Towards a new Descriptive Geometry

Although Descriptive Geometry owes its name to Gaspard Monge (1794), it is a very ancient science that has given scientific support to engineers and artists of every time. It allows, in fact, the control of the three-dimensional shapes within a virtual space, both if merely graphical, as it happened in the past, as well as if digital, as it happens nowadays. Descriptive geometry exercises this control in two phases, which are not subsequent but interactive: the project of geometric operations; the representation of such operations. The construction of a ruled surface, for instance, happens first as construction of the straight lines that lay on three oblique directrices, then as representation of the construction operations that bring to the representation of the same surface. Once represented the surface, it is possible to model it, for instance extending it or sectioning it, and the cycle restarts. This process is typical of the project, both for the architecture, as well as for the design, and it can in fact be considered an abstract exercise of these arts. Descriptive geometry possesses, so, a heuristic potential that unfolds not only in the project but also in the research. Gino Loria (1935) counts among the Mathematical Methods "La costruzione come metodo di dimostrazione esistenziale" (Construction as existential demonstration method). And. likewise, René Thom acknowledges in the model an essential tool of the human mind even in the most abstract research. This stated, it is necessary also to observe how during the last years this science, once alive and vital, has suffered a serious crisis, induced by the advent of the computer-aided drawing and modelling applications. These applications allow, in fact, the automatic realization of two classes of operations: the construction of simple and derived figures (as surfaces and their intersections) and the dynamic visualization of the same and, as it can be observed, these are the same performances of the ancient science of the representation. We could therefore conclude, but only at a first superficial examination, that descriptive geometry has de facto converted itself into the CAD. We believe, instead, that it is not so. We think that the CAD is a tool, new and powerful, which is added to the traditional tools (rule, compasses and other devices of the technical drawing). But the CAD doesn't replace, because it doesn't include, that large group of theoretical principles and consequent procedures that allow the control and the modelling of the shapes of the space. Nevertheless, being a powerful tool, much more powerful than those traditional, the CAD can provide to descriptive geometry the means to carry out procedures always more simple and general, and the means to face those problems that, before, were ignored because of their complexity. For instance: the solution of the Apollonian problem in the space finds in the accuracy of the digital representation a solution absolutely general that was impracticable with rule and compasses. The digital representation could therefore be, for descriptive geometry, what the telescope was for the astronomy in Galileo’s times and lead to an analogous process of renewal. In conclusion, the crisis of descriptive geometry is not due to obsolescence, but rather to the lack of a strong connection or better of integration with the digital technologies, software and hardware. This integration process has already begun in the applications of descriptive geometry, even though it is far from being completed; as far as the basis research is concerned it is still in the years to come. This book presents some first results of a research project that was carried out at Italian universities during the period 2010 - 2012 and that aims to make a contribution towards the above said integration and development process, in order to reach a renewal of the studies of the scientific representation for the architectural project and the design. The spin-offs of the results are various and include the development of new and more efficient solutions to the classic problems of descriptive geometry, the definition of standards for the interfaces of the computer-aided modelling systems and also the realization of innovative applications.