The synthetic approach, as is known, addresses geometry problems without recourse to analytical methods; especially for architects, the synthetic approach relies on drawing and models, 2D graphics yesterday, 3D digital graphics today. The digital revolution has brought significant changes to the study of geometry both in education, and in research. If, for a long time, the instruments were rulers and compasses, today the main tools are computers. Currently, we can draw directly into space with an accuracy never before achieved, and we can use, in geometric constructions, forms far more complex than those represented by ruler and compass. This has enhanced the heuristic capabilities of the synthetic approach. There is a vast repertoire of geometry problems belonging to the Monge school that, for several years now is no longer studied in engineering and architecture schools: however, in the light of new digital tools, it is still a precursor to new ideas for research. This contribution aims to show how this heritage can be updated and expanded through the digital synthetic method.

]]>

The proposed article aims to provide a methodological contribution to reading historical/scientific treatise on representation. In philology, textual criticism is the critical analysis of literary texts aimed at a deep understanding of the text and the

author’s intentions. In historical/scientific treatises on representation, the work is composed of two “texts” of equal importance: one is literary, the other graphic. In these cases therefore, textual criticism should be applied to both “texts” at the same time. The case study chosen to investigate the potential of the research method is the first illustrated treatise dedicated to perspective: De Prospectiva Pingendi by Piero della Francesca. While this is a work aimed at painters, the theoretical principles and applications of perspective are also addressed, contributing to a definition of the scientific basis of descriptive geometry. The bibliography is very rich, including essays by C. Winterberg (1899), G. Nicco Fasola (1942), M. Kemp (1994), and K. Andersen (2007). However, it should be noted that graphics that could explain the reasoning behind Piero della Francesca’s drawing are not always included, especially in critical editions. Among the pages of the treatise, proposition 8 in book III holds particular importance for the proposed investigation. Here, Piero constructs the perspective of a human head (Figure 1), which is first positioned straight (frontal position), and then looks upwards (generic position).

This proposition represents a major step in that for the first and only time, the author addresses the theme of free forms. To read this proposition critically, it is necessary to refer to a hybrid digital model (polygonal mesh, subdivision surface,

and NURBS) developed by studying the codices housed in the Palatina Library in Parma and the Municipal Library in Bordeaux. Construction of the digital model allows us to understand both Piero’s refined spatial reasoning and its relative plane rendering (Figure 2) in an original reverse path that highlights the scientific nature of the entire treatise and the potential of digital representation as applied to the textual criticism of drawings.

by Riccardo Migliari, Jessica Romor, Marta Salvatore

Traditional teaching of perspective wants the perspective image to be generated with various procedures, which make use of the orthogonal projections of the object that is to be represented. It is nevertheless well-known that the perspective image also can be generated autonomously, that is to say, without resorting to the orthogonal projections, as part of a method known as ‘central projection’.

In many schools and in many textbooks these two paths, which both lead to the genesis of the perspective image, remain distinct, as if they were two different methods, if for no other reason than their vocation; the first, also called, improperly, ‘the architect’s method’, which only focuses on the achievement of the result: an image similar to the natural vision of the space; the second, conceptual, devoted to the study of the central projection in itself and its applications of projective nature: from the genesis of the quadrics to the homography.

In the Roman school, yet, as from the second half of the twentieth-century, it was attempted to bring together into one single method the two above-mentioned approaches to perspective, giving a happy ending to a history that for centuries has seen the perspective split between artists and mathematicians.

In this paper, after a short presentation of the characteristics of the ‘perspective as a representation method’, is highlighted the advantages of the aforesaid method in academic teaching. These are, precisely: first of all the possibility to see in the perspective the generalization of the representation methods, following on from the thought of Wilhelm Fiedler (1832-1912); then the possibility to easily add the concepts relating to infinity (points, straight-lines and the improper plane); and, the possibility to establish a relationship that is not general, but operational, between the graphical perspective and the digitally rendered perspective.

The objective of the research project is the preservation, the study and the popularisation of the Architectural Perspectives (AP). The project pursues this aim through the development and the application of the most advanced communication technologies (ICT). The main topic of this research project, namely the Perspective, has a strongly European character, and as such it is acknowledged and shared.

Perspective is transversal in Europe: worldly speaking, because it starts from the evidences of Roman wall paintings, arriving at the Baroque illusions; spatially speaking, because there is no region or European language which has not employed the Perspective, often borrowing solutions from one country to another; culturally speaking, because Perspective is the synthesis of science and art, and it is always developed interchanging the geometric and projective theoretical speculations, and the practising of the drawing and the architecture.

The Perspective, which this research project considers a privileged case study, is that of the geometric structures that, painted onto walls (2D) or carried out in dimensions (3D), are able to evoke an illusory depth of the architectural space. A good example of the 2D kind is the fresco painted by Agostino Tassi at Lancellotti Palace, in Rome (1621 – 1623).

Italy and all Europe have a cultural heritage of Architectural Perspectives (AP) which extent, complexity and historical importance, for science and for art, is,currently, inestimable.

This project foresees, among its main activities, the study and the validation of survey techniques and procedures, of registration, of classification of the Architectural Perspectives, at the end of achieving a first global valuation, and inductive, of the Italian cultural heritage. This also explains the very high number of Operational Units and their distribution on the territory.

Among these techniques is the Photography in ultra high resolution, which this research project will give a reliable measurement quality, such as to guarantee a documentation that is useful for the preservation. The urge of this kind of experimentation is evidenced, just to cite one example among many, by the fragile and precarious condition of Pompei.

The gathering and the diffusion of the data, do also have another value, because all these Perspectives are ‘treatises without words’ of geometrical science and their comparative examination is therefore an innovative tool for the development of the knowledge at historical scientific level. In particular, the project foresees the application of statistical methods to the analysis of the Perspective structures, like those already tested with success at the measurements of the double ration in the frescoes of Augustan age.

The digital techniques for the Perspective rendering and 3D modelling will give us the opportunity to study the proportional, compositional and stylistic aspects of the represented illusory spaces. These features do often have close relationships with the environment that houses the Perspective scenes, as these reconstructions highlight. The 3D documentation will also be useful to ensure the diffusion of the results.

Although the architectural Perspectives have always amazed the non-expert visitor, these kind of Perspectives are still not very well-known, maybe also because a systematic study, like the one this research project would like to contribute to, has never been undertaken. It is therefore not the least important part of this research project, the experimentation and the validation of innovative techniques of broad-spectrum communication of the essential data, collected and interpreted. With this objective in mind, and taking advantage of the experience matured through the previous research projects (2005 and 2008), techniques of interactive navigation of the spaces will be used, as well as techniques of Augmented reality implemented on tablets and smartphones.

Results

It is common knowledge how the projection from a central point onto a plane can be used to generate conic sections as transformations of the circle. And how these transformations can be carried out, in graphic form, with a simple and repetitive procedure. The creation of the conics as plane sections of the cone requires a more advanced level of knowledge and graphic ability, since it involves the use of descriptive geometry. With the advent of the information technology and the ensuing possibility of constructing virtual spaces of three dimensions, the technology offers today to the researcher and the teacher the possibility to

extend the above said constructions to the space. In this paper, we first describe the solid perspective in its theoretical basis and in its workability. In particular, we determine the biunivocal perspective relationship between two spaces: the real space, isotropic, and the contracted and anisotropic space of the solid perspective. Francesco Borromini’s Palazzo Spada Gallery is taken as case study to highlight how this perspective machine is capable of transforming architecture of regular shapes into the three-dimensional scenography of the same, and vice versa. We then present a sphere, studying its projective transformations into ellipsoid, paraboloid and hyperboloid. These transformations are finally examined from the canonical point of view of the projective geometry. Nowadays, thanks to the digital representation, it is possible to experiment directly in space the projective genesis of the ruled quadrics. Given two sheaves of planes corresponding in a perspectivity in space, these determine a surface which is a ruled hyperboloid or a quadric cone. It is possible to untie the two sheaves and freely move them about in space observing that the projectivity is preserved and that the two projective sheaves, in their

new positions, determine a new ruled quadric.

The study illustrates the solution of the three point space resection problem, treated by Gaspard Monge in Section V of Leçons de Geometrie Descriptive.

The problem entails the construction of the intersection curves of three tori. To solve this problem, Monge introduces several simplifications but, nevertheless, makes a mistake; this mistake has al-ready been pointed out by Gino Loria regarding the number of solutions allowed by the problem [11]. The mathematical representation, thanks to its high level of accuracy, today permits not only an efficacious solution of the general case, it also highlights without difficulty the right number of solutions.

We applied this theory to a case of photogrammetric rendering, difficult to carry out by means of the tools offered by commercial software. Case in question concerns the reconstruction of the ar-chitectural volumes, now lost, which were located along the road that crosses a village, near Rome. As is known, the reconstruction of points in space from two images is possible if these images are projective and we have at least two projective orientated stars. The first image is a vintage photo-graph (1892), the second image is a surveyed plan of the masonry still present at the site. Therefore, one of the two projective stars is assimilated to a class of vertical straight lines. With regard to photography, the problem is articulated in two typical phases of photogrammetric processes: inter-nal orientation and absolute orientation. For the absolute orientation we used the pyramid vertex method, in use since the Eighteenth Century, which consists in determining the projection center from three given points of which the positions in space are known.

The solution to the problem posed by the case study can be considered as a useful result. More in-teresting, however, is the result of the intersection of the three tori with the incident axes (fig. 1). It is, in fact, a graphic process that Gaspard Monge had already proposed in 1798 as a suitable alter-native to a system of equations that he considered difficult to solve. In particular, Monge explains how the descriptive geometrical procedure, involving the vision of the represented forms, allows you to exclude in a simple and direct manner the solutions that resolve the problem from theoretical point of view, but do not solve it in the real case because they lead to unrealistic placements of the projection center. Thus, the symbiosis between calculation and analog description, Monge had pre-dicted in these words: «[…] la géométrie descriptive porteroit dans les opérations analytiques le plus compliquées l’évidence qui est son caractère, et, à son tour, l’analyse porteroit dans la géo-métrie la generalité qui lui est propre […]» [18].