This study shows a contemporary approach to the disciplines of drawing and mathematics. The goal is to underline the existing relations, between graphic and analytic representation, nowadays necessary for the cultural training of future architects. The paper shows the state of the art of the experimental work carried on by the authors in the School of Architecture of Roma Tre University to unify research and didactics on this topic. The authors started from two parallel methods: a theoretic approach equipped with analytical proofs, and a laboratory approach. The object of the study is the construction of historical drawing instruments: historical instruments to draw conic curves and those used to trace curves in construction yards during 1800 in Italy. The graphic construction of a curve with ruler and compass will be followed by the analytical representation with parametric ad cartesian equations. During laboratorial sessions students will build mathematical drawing machines such as ellipsographs, hyperbolographs, parabolograph and use them to draw and explore the curves. The goal is to stress the meaning of the characteristic parameters of each curve, to experiment the variations of the shape of a curve in a conscious way. When using a drawing machine, students test with their own hands how the initial “setting” influences the shape of the curve. At the same time they visualize the curve and the corresponding analytical representation. Two-dimensional sections of three dimensional objects are curves. By creating themselves a huge collection of curves students will manage and represent many complex three dimensional objects. First of all conic curves will be presented as plane sections of a cone or, in other words, as a projection of a circle by showing 3d digital models. Then each conic section will be defined as a geometric locus, followed by the description of its parametric and cartesian equation. The first machines approached are the tightened thread type, since they show clearly the geometric locus. Then students will explore instruments with linkages. There exist many different machines to draw the same conic, for example there exist at least six different ellipsographs. Different machines can be exploited to show how each drawing method stresses or hides some of the features of the corresponding curve. Then also other curves (and the related machines) will be explored, such as cycloids and epycicloids. The interdisciplinary goals of this course are: develop the attitude of students to understand and foresee the features of a figurative project on a two-dimensional support, from the beginning of its initial representation; provide scientific and cultural basis to handle digital modelling; strengthen their ability to integrate knowledge coming from different disciplines. The two authors experimented this topic in a “Progetto Lauree scientifiche” (supported by the Italian Ministry of Education and University), entitled “Conic and caustic curves: relations and comparisons between graphic and mathematical methods” with high school students of “Liceo Classico Vivona” in Rome. They will start a course in the School of Architecture of Roma Tre University, during academic year 2014/2015.

Farroni, L., Magrone, P. (2014). Mathematical Drawing Machines: Historic Drawing from a Parametric Point of View. The Case of Conic Curves. In Libro de Actas del V Congreso Internacional de Expresión Gráfica. XI Congreso Nacional de Profesores de Expresión Gráficaen Ingeniería, Arquitectura y Áreas Afines, Egrafia 2014 (pp.130-137). Rosario : Editor CUES.

### Mathematical Drawing Machines: Historic Drawing from a Parametric Point of View. The Case of Conic Curves

#####
*Farroni, Laura;Magrone, Paola*

##### 2014-01-01

#### Abstract

This study shows a contemporary approach to the disciplines of drawing and mathematics. The goal is to underline the existing relations, between graphic and analytic representation, nowadays necessary for the cultural training of future architects. The paper shows the state of the art of the experimental work carried on by the authors in the School of Architecture of Roma Tre University to unify research and didactics on this topic. The authors started from two parallel methods: a theoretic approach equipped with analytical proofs, and a laboratory approach. The object of the study is the construction of historical drawing instruments: historical instruments to draw conic curves and those used to trace curves in construction yards during 1800 in Italy. The graphic construction of a curve with ruler and compass will be followed by the analytical representation with parametric ad cartesian equations. During laboratorial sessions students will build mathematical drawing machines such as ellipsographs, hyperbolographs, parabolograph and use them to draw and explore the curves. The goal is to stress the meaning of the characteristic parameters of each curve, to experiment the variations of the shape of a curve in a conscious way. When using a drawing machine, students test with their own hands how the initial “setting” influences the shape of the curve. At the same time they visualize the curve and the corresponding analytical representation. Two-dimensional sections of three dimensional objects are curves. By creating themselves a huge collection of curves students will manage and represent many complex three dimensional objects. First of all conic curves will be presented as plane sections of a cone or, in other words, as a projection of a circle by showing 3d digital models. Then each conic section will be defined as a geometric locus, followed by the description of its parametric and cartesian equation. The first machines approached are the tightened thread type, since they show clearly the geometric locus. Then students will explore instruments with linkages. There exist many different machines to draw the same conic, for example there exist at least six different ellipsographs. Different machines can be exploited to show how each drawing method stresses or hides some of the features of the corresponding curve. Then also other curves (and the related machines) will be explored, such as cycloids and epycicloids. The interdisciplinary goals of this course are: develop the attitude of students to understand and foresee the features of a figurative project on a two-dimensional support, from the beginning of its initial representation; provide scientific and cultural basis to handle digital modelling; strengthen their ability to integrate knowledge coming from different disciplines. The two authors experimented this topic in a “Progetto Lauree scientifiche” (supported by the Italian Ministry of Education and University), entitled “Conic and caustic curves: relations and comparisons between graphic and mathematical methods” with high school students of “Liceo Classico Vivona” in Rome. They will start a course in the School of Architecture of Roma Tre University, during academic year 2014/2015.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.