Intermediate Level

Architecture

The application evidences the generation of a surface through a straight line that moves parallel to a plane, supported simultaneously on a straight segment and on a circumference.

A surface with these characteristics is a conoid. It is a ruled surface, since for each of its points passes a straight line lying in it.

More generally, a conoid is a surface generated by a straight line, the generatrix, that moves parallel to a plane, the directrix plane, while simultaneously intersecting a straight line segment, the axis of the conoid, and a line, the diretrix, which can be open or closed.

When the axis of the conoid is perpendicular to the directrix plane, the conoid is said to be straight.

The figure shows the generation of a straight conoid whose directrix is a circumference.

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The use of ruled surfaces in the roof of buildings is associated with renowned architects such as Antoni Gaudí, Felix Candela and Santiago Calatrava.

 The conoids with a sinusoidal directrix were used by Gaudí on the roofs of the Sagrada Familia Schools in Barcelona. Also the side walls of this building use the same shape, which has contributed to a considerable reduction of construction costs.
 Calatrava also used these surfaces in a mobile sculpture, "Wave", at the Meadows Museum in Dallas (USA)
 and in the roof of Bodegas Ysios in Laguardia (Spain).

These surfaces, with equation $$z = a x sin({y \over{b}})$$, are called Gaudí's surfaces, since he was the first architect to use them in his works.

References

• Catálogo da exposição Formas & Fórmulas (MUHNAC e FCUL)
• J. Dolezal, The Story of a Right Wavelet Conoid, WDS’11 Proceedings of Contributed Papers, Part I, 72–77, 2011
• L. Mateus, Linhas e Superfícies – Texto de apoio à disciplina Modelação Geométrica, FAUTL – Ano lectivo 2012/2013
• C. Alsina, http://textos.pucp.edu.pe/pdf/413.pdf
• Centro Virtual Cervantes, Naturaleza y geometria, http://cvc.cervantes.es/actcult/gaudi/geometria.htm