Solid Perspective as a Tool for the Generation and the Study of Quadric Surfaces

by Riccardo Migliari and Federico Fallavollita

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.

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The Monge Three Point Space Resection Problem

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 [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].

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Perspective: Theories and Experiments on the “Veduta vincolata” (Restricted Sight)

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by Riccardo Migliari and Jessica Romor

Erwin Panofsky’s work on ‘Perspective as symbolic form’, has had a powerful impact not only on the art critics and therefore on the artistic-historical literature, but also on the studies that deal with the theme of perspective from the scientific point of view. The reflections stated by Panofsky in the incipit of his essay, in fact, were considered in an uncritical and superficial way to say that the perspective describes an image of the space that it represents, similar to that of the human vision only if: the eye of whoever is looking is positioned exactly in the projection centre used to generate the perspective image; the same eye remains motionless and therefore with the direction of the gaze perpendicular to the picture. This condition of observation of the perspective is known to the Italian scholars as ‘vedutavincolata ‘. Recent studies have proved, theoretically and experimentally, that, on the contrary: the eye of whoever is looking at a perspective can move in an area around a projection centre without causing a collapse of the perspective illusion, or better, of the sense of visual depth evoked by the perspective; the eye of whoever is looking can freely move around, in every possible direction, without compromising the effects of the perspective. These studies are expounded in this paper, first of all describing the theories and the experimentations that have given the above mentioned results. Secondly, it describes the verifications carried out on important perspectives painted on walls and entire rooms, in which the conditions of one single projection centre are respected. It also describes other works, in which appropriate solutions permit to dilate the area of the vedutavincolata. Finally, in a quick re-reading of the first pages of Panofsky’s essay, this paper aims at an interpretation of it, which attempt to overcome the conflict between the advocates and the detractors of perspective as legitimate form.

16TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS©2014 ISGG

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Scientific representation: research and teaching

This essay discusses the importance of didactics and the textbooks that represent the educational tool of it, being these an activity that is capable to integrate the results of research into the body of the Science to which they belong. The textbook, in fact, unrolls the leading thread of History while essays, articles and conference proceedings only shed light on a short stretch of that historical line. The essay also examines the possible criteria that permit to distinguish between a commercial publication and innovative works, which deserve no less attention than the results of scientific studies, transmitted in the manners that are exclusive of the academic communication.

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In praise of theory

by Riccardo Migliari

The disciplines of the field of Representation have a great quality which becomes a fault, and namely: they are shared. The teacher in architectural design draws and represents, where by the first term I refer to the invention drawing and by the second to the coded geometric model. So does the teacher in architectural survey, indeed, he does not only create projects, he also surveys. And so on, I could mention almost all the disciplines that form an architect, except for, maybe, the mathematics.

This characteristic of our science could lead to a great advantage: the possibility to easily interact with any other field of study of engineering and of architecture, in order to develop interdisciplinary research. But this opportunity is not well utilised, because of a fault, which is what could be called ‘the other side of the coin’. This fault consists in a widespread prejudice which says that the disciplines of the field of representation, exactly because they are shared, are also within reach of those who practise the disciplines without better studying them, relying only on the knowledge gained during the formative studies.

If to this prejudice we add the bad habit of not collecting information on the results obtained during the researches carried out by other research units, maybe in the room next to ours, then we have completed the picture of a hidden underestimation of our scientific and didactic contribution within the field of the respective schools.

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