This study proposes a synthetic solution to the problem of the construction of the axes of a quadric cone. It also proposes the extension of this method to the construction of the elliptic hyperboloid given three skew lines. The above problems are particularly complex when addressed in Descriptive Geometry (DG) with classical graphic methods of representation. Today, the introduction of digital methods of representation and the ability to draw directly into space with high levels of accuracy, allows a revisitation of some of the problems of the traditional repertoire of DG, studying them in space, and developing simple solutions. This is the case study presented here. The construction of the axes of the cone is followed by a historical-critical survey of an experimental nature regarding some alternative solutions of the traditional heritage of the DG. In particular, two solutions are elaborated: one graphical, the other theoretical, by, respectively, Théodore Olivier and Otto Wilhelm Fiedler. The paper concludes with the extension of the construction of the axes of the cone to the construction of the elliptic hyperboloid given three skew lines. This study is an opportunity to show the potential of digital tools in the renewal of DG. As, in fact, Gaspard Monge teaches: «DG has two main objectives. The first is to accurately represent, in the drawings that have only two dimensions, objects that have three [...]. The second objective of DG is to deduce, from the exact description of the bodies, all that necessarily follows from their respective positions and shapes». In other words, to visualize forms in space through knowledge of the methods of representation and to construct and understand geometric entities, their properties, their relationships. «In this sense – continues Monge - it is a means of seeking truth and it provides examples of the continuous passage from the known to the unknown». Today, information technology has rendered automatic the initiatial objective, which is the visualization and, at the same time, has expanded the second one, the construction. The ability to directly represent in space, with micron level accuracy, strengthens the heuristic value of the design and enhances the experimental nature of the same that includes discovery/invention, i.e., the passage from the known to the unknown.
Construction of the three principal axes of quadric ruled surfaces
Salvatore, Marta
2012
Abstract
This study proposes a synthetic solution to the problem of the construction of the axes of a quadric cone. It also proposes the extension of this method to the construction of the elliptic hyperboloid given three skew lines. The above problems are particularly complex when addressed in Descriptive Geometry (DG) with classical graphic methods of representation. Today, the introduction of digital methods of representation and the ability to draw directly into space with high levels of accuracy, allows a revisitation of some of the problems of the traditional repertoire of DG, studying them in space, and developing simple solutions. This is the case study presented here. The construction of the axes of the cone is followed by a historical-critical survey of an experimental nature regarding some alternative solutions of the traditional heritage of the DG. In particular, two solutions are elaborated: one graphical, the other theoretical, by, respectively, Théodore Olivier and Otto Wilhelm Fiedler. The paper concludes with the extension of the construction of the axes of the cone to the construction of the elliptic hyperboloid given three skew lines. This study is an opportunity to show the potential of digital tools in the renewal of DG. As, in fact, Gaspard Monge teaches: «DG has two main objectives. The first is to accurately represent, in the drawings that have only two dimensions, objects that have three [...]. The second objective of DG is to deduce, from the exact description of the bodies, all that necessarily follows from their respective positions and shapes». In other words, to visualize forms in space through knowledge of the methods of representation and to construct and understand geometric entities, their properties, their relationships. «In this sense – continues Monge - it is a means of seeking truth and it provides examples of the continuous passage from the known to the unknown». Today, information technology has rendered automatic the initiatial objective, which is the visualization and, at the same time, has expanded the second one, the construction. The ability to directly represent in space, with micron level accuracy, strengthens the heuristic value of the design and enhances the experimental nature of the same that includes discovery/invention, i.e., the passage from the known to the unknown.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
