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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