Nexus Network Journal

, Volume 13, Issue 3, pp 701–714

Developable Surfaces: Their History and Application

Research

Abstract

Developable surfaces form a very small subset of all possible surfaces and were for centuries studied only in passing, but the discovery of differential calculus in the seventeenth century meant that their properties could be studied in greater depth. Here we show that the generating principles of developable surfaces were also at the core of their study by Monge. In a historical context, from the beginning of the study of developable surfaces, to the contributions Monge made to the field, it can be seen that the nature of developable surfaces is closely related to the spatial intuition and treatment of space as defined by Monge through his descriptive geometry, which played a major role in developing an international language of geometrical communication for architecture and engineering. The use of developable surfaces in the architecture of Frank Gehry is mentioned, in particular in relation to his fascination with ‘movement’ and its role in architectural design.

Keywords

developable surfaces Gaspard Monge descriptive geometry ruled surfaces Frank Gehry Leonhard Euler 

References

  1. Aristotle. 2004. De Anima. H. Lawson Tancred, ed. London: Penguin.Google Scholar
  2. Amar Ben M., Pomeau Y. (1997) Crumpled paper. Proceedings of the Royal Society 453: 719–755Google Scholar
  3. Cajori F. (1929) Generalisations in Geometry as Seen in the History of Developable Surfaces. The American Mathematical Monthly 36(8): 431–437CrossRefMATHMathSciNetGoogle Scholar
  4. Cayley Arthur. (1850) On the developable surfaces which arise from two surfaces of the second order. Cambridge and Dublin Mathematical Journal V: 46–57Google Scholar
  5. Cayley, Arthur. 1862 On Certain Developable Surfaces. Proceedings of the Royal Society of London 12: 279–280.Google Scholar
  6. Clairaut, Alexis–Claude. 1731. Recherches sur les courbes à double courbure. Paris.Google Scholar
  7. Collidge Julian Lowell. (1940) A history of geometrical methods. Oxford University Press, OxfordGoogle Scholar
  8. Darboux Gaston. (1904) The Development of Geometrical Methods. The Mathematical Gazette. 3(48): 100–106CrossRefMATHGoogle Scholar
  9. Delcourt Jean. (2011) Analyse et géométrie, histoire des courbes gauches De Clairaut à Darboux. Archive of the History of Exact Sciences 65: 229–293CrossRefMATHMathSciNetGoogle Scholar
  10. Domingues João Caramalho. (2008) Lacroix and the Calculus. Birkhaüser., BerlinMATHGoogle Scholar
  11. Dupin Charles. (1819) Essai historique sur les services et les travaux scientifiques de Gaspard Monge. Bachelier, ParisGoogle Scholar
  12. Euler, Leonhard. 1772. De solidis quorum superficiem in planum explicare licet (E419). Novi commentarii academiae scientiarvm imperiatis Petropolitanae XVI (1771): 3–34. Rpt. Leonhardi Euleri Opera Omnia : Series 1, vol. 28, pp. 161–186.Google Scholar
  13. Frezier, Amédée–François. 1737–1739. La théorie et la pratique de la coupe des pierres et des bois ... ou traité de stéréotomie a l’usage de l’architecture. Strasbourg–Paris: Jean Daniel Doulsseker–L. H. Guerin.Google Scholar
  14. Gehry Frank O. (1996) Current and Recently Completed Work. Bulletin of the American Academy of Arts and Sciences 49(5): 36–55CrossRefGoogle Scholar
  15. Glaeser G., Gruber F. (2007) Developable surfaces in contemporary architecture. Journal of Mathematics and the Arts 1(1): 59–71CrossRefMATHMathSciNetGoogle Scholar
  16. Hall A H. (1980) Philosophers at War: The Quarrel between Newton and Leibniz. Cambridge University Press, New YorkCrossRefMATHGoogle Scholar
  17. Hawney, W. 1717.The Complete Measurer . London.Google Scholar
  18. Kline Morris. (1990) Mathematical Thought from Ancient to Modern Times, 3 vols. Oxford University Press, OxfordGoogle Scholar
  19. Lawrence, Snežana. 2002. Geometry of Architecture and Freemasonry in 19th century England. Ph.D Thesis, Open University UK.Google Scholar
  20. Lawrence, Snežana. 2010. Alternative ways of teaching space: a French geometrical technique in nineteenthcentury Britain. Pp. 215–224 in Echanges entre savants français et britanniques depuis le XVIIe siècle, Robert Fox and Bernard Joly, eds. Oxford: College Publications.Google Scholar
  21. Lebesgue H. (1899) Sur quelques surfaces non réglées applicables sur le plan. Comptes Rendus de l’Académie des Sciences(CXXVIII): 1502–1505Google Scholar
  22. Liu Y., Pottman H., Wallner J., Yang Y-L., Wang W. W. (2006) Geometric Modeling with Conical Meshes and Developable Surfaces. ACM Transactions on Graphics 25(3): 681–689CrossRefGoogle Scholar
  23. Monge, Gaspard. 1769. Sur les développées des courbes à double courbure et leurs inflexions. Journal encyclopédique (June 1769): 284-287.Google Scholar
  24. Monge, Gaspard. (1780) Mémoire sur les propriétés de plusieurs genres de surfaces courbes, particulièrtement sur celles des surfaces d′eveloppables, avec une application à la theorie des ombres et des pénombres. Mémoires de divers sçavans(9): 593–624 (written 1775)Google Scholar
  25. Monge, Gaspard. (1785) Mémoire sur les développées, les rayons de courbure, et les différents genres d’inflexions des courbes à double courbure. Mémoires de divers sçavans(10): 511–520 (written 1771)Google Scholar
  26. Monge, Gaspard. 1795a. Feuilles d’analyse appliquée à la géométrie. Paris. Second ed. with the title Applications de l’analyse à la géométrie. Paris : Baudouin, 1811.Google Scholar
  27. Monge, Gaspard. 1795b. Géométrie descriptive. First ed. in Les Séances des écoles normales recueillies par des stégraphes et revues par des professeurs, Paris. Re–edition pp. 267–459 in L’Ecole normale de l’an III, Leçons de mathétiques, Laplace, Lagrange, Monge, Jean Dhombres, ed. Paris: Dunod, 1992.Google Scholar
  28. Monge, Gaspard. 1850. Application de l’analyse a la géométrie, 5th ed. Paris.Google Scholar
  29. Picard, E. 1922.Traite d’analyse, 3rd ed. Paris: Gauthier–Villars et fils.Google Scholar
  30. Reich, Karin. 2007. Euler’s Contribution to Differential Geometry and its Reception. Pp. 479– 502 in Leonhard Euler: Life, Work and Legacy, Robert E. Bradley and C. Edward Sandifer, eds. Amsterdam: Elsevier.Google Scholar
  31. Sakarovitch, Joël. 1989. Theorisation d’une pratique, pratique d’une theorie: Des traits de coupe des pierres à la géométrie descriptive. Ph.D. thesis, Ecole d’Architecture de Paris La Villette.Google Scholar
  32. Sakarovitch, Joël. 1995. The Teaching of Stereotomy in Engineering Schools in France in the XVIIIth and XIXth centuries: An Application of Geometry, an ‘Applied Geometry’, or a Construction Technique? Pp. 205–218 in Between Mechanics and Architecture, Patricia Radelet–de Grave and Edoardo Benvenuto, eds. Basel: Birkhäuser.Google Scholar
  33. Sakarovitch, Joël. (1997) Epures D’architecture. Birkhäuser, BaselGoogle Scholar
  34. Sakarovitch, Joël. 2009. Gaspard Monge Founder of ‘Constructive Geometry’. Pp. 1293–1300 in The Proceedings of the Third International Congress on Construction History, Cottbus. http://www.formpig.com/pdf/formpig_gasparmongefounderofconstructivegeometry_sakarovitch.pdf (last accessed 30 June 2011).
  35. Struik Dirk J. (1933) of a History of Differential Geometry I. Isis 19(1): 92–120CrossRefMATHGoogle Scholar
  36. Struik Dirk J. (1933) Outline of a History of Differential Geometry II. Isis 20(1): 161–191CrossRefGoogle Scholar
  37. Taton, Rene. 1951. LŒuvre scientifique de Monge. Paris: Presses Universitaires de France.Google Scholar
  38. Taton, Rene. (1966) La première note mathématique de Gaspard Monge. Revue d’histoire des sciences et de leurs applications 19(2): 143–149CrossRefMathSciNetGoogle Scholar
  39. Velimirovic, L. and M. Cvetkovic. 2008. Developable surfaces and their applications. Pp. 394– 403 in Mongeometrija 2008, Proceedings of the 24th International Scientific Conference on Descriptive Geometry, Serbia, September 25–27, 2008.Google Scholar

Copyright information

© Kim Williams Books, Turin 2011

Authors and Affiliations

  1. 1.School of EducationBath Spa UniversityBathUnited Kingdom

Personalised recommendations