15º Simpósio Nacional de Geometria Descritiva e Desenho Técnico IV International Conference on Graphics Engineering for Arts and Design São Paulo, Brasil œ 59 Novembro de 2001
TEACHING DESCRIPTIVE GEOMETRY FOR ARCHITECTS: DIDACTIC PRINCIPLES AND EFFECTIVE METHODS DEMONSTRATED BY THE EXAMPLE OF MONGE PROJECTION
Claus Pütz^{1 }RWTH Aachen, Germany Institute for Geometry and Applied Mathematics
ABSTRACT
The main task of teaching Descriptive Geometry (DG) for architects at university is to train three dimensional thinking. Nowadays the German school system completely neglects to encourage this. Great efforts are required to prepare first year students for their task to design in three dimensional space. Abstract thinking in DG is a perfect medium for this purpose. Since the students will realise this only in retrospect, topics and tasks are to be used which are naturally related to architecture. This will maintain a high level of motivation that is essential for successfully building up threedimensional imagination and thinking. The main concern of this paper is the demonstration of various didactic and methodical principles which optimise a thorough learning of Monge projection: the segmentation of a complex problem into abstract basic tasks, the visualisation of all phenomena in models and realistic illustrations, the computer animated movement (interactive variation of the basic elements of a drawing to demonstrate their significance), the following of the lecturer‘s presentation and at the same time the copying of the constructive development by the students, the transfer into their homework and the subsequent conversion into models.
Keywords: descriptive geometry, Monge projection, architecture, didactic principles.
Introduction
Being a compulsory subject for architects at RWTH Aachen with only two weekly hours for one year Descriptive Geometry is having an extremely small teaching load at disposal. In addition this class is annually attended by about 250 students. To be able to supply a qualified contribution to the course of studies on architecture which is also approved by the students, the faculty of mathematics has to develop a very sophisticated and broad teaching concept under disadvantageous conditions. In order to prevent DG from being removed from the syllabus great acceptance by both teaching and learning architects has to be called for. A smooth organisation of the lessons is a must if we want DG to appear useful to the students and keep them motivated during class. It is also essential, that the students` energies go completely into their dealing with the subject, so that counterproductive frustrations do not arise. The success of this concept is indicated by course evaluations: e.g. in July 2000 the DG course achieved by far the best score with a mark of 1.4 (on a scale from 1 (best) to 6 (worst)).
^{1} email: c@dgac.de
2 Aims of DG within the architectural education
The main task of DG as subject in the architectural education is in my opinion the training of threedimensional imagination. To achieve this aim a great deal of motivation is necessary, which can be maintained and even increased, if e.g. contents [1] and exercises with an obvious reference on architecture are employed. An additional aim is dealing with drawings, which is best learned by drawing by hand. Attending the course of exercises developed by us the necessary competence is built up during a few weeks [2]. An additional emphasis is on the conscious dealing with different ways of projection, for architects should always be able to choose the most adequate and reasonable way of projection suiting the respective purpose; within this the different options on parameters are very important [3]. The application of CAD at the beginning of the study is a useful addition, but from the didactic point of view it cannot replace the classic DG.
3 The concept of the lectures for architects
Unfortunately lectures are among the first events avoided by the students, if the temporal burden becomes uncomfortable. Therefore the lecture has to be designed in a form, in which the prospective architects clearly can recognize their profit. Also the concentration during 90 minutes should not be overestimated. To maintain the students‘ motivation architectural examples illustrating the key point of the respective formulation are always demonstrated by means of a second overheadprojector.
The basic tasks introduced in the lecture are deepened during the presentation and explanation of the exercises. Within the exercise sessions the students have to apply the basic tasks to more complex problems. The basis for this is an exercise booklet which contains detailed descriptions of the tasks and several special worksheets for each exercise. The booklet is distributed as a file, so that single pages needed for specific lectures can easily be extracted.

Figure 1: Worksheet before and after completion by the student.
Due to the concept of drawing on the prepared worksheets, the students are required to attend the lectures, because the worksheets cannot be dealt with without the explanations given during the lecture. In this way students have a reference book about the treatment of geometrical tasks to their disposal they have worked out by themselves. By drawing during the lecture the students‘ attention is kept at a high level; even if some students only repeat the constructions in a mechanical way, they are still exposed intensely to the contents.
By using colours wherever possible repeated elements in the construction are emphasised. Students are enabled to identify them easily in complex drawings and to find the explanations on the corresponding geometrical background within the booklet.
Apart from training threedimensional imagination I consider it essential to work out clearly the specific transfer into a stepbystep construction and  if reasonable œ even to offer —recipes“. The experiences of professional architects show that the combination of the recipes and the geometrical knowledge acquired during their studies provide them a lasting ability to tackle various geometrical tasks efficiently.
4 The concept of the exercise sessions for architects
The theoretical basis of the exercises is provided in the lectures. The lectures‘ structure is as far as possible fitted with the systematic sequence of the exercises. For easier integration of DG into the architectural course of studies examples of relevant architecture reduced to their essentials are employed as basis of the exercises wherever possible.
An 90 minute introduction to the exercises is given in the lecture room during which the students participate by drawing on the prepared worksheets using different colours. The basis for this is an exercise booklet with about 90 pages which includes detailed descriptions of the tasks and several special worksheets for each exercise. In case there is a loss of clarity due to too many constructions on a single worksheet, the next worksheet shows the steps already completed and the construction can easily be continued. Furthermore the students who got lost in the construction can join again at a later stage without any problems.
The basic tasks introduced in the lecture are deepened during the presentation and explanation of the exercises. Within these exercises the students have to apply the basic tasks to more complex problems. This happens with obvious reference to the lecture without repeating its contents. As a side effect the students‘ motivation to attend the lectures is increased.
In total, there are nine exercises to be successfully completed by the students. Among these are 5 exercises to be done in homework and 4 exercises to be done during class.
The homework exercises have to be completed by each student on his own, developing and working out his own sketch within the set guidelines. During the homework each student will experience individual problems, so that the solution of the exercises is beyond the mechanical application of constructional recipes. To prepare the students as versatile as possible the presented examples are in general more challenging than the ones given to the students later on. The homework exercises have to be completed within 10 days after their presentation using pencils on several A2 sized sheets of tracing paper.
The exercises to be done in class are prepared on A4 sized sheets and have to be completed within 90 minutes; handing out slightly different versions to each student ensures that each one handles his exercise on his own. Only a sound command of the tasks allows the student to deal with the relatively challenging exercises within the limited period of time.
4.1 The structure of the exercises‘ sequence
The sequence of exercises is designed in reference to increasing complexity of the geometrical phenomenon.
The lectures begin with the easily understandable application of cuboids in axonometries (oblique parallel projection); these exercises are followed by abstract geometrical basic task of Monge projection (orthogonal parallel projection) and the more complex connections of the central projections especially the perspective application of cylinders; the course ends up with the construction of shadows (parallel projection) in perspective images (central projection).
The course starts with the birdseyeview, since this requires a rather simple construction and the underlying geometrical principle is easy to understand. Experience shows that the main difficulty of the first exercise, apart from the ability to think in three dimensions, is the skillful handling of the drawing tools. The second exercise is the reproduction of a complex building, where the required dotting of hidden lines presents a challenge in threedimensional imagination.
The results of both exercises represent a sound basis for the development of the topic “Monge projection—: On the one hand, this topic is especially difficult for students to understand because of its high level of abstraction, on the other hand, it is the main basis of the architect‘s daily work. Therefore, the treatment of “Monge projection— should take the key position within the DG lectures. The students benefit immensely from this solid knowledge when dealing with cones, cylinders, perspective and construction of shadows later on.
4.2 The homework exercise on the subject of Polyhedra within the Monge projection
Having completed the first two exercises the students are able to draw exactly and therefore are prepared to deal with the more demanding constructions of Monge projection.
Figure 24: —Tree houses“ in Helmond by the architect Piet Blom.

The third exercise contents the development of a so called —tree house“ which consists of a cuboid standing on one of its corners and which is blended with an threesided prism. The exercise‘s difficulty lies within the threedimensional imagination, thinking and constructing.
At first one has to determine a cuboid, whose edges have integer lengths, in a general location in both ground plan and elevation; therefore one needs to master the phenomenon of lines parallel to the elevation plane, lines parallel to the ground plane, normal of a plane and the determination of the true lengths.

Figure 5: Choosing one of the cuboid‘s corners (before and after completion).

Figure 6: Completing the cuboid (before and after completion).
The plane with its normal is to be completed as a cuboid using the parallelism of the cuboids‘ edges. The clarifying of visibility is especially difficult in the elevation, because in this case two facets of the cuboid appear each as one line only.