The objective of this study is to reveal some properties of ruled surfaces through virtual laboratory and, overall, to try to contribute to the rereading and the renewal of Descriptive Geometry (DG). Precisely the paper describes a new scientific method to construct an elliptic hyperboloid, given three skew lines, through mathematic representation, and illustrates the properties of Chasles’s theorem [1 (figure 1). First of all I need to introduce and explain some words and ideas that I will use in this paper. In the circle school of doctorate of Sapienza University of Rome, some researchers, under the direction of Riccardo Migliari, intend to renew the study of DG through the following means: a) the addition of the mathematical representation method to the classical methods (perspective, orthographic projection, axonometric projection, etc.) [2]: a concept that we could also find in Monge’s intuitions revealing that the synergy between informatics calculation and codified representation can be the best research instrument in geometry; b) a rereading of the classical problems of DG and the study of more simple solutions thanks to the above mentioned synergy. By “mathematical representation method” we mean representation through classical equations, as well as NURBS, reaching a degree of accuracy in the range of microns (I have used software as rhinoceros4.0 and thinkdesign2009.1). By “virtual laboratory” we refer to the 3D computer graphics experimentation which substitutes, in the renewed DG, the 2D traditional representation. Within this laboratory curves and surfaces are constructed directly in the space. In addition, the virtual laboratory allows to draw through the use not only of circles and straight lines (rule and compasses) but also of conic curves and quadratic surfaces as well as more complex forms. DG, as well known, is a branch of mathematics which has three essential objectives: 1)The visualization of three-dimensional shapes; 2) the construction of the shapes; 3) and finally the invention/discovery of shapes and their properties and relations. Monge reminded us that DG is “an instrument to investigate the truth; it offers constant examples of passages from known to unknown”. For centuries man has been using solely the rule and compasses as essential tools to represent shapes. The informatics revolution has now introduced two new instruments for representation: the mathematical representation method and the numerical representation method. This epochal change allowed the first objective of DG, visualization, to become automatic. At the same time it enhanced the potentials of the construction and invention moments. This change made possible and desirable a reviewing of some of the problems of DG. The new mathematical representation method allows to visualize easily some properties that in the past were only enunciated in literature, but never represented. Good examples are the properties of the elliptical hyperboloid. The reasons of this gap in classical geometry literature are to be found in the complexities of these shapes that could only be represented with the instruments of classical descriptive geometry. Nowadays informatics revolution gives to mathematicians and to the DG researchers the possibility to examine and discover the properties of shapes straightly into space. Digital geometrical drawing is, above all, an instrument of logics and thought. Most probably we are facing a new possible development of the science, equal to the one occurred after French revolution thanks to the codification of Monge’s method.

### A ruled surface through three given skew straight lines and the Chasles’s theorem: a rereading through virtual laboratory

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*FALLAVOLLITA, FEDERICO*

##### 2010

#### Abstract

The objective of this study is to reveal some properties of ruled surfaces through virtual laboratory and, overall, to try to contribute to the rereading and the renewal of Descriptive Geometry (DG). Precisely the paper describes a new scientific method to construct an elliptic hyperboloid, given three skew lines, through mathematic representation, and illustrates the properties of Chasles’s theorem [1 (figure 1). First of all I need to introduce and explain some words and ideas that I will use in this paper. In the circle school of doctorate of Sapienza University of Rome, some researchers, under the direction of Riccardo Migliari, intend to renew the study of DG through the following means: a) the addition of the mathematical representation method to the classical methods (perspective, orthographic projection, axonometric projection, etc.) [2]: a concept that we could also find in Monge’s intuitions revealing that the synergy between informatics calculation and codified representation can be the best research instrument in geometry; b) a rereading of the classical problems of DG and the study of more simple solutions thanks to the above mentioned synergy. By “mathematical representation method” we mean representation through classical equations, as well as NURBS, reaching a degree of accuracy in the range of microns (I have used software as rhinoceros4.0 and thinkdesign2009.1). By “virtual laboratory” we refer to the 3D computer graphics experimentation which substitutes, in the renewed DG, the 2D traditional representation. Within this laboratory curves and surfaces are constructed directly in the space. In addition, the virtual laboratory allows to draw through the use not only of circles and straight lines (rule and compasses) but also of conic curves and quadratic surfaces as well as more complex forms. DG, as well known, is a branch of mathematics which has three essential objectives: 1)The visualization of three-dimensional shapes; 2) the construction of the shapes; 3) and finally the invention/discovery of shapes and their properties and relations. Monge reminded us that DG is “an instrument to investigate the truth; it offers constant examples of passages from known to unknown”. For centuries man has been using solely the rule and compasses as essential tools to represent shapes. The informatics revolution has now introduced two new instruments for representation: the mathematical representation method and the numerical representation method. This epochal change allowed the first objective of DG, visualization, to become automatic. At the same time it enhanced the potentials of the construction and invention moments. This change made possible and desirable a reviewing of some of the problems of DG. The new mathematical representation method allows to visualize easily some properties that in the past were only enunciated in literature, but never represented. Good examples are the properties of the elliptical hyperboloid. The reasons of this gap in classical geometry literature are to be found in the complexities of these shapes that could only be represented with the instruments of classical descriptive geometry. Nowadays informatics revolution gives to mathematicians and to the DG researchers the possibility to examine and discover the properties of shapes straightly into space. Digital geometrical drawing is, above all, an instrument of logics and thought. Most probably we are facing a new possible development of the science, equal to the one occurred after French revolution thanks to the codification of Monge’s method.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.