Risorse bibliografiche
Risorsa bibliografica obbligatoria
Risorsa bibliografica facoltativa
Scheda Riassuntiva
Anno Accademico 2014/2015
Scuola Scuola di Architettura e Società
Docente Cocchiarella Luigi
Cfu 4.00 Tipo insegnamento Monodisciplinare

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Nome Sezione Insegnamento

Programma dettagliato e risultati di apprendimento attesi

Geometrical Complements of Graphic Representation (4 ECTS)


Teacher: Luigi Cocchiarella





Geometrical Complements of Graphic Representation is an advanced optional course in the field of architectonic representation, devoted to interested 10th semester students who are concluding their Laurea Magistrale (Master of Science) curricula. Inspired by the Complements of projective and descriptive geometry courses traditionally included in the polytechnic programs, but at the same time embodying new contents according to an updated approach, the course aims to complete the disciplinary learning trail started in the first semester with Geometrical foundations of graphic representation courses (Laurea) and Architectural Representation studios (Bachelor of Science).

To be consistent with the last semester learning requirements, the main focus will concern a detailed investigation on the relationships among physical space, geometrical models and graphic image in the architectural design processes.

Starting from a specific analysis of projective and non projective methods, techniques and instruments, some significant classic descriptive problems (spatial visualizations by means of the image) and constructive problems (spatial transformations by means of the image) will be revisited and discussed in deepth, especially dwelling on some specialized subjects and applications generally not included in the basic representation curricula.

Looking at the long evolution of geometrical representation, and focusing on some significant steps in the history of spatial concepts, geometrical theories and graphic means from the ancient optical theories to the computer graphics era, the course would also achieve a twofold purpose: a disciplinary (or internal) one, trying to point out the relationships between analogical and digital representation; an interdisciplinary (o external) one, trying to point out the strong connections of both these representative approaches with design thinking, design processes, and the built architectural and environmental spaces.




The course will consist of lessons (19 hours) and applications (23 hours), and the presence in class is requested.

During the lessons, both theoretical contents and practical keys to solve graphic problems will be provided. Digital slides and drawn schemes on traditional blackboard will be used to explain, analyze and discuss in detail geometrical properties and graphic procedures, and to present significant applications. Students have to record and critically rework the contents developed in class. Although lessons will be not entirely replaced on website, some significant materials will be available on the “Corsi on line” web page and, of course, calendar, program, additional advice. Teachers from other universities could be invited to give additional lectures.

About the applications, although the course is mainly focused on the drawing and modeling processes, the aim of the work will be also preliminarily summarized by the students in a written abstract about 500 words long with a maximum of 2 images, while a more detailed comment will be presented in a 5.000 words text with a maximum of 10 images, according to a given layout. In order to foster discussion and support materials collection, groups of two students will be admitted, but the final work and the exam will be individual. Graphic and written work will be discussed and prepared in class, and completed at home by students. Drawings and models will be realized by using traditional and/or digital tools, according to the specific needs and to the students preferences and abilities. Each paper will be presented in class by the author in a 15 minutes public lecture (speaking time), and discussed by classmates in the following 5 minutes (question time). Lectures will be scheduled basing on the number of students attending the course.

Some tutorial extra time, for students who need additional discussions and revisions, will be provided every week, according to students and teacher availabilities.


Proposed Syllabus


A)  Lessons will deal with the following topics: 


Historical References

The course will preliminarly refer to the history of geometric representation and architecture, as a general background where theories, methods, tools, meanings and applications can be found and related to their cultural context. It will also point out the close relationships between science and art in the interdisciplinary field of architectural representation. Students are thoroughly recommended to consider the following points in the history of geometric drawing. For a first guide, see the Bibliography.


  • Geometrical measuring and geometrical imaging: science, art and architectural configuration from Euclid to Brunelleschi
  • Pictorial approach and mathematical approach: science, art and architectural configuration from  Piero Della Francesca to Poncelet
  • Analogical approach and digital approach: science, art and architectural configuration from Gauss to Boole  


Theoretical Literature

This section aims to deal with the substance of geometrical representation, particularly focusing both on the “veridicality” and on the “illusoriness” of the image. To this aim, projective transformations will be examined in deep, especially insisting on some interesting connections between the notions of space, figure and measure. Anyway, according to a modern approach, this branch will be also related to other geometries, first of all to Optics and Euclidean, Topological, Fractal geometries, with some additional mentions to other geometrical fields. Then, some remarks on the educational value of Geometry and Graphics in Architecture will be given, and discussed. Theoretical lessons will mainly focus on the following topics. For a first guide, see the Bibliography.


          Main topics

  • The modern approach, since the Erlangen Program: system of Spaces and system of Geometries
  • Projective geometry: entities, transformations, invariants
  • The key transformations: perspectivity, projectivity, homology
  • Projective and Descriptive Geometry: geometrical foundations and graphic methods
  • Geometry and Graphics in the computer body: the notions of space, image, model and transformation, between tradition and innovation


          Special topics

  • From image to measure: inverse perspective and photogrammetric reconstruction
  • Between reality and illusion: solid perspective and theatrical configurations
  • Ambiguous images, spaces, objects: anamorphoses
  • Rewriting treatises and manuals: traditional and digital philosophy in approaching classical theories and methods


Geometry and Graphics Education

Although the advent of digital graphics, maybe the greatest revolution in the field of graphic representation since Renaissance, has enormously increased our chances in managing and imagining geometric and graphic structures, projective and descriptive geometry courses have paradoxically diminished, and sometimes disappeared, from some architectural curricula. Then, in order to avoid the risk of losing a millennial knowledge, it may be necessary to rethink and update some teaching techniques and to criticize in deep the purposes of the matter. Here some topics proposed to the students. For a first guide, see Bibliography.


  • Reloading geometry and graphics: educational goals and tasks in the computer era
  • Re-presenting architectural representation: from descriptive geometry to geometrical description in the teaching approach to geometry and graphics curricula


B)  Applications will deal with the following themes:


 Applied Geometry and Graphics

The applications will concern cases study connected with theoretical contents. Each student has to decide which one of the mentioned topics best fits his interest, trying to approach it critically and to develop it with intellectual autonomy. The choice will be preliminarily discussed with and approved by the teacher, then students will be encouraged to collect appropriate materials about the selected topic. Applications will mainly revolve around the graphic analyses of some theoretical configurations included in treatises and manuals, or around graphic reconstructions of significant architectural, theatrical, sculptural or pictorial spaces, photographic or cinematographic sceneries, integrated by scaled models if necessary. Here a merely indicative list of what could be achieved is given. The specific bibliography will be provided by students under the guidance of the teacher.


  • Graphic reconstruction/modeling of classical configurations enclosed in treatises and manuals (i.e., spatial location and graphic representation of the geometrical elements described in the Euclid’s Optics statements, or in Girard Desargue’s theorem,…)
  • Graphic reconstruction/modeling of architectural projects enclosed in treatises and manuals (i.e., spatial investigation and new visualizations of some non realized projects, from Andrea Palladio to Peter Eisenman…)
  • Graphic reconstruction/modeling of stereotomic designed and built systems (i.e., comparison between drawings and building in the Philibert Delorme’s Trompe d’Anet,…)
  • Graphic reconstruction/modeling of architectural and theatrical sceneries from paintings, sculptures, pictures, movie frames (i.e., spatial investigation and new visualizations of the architecture represented in the paintings by Raffaello Sanzio, or in the reliefs by Donatello, or in the pictures by Cartier Bresson, or in the movies by Pier Paolo Pasolini ,…)
  • Graphic analysis/modeling of illusive distorted spaces (i.e., spatial investigation of Borromini solid perspective in the Palazzo Spada Gallery, or of Adalbert Ames anamorphic visual illusion in the Distorted Room…)
  • Graphic analysis/modeling of illusive distorted images (i.e., spatial investigation of Jean François Niceron or Andrea Pozzo anamorphoses…)
  • Graphic analysis/modeling of lights and shadows (i.e., applications on lights, shade and shadows, including reflection and catopric images, emphasizing these phenomena as the basic physical models for the birth and the development of projective geometry, or rather, applications on the use of light and shadows in solving both projecting and metrical problems, or even, on the use of lights and shadows in theatrical illusionism,…)
  • Graphic analysis/modeling related to cartographic projections (i.e., conical, cylindrical and mixed projections in representing earth and heaven, and their influence both in fixing the notions of geometrical infinity, and on the accepted “images of the world” along the History…)
  • Analogical representation vs digital representation: meaning and construction methods (i.e., comparison between geometric and semantic dimension by juxtaposing and analyzing some significant geometric analogical drawings/projects and some significant geometric digital models/projects,…)
  • Theories, methods, tools: geometrical devices (i.e., analysis of the basic connections between projective geometry and some important visual and optical measuring instruments, from the ancient baculus to the modern laser scanner,…)
  • Teaching, learning, performing: visual literacy between tradition and innovation (i.e., comparing traditional and digital philosophy in approaching classical theories and methods, prepare a step by step lesson on classical topics, for example perspective projection or other, by taking advantage of digital models and animations in showing the spatial location of the elements and their projection on the picture plane to form the final image, …)   



Expected Results  


As an advanced optional course located at the end of the degree curriculum, Geometrical complements of graphic representation aims both to increase technical skills and to stimulate research aptitudes in the field of Geometry and Graphics for Architecture, as nowadays strongly required both by professional and by scientific communities. In order to achieve this purposes, without denying or diminishing the primacy of the advanced drawing and modeling activities and the related results, students will be also requested to develop individual papers, including original drawings, models and written texts, which will be presented and discussed in class as short lectures, evaluated as exam materials, and possibly sent to conferences or journals reviewers according to the teacher's advice. In order to promote a wider knowledge sharing, international relationships will be encouraged, both by means of free contacts with other universities students and/or teachers, and by using the students and staff mobility opportunities connected with the Erasmus projects. Last, prospective interested students will be supported in developing disciplinary oriented final degree thesis.


Bibliography (basics)


A)  Related to lessons:


     Historical References

Evans, R., The Projective Cast. Architecture and Its Three Geometries, The MIT Press, Cambridge/London 1995

Kemp, M., The Science of Art. Optical themes in western art from Brunelleschi to Seurat, Yale University Press, New Haven and London 1990

McCullough M., Mitchell W.J., and Purcell P., The electronic design studio : architectural knowledge and media in the computer era, The MIT press, Cambridge (Mass.) 1990


     Theoretical Literacy

     (Main Topics)

Coxeter, H. S. M., Projective Geometry, Springer, New York .etc., 1987

Pirenne M.H.L., Optics painting & Photography, The University Press, Cambridge 1970 (or Pirenne M.H.L., Percezione visiva: ottica, pittura e fotografia, Muzzio, Padova 1991)

Hilbert D., Cohn-Vossen S., Geometry and Imagination, AMS Chelsea Publishing, Providence R.I. 1999 (or Hilbert D., Cohn-Vossen S., Geometria intuitiva. Complemento: I primi fondamenti della topologia di Pavel Sergeevic Aleksandrov, Boringhieri, Torino 1972)

Pottmann H., Asperl A., Hofer M., Kilian A., Architectural Geometry, Bentley Institute Press, Exton, PA, 2007 

     (Special Topics)

Blunt A., Philibert de l'Orme,Zwemmer, London 1958 (or Potie P., Philibert de l'Orme : figures de la pensee constructive, Parentheses, Marseille 1996)

Baltrusaitis J. (translated by Strachan W. J.), Anamorphic art, Chadwyck-Healey, Cambridge 1977 (or Baltrusaitis J., Anamorphoses ou perspective curieuse, O. Perrin, Paris 1955, or Baltrusaitis J., Anamorfosi, o Thaumaturgus opticus, Adelphi, Milano 1990)

Langford M., Basic Photography, Focal Press, Oxford and..., 2000


     Geometry and Graphics Education

Hemmerling M., Tiggemann A., Digital Design Manual, DOM Publisher, Berlin 2011

Mammana C., Villani V. (editors), Perspective on the teaching of geometry for the 21st century: an ICMI study, Kluwer, Dordrecht 1998) 

Journal for Geometry and Graphics, Heldermann Verlag, Lemgo (http://www.heldermann.de/JGG/iggcover.htm


B)  Related to applications:

     Applied Geometry and Graphics

As written above in the program, the bibliography for the individual applications will be provided by students under the guidance of the teacher. Anyway, as a first reference here the links to the Politecnico di Milano OPAC Library and to the National SBN Catalogue websites:



Note Sulla Modalità di valutazione

The exam is individual, and it will consist of a final discussion focusing on the work developed by the student and on the theoretical contents of the course program. The grade will be given basing on the quality of paper, drawings and models, lecture, final discussion. If necessary, preliminary intermediate evaluations will be scheduled during the semester. A digital copy of the work, including rough and fair copy of notes, lessons, drawings, models, maquettes, writings, abstracts, papers, presentation, will be put on CD-ROM or DVD-ROM and delivered to the teacher one week before the exam.


The evaluation aims to check the following educational goals:


_oral text: knowledge and understanding; making judgements; communication skills

_class work evalutaion: applying knowledge and understanding; making judgements; learning skills; communication skills


Since sudents will be required to work in groups on specific tasks (i.e. measurements, model making, photography, workflow planning) the aptitude and the ability to work in team will be tested as well.


Risorsa bibliografica obbligatoriaKemp, M., The Science of Art. Optical Themes in Wetern Art from Brunelleschi to Seurat, Editore: Yale University Press, New Haven and London, Anno edizione: 1990
Risorsa bibliografica obbligatoriaEvans, R., The projective cast. Architecture and its three geometries, Editore: The MIT Press, Cambridge-London, Anno edizione: 1995
Risorsa bibliografica obbligatoriaPottmann H., Asperl A., Hofer M., Kilian A., Architectural Geometry, Editore: Bentley Institute Press, Exton, PA, Anno edizione: 2007
Risorsa bibliografica obbligatoriaHemmerling M., Tiggemann A., Digital Design Manual, Editore: DOM Publisher, Berlin, Anno edizione: 2011
Risorsa bibliografica obbligatoriaJournal for Geometry and Graphics, Editore: Heldermann Verlag, Lemgo http://www.heldermann.de/JGG/jggcover.htm

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Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua Inglese
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Disponibilità di libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese
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