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.
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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.
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by Riccardo Migliari, Federico Fallavollita, Marta Salvatore
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 . 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 […]» .