FROM THE DESCRIPTIVE GEOMETRY TO THE INFORMATICS LANGUAGE.

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Abstract

This study is centred in the environment of the research into solutions of the problems of graphic representation, expressing theoretical considerations which allow the carrying out of geometric-descriptive procedures decoded in informatics language, for implementing CAD commands. The solution of these problems requires a reliable command of stereometry and descriptive geometry and it can extend remarkably the level of external geometric competences. The implementation of the means of ICT has brought a higher effectiveness of graphing mapping processes. In the informatics field, the advent of software dedicated to computerized representation has increased the possibilities of investigation of the forms of spatial objects, analysing their properties and reciprocal relationships has considerably simplified their graphic elaboration, the rapidity of execution, the processes of iteration, and the precision and sureness of the results. From our didactic experiences, matured within the Design course given at the Engineering Faculty of the University of Palermo, we have ascertained that the use of the informatics tool is a valid technical aid for the comprehension and the critical analysis of the geometric figures in three dimensional space. From the exercises carried out in the didactic laboratories, the need to introduce, within the computer aided design software, commands with new functions with the goal of simplifying descriptive geometry and projective applications. Focusing our attention on the AutoCAD software, it is possible, with the commands available today, to design only ellipse conic with the command “ellipse”, if the coordinates of the axes end points are known. On the other hand, it is not possible to draw conics (ellipse, parabola, hyperbola) in which two conjugate diameters are known, or one diameter and a conjugate chord, or two conjugate chords, or five elements amongst points and tangents; these conditions are very frequent in the applications of the different methods of representation. Our research, starting from the operative limitations highlighted above, intends to translate into programming language AutoLISP, structured with functions of analytical and elementary geometry, the algorithm of descriptive and projective geometry, using procedures and methods proposed by the most illustrious treatisers of the Science of Representation for the construction of conics (A.F. Frézier, M. Chasles, K. Pelz, J.V. Poncelet, J. Steiner). In this paper we report the translation of an algorithm of the construction of an conic from its five known points with projective and homological process . We proposes an application in architecture, aimed to underline the operative advantages of the CAD function created for the fast and rigorous resolution of graphic problems, linked to the general construction of conics. The discussed example is referred to a perspective to a horizontal picture plane representation of a Roman cross vault on a square plant with a springer plane coincident with the π picture plane. An opportune position of the center of projection V assigned, the perspective image of the circular directrix of the two equal cylinders -belonging to planes perpendicular to the π picture plane, passing for the sides of the springer plane- and of the diagonal ellipses -intersections of the two cylinders- it can respectively be an ellipse, a parabola or an hyperbole in relationship to the position of these planed curves in comparison to limit plane respectively, external, tangent or secant plane to the cylinders with circular section. For the geometric construction of the conics it is enough to determine, employing resolute methods for the affine orthogonal homology and with a proper center, only five points of these conics to
Lingua originaleEnglish
Stato di pubblicazionePublished - 2010

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Descriptive geometry
Ellipse
Chord or secant line
Parabola
Tangent line
Conjugate diameters
Software
Hyperbola
Projective geometry
Language
Coincident
Computer-aided Design
End point
Perpendicular
Exercise
Programming Languages
Homology
Figure
Horizontal
Intersection

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title = "FROM THE DESCRIPTIVE GEOMETRY TO THE INFORMATICS LANGUAGE.",
abstract = "This study is centred in the environment of the research into solutions of the problems of graphic representation, expressing theoretical considerations which allow the carrying out of geometric-descriptive procedures decoded in informatics language, for implementing CAD commands. The solution of these problems requires a reliable command of stereometry and descriptive geometry and it can extend remarkably the level of external geometric competences. The implementation of the means of ICT has brought a higher effectiveness of graphing mapping processes. In the informatics field, the advent of software dedicated to computerized representation has increased the possibilities of investigation of the forms of spatial objects, analysing their properties and reciprocal relationships has considerably simplified their graphic elaboration, the rapidity of execution, the processes of iteration, and the precision and sureness of the results. From our didactic experiences, matured within the Design course given at the Engineering Faculty of the University of Palermo, we have ascertained that the use of the informatics tool is a valid technical aid for the comprehension and the critical analysis of the geometric figures in three dimensional space. From the exercises carried out in the didactic laboratories, the need to introduce, within the computer aided design software, commands with new functions with the goal of simplifying descriptive geometry and projective applications. Focusing our attention on the AutoCAD software, it is possible, with the commands available today, to design only ellipse conic with the command “ellipse”, if the coordinates of the axes end points are known. On the other hand, it is not possible to draw conics (ellipse, parabola, hyperbola) in which two conjugate diameters are known, or one diameter and a conjugate chord, or two conjugate chords, or five elements amongst points and tangents; these conditions are very frequent in the applications of the different methods of representation. Our research, starting from the operative limitations highlighted above, intends to translate into programming language AutoLISP, structured with functions of analytical and elementary geometry, the algorithm of descriptive and projective geometry, using procedures and methods proposed by the most illustrious treatisers of the Science of Representation for the construction of conics (A.F. Fr{\'e}zier, M. Chasles, K. Pelz, J.V. Poncelet, J. Steiner). In this paper we report the translation of an algorithm of the construction of an conic from its five known points with projective and homological process . We proposes an application in architecture, aimed to underline the operative advantages of the CAD function created for the fast and rigorous resolution of graphic problems, linked to the general construction of conics. The discussed example is referred to a perspective to a horizontal picture plane representation of a Roman cross vault on a square plant with a springer plane coincident with the π picture plane. An opportune position of the center of projection V assigned, the perspective image of the circular directrix of the two equal cylinders -belonging to planes perpendicular to the π picture plane, passing for the sides of the springer plane- and of the diagonal ellipses -intersections of the two cylinders- it can respectively be an ellipse, a parabola or an hyperbole in relationship to the position of these planed curves in comparison to limit plane respectively, external, tangent or secant plane to the cylinders with circular section. For the geometric construction of the conics it is enough to determine, employing resolute methods for the affine orthogonal homology and with a proper center, only five points of these conics to",
keywords = "programming language AutoLISP, descriptive geometry, representation of conics",
author = "Pizzurro, {Maria Rita} and {Di Paola}, Francesco",
year = "2010",
language = "English",

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T1 - FROM THE DESCRIPTIVE GEOMETRY TO THE INFORMATICS LANGUAGE.

AU - Pizzurro, Maria Rita

AU - Di Paola, Francesco

PY - 2010

Y1 - 2010

N2 - This study is centred in the environment of the research into solutions of the problems of graphic representation, expressing theoretical considerations which allow the carrying out of geometric-descriptive procedures decoded in informatics language, for implementing CAD commands. The solution of these problems requires a reliable command of stereometry and descriptive geometry and it can extend remarkably the level of external geometric competences. The implementation of the means of ICT has brought a higher effectiveness of graphing mapping processes. In the informatics field, the advent of software dedicated to computerized representation has increased the possibilities of investigation of the forms of spatial objects, analysing their properties and reciprocal relationships has considerably simplified their graphic elaboration, the rapidity of execution, the processes of iteration, and the precision and sureness of the results. From our didactic experiences, matured within the Design course given at the Engineering Faculty of the University of Palermo, we have ascertained that the use of the informatics tool is a valid technical aid for the comprehension and the critical analysis of the geometric figures in three dimensional space. From the exercises carried out in the didactic laboratories, the need to introduce, within the computer aided design software, commands with new functions with the goal of simplifying descriptive geometry and projective applications. Focusing our attention on the AutoCAD software, it is possible, with the commands available today, to design only ellipse conic with the command “ellipse”, if the coordinates of the axes end points are known. On the other hand, it is not possible to draw conics (ellipse, parabola, hyperbola) in which two conjugate diameters are known, or one diameter and a conjugate chord, or two conjugate chords, or five elements amongst points and tangents; these conditions are very frequent in the applications of the different methods of representation. Our research, starting from the operative limitations highlighted above, intends to translate into programming language AutoLISP, structured with functions of analytical and elementary geometry, the algorithm of descriptive and projective geometry, using procedures and methods proposed by the most illustrious treatisers of the Science of Representation for the construction of conics (A.F. Frézier, M. Chasles, K. Pelz, J.V. Poncelet, J. Steiner). In this paper we report the translation of an algorithm of the construction of an conic from its five known points with projective and homological process . We proposes an application in architecture, aimed to underline the operative advantages of the CAD function created for the fast and rigorous resolution of graphic problems, linked to the general construction of conics. The discussed example is referred to a perspective to a horizontal picture plane representation of a Roman cross vault on a square plant with a springer plane coincident with the π picture plane. An opportune position of the center of projection V assigned, the perspective image of the circular directrix of the two equal cylinders -belonging to planes perpendicular to the π picture plane, passing for the sides of the springer plane- and of the diagonal ellipses -intersections of the two cylinders- it can respectively be an ellipse, a parabola or an hyperbole in relationship to the position of these planed curves in comparison to limit plane respectively, external, tangent or secant plane to the cylinders with circular section. For the geometric construction of the conics it is enough to determine, employing resolute methods for the affine orthogonal homology and with a proper center, only five points of these conics to

AB - This study is centred in the environment of the research into solutions of the problems of graphic representation, expressing theoretical considerations which allow the carrying out of geometric-descriptive procedures decoded in informatics language, for implementing CAD commands. The solution of these problems requires a reliable command of stereometry and descriptive geometry and it can extend remarkably the level of external geometric competences. The implementation of the means of ICT has brought a higher effectiveness of graphing mapping processes. In the informatics field, the advent of software dedicated to computerized representation has increased the possibilities of investigation of the forms of spatial objects, analysing their properties and reciprocal relationships has considerably simplified their graphic elaboration, the rapidity of execution, the processes of iteration, and the precision and sureness of the results. From our didactic experiences, matured within the Design course given at the Engineering Faculty of the University of Palermo, we have ascertained that the use of the informatics tool is a valid technical aid for the comprehension and the critical analysis of the geometric figures in three dimensional space. From the exercises carried out in the didactic laboratories, the need to introduce, within the computer aided design software, commands with new functions with the goal of simplifying descriptive geometry and projective applications. Focusing our attention on the AutoCAD software, it is possible, with the commands available today, to design only ellipse conic with the command “ellipse”, if the coordinates of the axes end points are known. On the other hand, it is not possible to draw conics (ellipse, parabola, hyperbola) in which two conjugate diameters are known, or one diameter and a conjugate chord, or two conjugate chords, or five elements amongst points and tangents; these conditions are very frequent in the applications of the different methods of representation. Our research, starting from the operative limitations highlighted above, intends to translate into programming language AutoLISP, structured with functions of analytical and elementary geometry, the algorithm of descriptive and projective geometry, using procedures and methods proposed by the most illustrious treatisers of the Science of Representation for the construction of conics (A.F. Frézier, M. Chasles, K. Pelz, J.V. Poncelet, J. Steiner). In this paper we report the translation of an algorithm of the construction of an conic from its five known points with projective and homological process . We proposes an application in architecture, aimed to underline the operative advantages of the CAD function created for the fast and rigorous resolution of graphic problems, linked to the general construction of conics. The discussed example is referred to a perspective to a horizontal picture plane representation of a Roman cross vault on a square plant with a springer plane coincident with the π picture plane. An opportune position of the center of projection V assigned, the perspective image of the circular directrix of the two equal cylinders -belonging to planes perpendicular to the π picture plane, passing for the sides of the springer plane- and of the diagonal ellipses -intersections of the two cylinders- it can respectively be an ellipse, a parabola or an hyperbole in relationship to the position of these planed curves in comparison to limit plane respectively, external, tangent or secant plane to the cylinders with circular section. For the geometric construction of the conics it is enough to determine, employing resolute methods for the affine orthogonal homology and with a proper center, only five points of these conics to

KW - programming language AutoLISP, descriptive geometry, representation of conics

UR - http://hdl.handle.net/10447/49582

M3 - Paper

ER -