Theories and Methods for Development of Developable Ruled Surfaces and Approximate Flattening of Non-developable Surfaces


  • Mara Capone DiArc Department of Architecture, University of Naples Federico II



Developable surfaces, Ruled surfaces, Double curvature surface, Parametric modeling


Geometric genesis of surfaces and knowledge of their properties are basis for solving many problems, both constructive and measurement. A developable surface can be manufactured starting from a flat “strip”, using a flexible and non-deformable material. This is a very important feature of the surface. Geometry studies the properties that don’t change and, therefore, the shape of the “strip” to obtain a certain configuration, after a series of rigid movements. The paper addresses different methods to define the development of developable surfaces and non-developable surface “flattening”, or approximate development. The aim is to study the relationships between methods, illustrated in some treatises, and the applications that can derive from the use of parametric tools. We are going to create an overview of different approaches, that have defined the bases of differential calculus for the study of ruled surfaces properties, and of different methodologies that allow to determine their development. Starting from the first definition of surface by Aristotle in De Anima (384-322 BC) and the ambiguous definitions by Amédée François Frézier (1682-1773), we analyzed the studies of Leonhard Euler (1707-1783) and Monge’s main work on developable surfaces (1795).


Euler, L. (1772). De solidis quorum superficiem in planum explicare licet. In Novi commentarii academiae scientiarum imperiatis Petropolitanae. XVI. pp. 3-34.

Fallavollita, F. (2008). Le superfici rigate e le superfici sviluppabili. Una rilettura attraverso il laboratorio virtuale. Tesi di dottorato di ricerca in Scienze della Rappresentazione e del Rilievo. Tutor: L. De Carlo, R. Migliari. Sapienza Università di Roma, Dipartimento di Rilievo, Analisi, Disegno dell’Ambiente e dell’Architettura.

Gauss, K.F. (1902). General Investigations of Curved Surfaces of 1827 and 1825, translated with notes and a bibliography by James Caddal Moreheas and Adam Miller Hiltebeitel. Princeton: The Princeton University Library.

Hachette, J.N.P. (1828). Traité de Géométrie Descriptive. Paris: Corby.

Lanzara, E. (2015). Paneling Complex Surfaces. Razionalizzazione di superfici complesse per l’industrializzazione. Tesi di dottorato di ricerca in Tecnologia dell’Architettura e rilievo e rappresentazione dell’architettura e dell’ambiente.

Tutor: M. Capone, cotutor: A. Picerno Ceraso. Università degli Studi di Napoli Federico II.

Leroy, C.F.A. (1846). Trattato di Geometria Descrittiva con una collezione di disegni composta da 60 tavole. Traduzione dal francese con note di Salvatore d’Ayala e Paolo Tucci, Napoli: Dalla Reale Tipografia della Guerra. (Prima ed. Traité de géométrie descriptive, Parigi 1842).

Liu, Y. et al. (2006). Geometric Modeling with Conical Meshes and Developable Surfaces. In ACM Transactions on Graphics, 25, 3, pp. 681-689.

Migliari, R. (2009). Geometria descrittiva. Roma: CittàStudi.

Monge, G. (1780). Mémoire sur les propriétés de plusieurs genres de surfaces courbes et particulièrement sur celles des surfaces développables avec une application à la théorie générale des ombres et des pénombres. En Mém. div. Sav., t. 9, pp. 382-440.

Monge, G. (1785). Mémoire sur les développées, les rayons de courbure, et les différens genres d’inflexions des courbes a double courbure. Paris: De l’Imprimerie Royale.

Monge, G. (1798). Géométrie Descriptive. Paris: Baudouin.

Monge, G. (1801). Feuilles d’analyse appliquée à la géométrie, appliquée à la géométrie à l’usage de l’Ecole Polytechnique. Paris: Baudouin.

Quintial, F.G., Barrallo, J., Artiz Elkarte, A. (2015). Freeform surfaces adaptation using developable strips and planar quadrilateral facets. In Journal of Facade Design and Engineering, 3, pp. 59-70.

Sereni, C. (1826). Trattato di Geometria Descrittiva. Roma: Nella Stamperia di Filippo e Nicola De Romanis.

Snežana, L. (2011). Developable Surfaces: Their History and Application, In Nexus Network Journal, 13, pp. 701-714.

Tang, C., Bo, P., Wallner, J., Pottmann, H. (2016). Interactive design of developable surfaces. In ACM Transactions on Graphics, 35, 2, Article 12: <> (consultato il 23 ottobre 2018).



How to Cite

M. Capone, “Theories and Methods for Development of Developable Ruled Surfaces and Approximate Flattening of Non-developable Surfaces”, diségno, vol. 1, no. 3, pp. 53–68, Dec. 2018.