The geometric fundamentals of design: towards a new descriptive geometry

Geometry, thus directed towards the needs of the project, is the geometry that, bringing together millennia of experience, has been systematised by Gaspard Monge and that received from him the name of Descriptive Geometry (1794), and as such it is still known. This science gathers the methods necessary to the visualization and to the measurement, as well as the theories and the procedures necessary to the construction of the conceived shape. Descriptive geometry (DG) also has another function, which is not as conspicuous as the ones already enunciated, but which is no less important for that: it, in fact, stimulates the invention. Monge himself admits that this science is a means to search for truth and he offers endless examples of the passage from the well-known to the unknown. As is common knowledge, at the end of the last century, there have been developed information technologies able to automate some of the tasks that until then were relying on descriptive geometry and, in particular: the visualization and the measurement are nowadays excellently accomplished by the information systems. At the same time, thanks to the profound changes that we have mentioned before, the design digital tools are much more efficient and accurate today than they were in the past and, consequently, the invention expresses itself more freely. Thus to the traditional representation methods, are therefore added the digital representation methods: mathematical representation and the numerical or polygonal representation. The digital representation methods have some other extremely innovative characteristics, which have remarkable impacts on the planning as well as on the theoretical evolution of descriptive geometry and these are: the accuracy and the spatiality. The accuracy is the capability of the method to describe the measure: a mathematical representation of good quality has an accuracy of some micron, namely of a few thousandths of a millimetre. The spatiality is the capability of the representation to simulate the space and thus allow the designer to act on the three-dimensional forms, as he could do it on their physical model. The extensive spatiality of the digital representations has another remarkable effect, too: it makes it possible to create geometric constructions that until only recently had a purely theoretical character. Such are, for instance, the genesis and the projective transformations of quadric surfaces, or construction of principal axes of quadric ruled surfaces. Another good example of the new characteristics of the descriptive geometry extended to the digital tools, which is presented in this paper, is given by the solution of the Apollonian problem in the space, which consists in constructing the sphere (or the spheres) that touches other four given spheres. The experience described is intended to show some characteristic aspects of the renewal process of study of DG. The union between the concise reasoning of the geometry and the electronic calculation, is able to create a synthesis between quantity and quality, producing results that, while they describe with audacious analogies and with great immediacy the reality of a phenomenon, they also allow a quantitative analysis with controlled accuracy