geometric modelling in education process - · pdf fileinternational conference on engineering...

International Conference on Engineering Education July 25–29, 2005, Gliwice, Poland. 1

Geometric Modelling in Education Process Authors: Ivana Linkeová, Czech Technical University in Prague, Karlovo n.13, Prague, [email protected] Abstract The paper presents novel teaching methods which are introduced into the subject ‘Constructive Geometry’ at the Faculty of Mechanical Engineering (FME) at Czech Technical University in Prague. These methods are focused on three-dimensional (3D) modelling of more complicated problems from descriptive geometry – e.g. intersection of surfaces of revolution, helicoidal surfaces and envelope surfaces. Constructive Geometry is taught as a basic introductory subject in the first year of study. It aims to provide students with the geometry knowledge, fundamentals and skills of the graphic representation in the extent needed for further studies. It shall, first of all, acquaint the students with the basics of geometric representation of real mechanical details, teach them to express graphically, draw and read drawings and particularly to draw sketches in order to be able to attend the lectures of special subjects during the following years of study. This knowledge is crucial for the students to enable them not only to record technical ideas and designs in their future expertise, but also discuss the problem with their fellow-workers. In the introductory phase of study, this subject also strongly influences the development of the spatial imagination as well as it deepens the responsibility, conscientiousness and interest in work and contributes to a higher degree of students’ graphical manifestation. The 3D models of important geometric problems make the connection between the real objects and their geometric representation easier for students who have not gained any practical experience and thus these increase the students’ motivation for their studies. Especially, these models try to strengthen their self-confidence and identification with their professional field. The 3D modelling contributes to decrease the number of the students who do not fully understand the matter and therefore they lose contact with the studies. Subsequently, they try to memorize the construction as well as the geometric representation without fully understanding the technical principle. The vivid approach aims to decrease the percentage of students that misunderstand the technical basis. Moreover, these students cannot follow the course and very often leave the study. Index Terms descriptive geometry, envelope surfaces, geometrical modelling, helicoidal surfaces, surfaces of revolution INTRODUCTION The paper presents a set of vivid 3D models of important geometric objects and geometric problems, solution of which belongs to the essential knowledge of the specialists in the field of mechanical engineering. The topics of created 3D models are based on from the experience when teaching subject Constructive Geometry in the FME.

Constructive Geometry teaches geometric fundamentals and properties of the objects from mechanical engineering, with their graphic representation and graphic solution of the mutual geometric relations, such as construction of meridians, curves of intersection, characteristic curves of envelope surfaces, etc. Constructive Geometry is taught in first year of study in the FME. This subject makes great demands on the student’s spatial imagination. The student without previous practical knowledge of descriptive geometry or technical drawing – without trained spatial imagination – can easily lose contact with the problems solved in Constructive Geometry and finally with the study. Establishing of the set of 3D models is a direct response to relatively high percentage of unsuccessful students in the first year of their study. Novel teaching materials continue to increase efficiency of pedagogical process at tuition itself as well as at the self-study.

Rhinoceros – NURBS modelling program for Windows was used as a working environment for 3D models creation. Software Rhinoceros provides perfect tools to construct, visualise and analyse the created models.

This paper is organised as follows: Section Role of Constructive Geometry deals with the function of constructive geometry, engineering drawing and descriptive geometry in the mechanical engineering education process. Section Novel Teaching Methods in Constructive Geometry summarizes the reasons for introducing 3D models into the Constructive Geometry lessons and enumerates the benefits of such an approach. The detailed procedure of 3D model creation by means of Rhinoceros software is given in the section 3D Modelling. In this section, the construction of a characteristic curve, envelope surface and meridian of envelope surface is described. Further 3D models of surfaces of revolution, helicoidal surfaces and envelope surfaces are presented in the section Examples together with the analysis of troubles that the students meet when solving these tasks. Section Conclusion summarises the experience with the novel teaching methods introduced into the course and gives an overview of contribution of such an approach.


ROLE OF CONSTRUCTIVE GEOMETRY Constructive Geometry, in association with Engineering Drawing, belongs to first subjects taught at technical universities. These subjects are essential if the students are to obtain a fundamental knowledge of the design and construction of machines and mechanisms. The basis of Constructive Geometry and Engineering Drawing consists of descriptive geometry which is the branch of geometry dealing with the planar representation of spatial objects. By means of such a representation, spatial geometric problems may be solved in the plane. The fundamental role of the descriptive geometry stems from the study of methods of projections and from the study of techniques for determining the forms, dimensions and relations of spatial objects by means of plane drawings.

In mechanical engineering courses, Constructive Geometry does not cover only problems included in descriptive geometry. Nowadays, Constructive Geometry solves many problems connected with the engineering applications, for instance the planar and spatial kinematic geometry, curves of intersections of various surfaces, properties of surfaces of revolution, helicoidal surfaces, envelope surfaces, developable surfaces.

Constructive Geometry serves to develop the ability to imagine objects in space which is essential for engineering design. Every engineer must know techniques used to represent objects by drawings. He must be able to think in three dimensions. Creative engineering without this ability is almost impossible. NOVEL TEACHING METHODS IN CONSTRUCTIVE GEOMETRY To enhance the efficiency of the teaching process in Constructive Geometry, the set of 3D models of important geometric problems have been created and used during lessons and published on the Internet (http://marian.fsik.cvut.cz/~linkeova) in a trial version. Some examples and exercises from the required textbooks [1], [2], [3] have been taken as a source of the topics of 3D models. For a trial version, challenging examples and exercises were chosen. This approach appeared to be very useful for these reasons: • The stage of familiarizing the students with the solved problems was noticeably compressed. • The students had the possibility to see the models presented during lessons on the Internet again. • The students had the possibility to check their solution during self-study immediately. 3D MODELLING In this section, the detailed procedure of 3D solution of envelope surface is presented. The envelope surface arises by one-parametric motion of a generating surface along a general trajectory. The generating and envelope surfaces touch at every position of generating surface along the so called characteristic curve. Generating and envelope surfaces have common tangent planes at every point of the characteristic curve. The characteristic curve can also be considered as the intersection curve of two infinitely close positions of generating surface.

It is known, that the envelope surface can be created by the same motion of the characteristic curve as the original generating surface. The scrutiny of the curves is easier than study of the surfaces; therefore the construction of characteristic curves belongs to the important problems of mechanical engineering. Geometric problems of envelope surfaces can be considered as basic problems of mechanical engineering in general, because each manufactured surface arises as the envelope surface of a moving cutting tool. At the same time, these problems are certainly the most difficult, because the characteristic curve is a spatial curve even in the simplest shapes of generating surfaces and trajectories. To imagine the real shape of the spatial curve is quite difficult and to imagine a surface which arises by motion of spatial curve is almost impossible.

In Constructive Geometry teaching, only a plane, sphere and surface of revolution are considered to be the generating surface. Only translation, rotation and helicoidal motion are taken into account as the possible motion; therefore a straight line, a circle and a helix can be considered as a possible trajectory. It is necessary to transform the characteristic spatial curve into the planar curve in the cases of rotation or helicoidal motion. This transformation lies in the construction of principle meridian of envelope surface. Task Setting and Solution in Monge Projection Task setting of our example is drawn in Figure 1 (a): truncated cone of revolution C is moving along the axis o by right-handed helicoidal motion along the helix h, lead of which is designated by v. The solution – characteristic curve k and principal meridian m – is drawn in Figure 1 (b). The theory of meridian construction is basic and it can be found in each textbook of descriptive geometry. The theory of point-wise construction of a characteristic curve is based on the enquiry of envelope surface generated by elementary motion of the elementary substitutive sphere and, furthermore, on the fact, that the point lies on the characteristic curve if and only if the tangent plane of generating surface at this point contains the tangent line of the trajectory. Description of this theory can be found in standard textbooks of descriptive geometry and it is not described here.


Modelling in Rhino In 3D modelling, the point-wise construction of characteristic curve is not elegant. In Rhinoceros, the characteristic curve is modelled as the intersection curve of two very close positions of generating surface. In Figure 1 (c), there are the 3D models of frontal, horizontal and profile projections of the truncated cone and the characteristic curve.

C2

o1

o2

v

x1,2

σ1

C1

σ1

C2

m2

m'2

m'1 m1

k1

k2h2

h1

o2

o1

C1

x1,2

v

a) Task setting in Monge projection b) Solution in Monge projection c) 3D model of characteristic curve

d) Task setting in Rhinoceros e) Array of the cones along the helix f) 3D final model

FIGURE 1 ENVELOPE SURFACE GENERATED BY HELICOIDAL MOTION OF THE TRUNCATED CONE OF REVOLUTION

The procedure of 3D modelling of our example includes the following steps and constructions, see Figure 1 (c) – (f): • Lines and curves modelling – it can be seen that the axis, the lead of helicoidal motion and the helix is modelled as a

pipe. This is for visualisation reasons, because only the straight lines and the curves are less noticeable in rendered 3D figure. This way of modelling lines and curves is used in all of the created models.

• Trajectory construction – the trajectory of helicoidal motion is a helix. Command for helices modelling requires the end points of an axis and the radius of the cylindrical surface on which the helix lies. The construction of the circle which touches the generating curve of a truncated cone as a top view of this cylindrical surface is sufficient. Here, the extraction of silhouette curves from truncated cone and their projection onto the horizontal plane of projection is used in order to obtain the top view of required generating curve of a cone. The touching circle is constructed by the tangent object snap. After these auxiliary constructions, the helix h as the trajectory of the motion can be modelled.


• Characteristic curve construction – first, it is necessary to construct the copy of the truncated cone moved infinitesimally along the trajectory. In Rhinoceros, there is a command that can create the array of the selected item along the curve. We can use this command for creation the array of truncated cones copied along the helix, as is shown in Figure (e), but there is a problem: either the required number of items along whole curve or the required distance between items can be given. Then, we split the helix at the point which is infinitesimally distant from the beginning of the helix and after that we create an array of two items along this very short helix. Finally, we can construct intersection of these two truncated cones to obtain the characteristic curve.

• Envelope surface construction – for this purpose, the command for creation of the surface by revolving a profile curve (characteristic curve) along a path curve (helix) is used.

• Principal meridian construction – principal meridian is constructed as an intersecting curve of the envelope surface and the plane which is perpendicular to the horizontal plane of projection and which passes through the axis of the helix.

• Rendering – to achieve the best image and good clarity of the model, it is necessary to adjust the orientation of the model in the space, colours, gloss and transparency of modelled objects and finally ensure the good lighting.

EXAMPLES To demonstrate the concept of the novel approach into the teaching Constructive Geometry, several interesting examples of three-dimensional models are shown in this section. • Intersections of surfaces of revolution

The surface of revolution arises when the generating curve rotates around the axis. The solution of intersecting curve of two surfaces of revolution depends on mutual position of the axes of surfaces. It is very hard to estimate and imagine the shape and position of the intersecting curve. The following facts appeared to be most difficult: intersecting curve has two branches (Figure 2), and degeneration of intersecting curve of two quadrics (Figure 3). • Helicoidal surfaces

Helicoidal surface is generated by a composed motion of generating curve: firstly, by rotation of generating curve around the axis and, simultaneously by translation of generating curve along the same axis. The angle of rotation and the distance of translation are proportional. The construction of axial section (especially of principal meridian) and construction of normal section are the basic problems of helicoidal surfaces. These constructions are not easy to solve in the case of simple motion, and in the case of composition of two motions it is even more difficult. Expample of helicoidal surface – Serpentine of Archimedes – is depicted in Figure 4. • Envelope surfaces

The problems about envelope surfaces were mentioned in section 3D Modelling. Here, an example of an envelope surface created by rotation of generating surface along the axis is given. Figure 5 shows the envelope surface generated by rotation of general surface of revolution. Characteristic curve and principal meridian are depicted, too.

o‘2

C2 o2

k2=k‘2

T2

T1 C1

k‘1

o1 o‘1

k1


FIGURE 2 INTERSECTION (k, k’) OF A CONE OF REVOLUTION (C) AND A THORUS (T)

C‘2C2

C‘1 C1

o‘2

o2

o‘1o1

k2 k‘2

k1

k‘1

FIGURE 3 DEGENERATED INTERSECTION (k, k’) OF TWO QUADRICS OF REVOLUTION

S2

S1

h2

h1

m2

m1

ρ1

σ1

r1

r2

k1

k2

o2

o1

FIGURE 4 CONTOUR (S), PRINCIPAL MERIDIAN (m) AND NORMAL SECTION (r) OF HELICOIDAL SURFACE (SERPENTINE OF ARCHIMEDES)

Serpentine of Achimedes is generated by helicoidal motion of a circle which lies in the plane perpendicular to the direction of the motion.


m1

E2

E1

G2

k2

k1

m2

o2

o1

a1

a2

S2

S1

G1

FIGURE 5 ENVELOPE SURFACE (E) GENERATED BY ROTATION OF THE SURFACE OF REVOLUTION (G), CHARACTERISTIC CURVE (k) AND PRINCIPAL MERIDIAN (m) CONCLUSION 3D modelling of geometric problems solved in Constructive Geometry is presented in this paper. 3D modes have been introduced in order to simplify and make the teaching and self-study more efficient. The connection between real shape and relationships of the objects and its graphical representations, which are drawn according to the descriptive geometry rules, requires extensive ability of spatial imagination. When reading the text where the constructions on a plane drawing are explained, it is necessary to visualize the space image of various operations used to solve the problem in the plane.

The trial version of 3D models has met the expectations. The increased interest of the students as well as easier understanding of the principles and procedures of constructions was noticed. The created 3D models were useful during consultation hours, too.

Additionally, the figures produced a great side effect – the students seemed to be excited. They wanted to create similar figures and similar models. This interest made us introduce the 3D modelling in Rhinoceros into facultative subject called ‘Geometry for CAD’. The content of practical part of Geometry for CAD was 3D modelling of the objects from mechanical engineering and every day objects besides geometric problems modelling. All modelled objects and geometric problems, which were created by students during Geometry for CAD teaching, are published on http://marian.fsik.cvut.cz/~linkeova.

The students chose the modelled object or geometrical problems themselves. It can be said that the spontaneous competition arose and the students approached this task with enthusiasm, creativity and enjoyment. This fact can be considered the most important impact of the introduction of geometrical modelling on teaching. ACKNOWLEDGEMENT The paper is supported by Czech Technical University in Prague, project No. CTU 0513112 – NURBS Representation of Curves and Surfaces in Maple. REFERENCES [1] Kargerová, M., "Geometry and Graphics", textbook, CTU in Prague Publishing, 2004.

[2] Kargerová, M., Mertl, P. "Konstruktivní geometrie", textbook, CTU in Prague Publishing, 2004

[3] Kopincová, E., Květoňová, B., "Cvičení z konstruktivní geometrie", textbook, CTU in Prague Publishing, 2003.

[4] Velichová, D., "Constructive Geometry", electronic book, http://www.km.sjf.stuba.sk/Geometria/KOGE/book.html, KM SjF STU, 2003.

[5] Chahly, A., T., "Descriptive Geometry", book, Higher School publishing House, Moscow, 1968.

[6] "Rhinoceros – NURBS Modelling for Windows", User’s guide.

geometric modelling in education process - · pdf fileinternational conference on engineering...

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