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New Properties About the Intersection of Rotational Quadratic Surfaces and Their Applications in Architecture

  • Andrés Martín-Pastor
  • Roberto Narvaez-Rodriguez
Geometer's Angle
  • 28 Downloads

Abstract

A graphic conjecture is presented based on a singular property stated by Archimedes [287–212 B.C.] in his work On Conoids and Spheroids. This ancient text constitutes the starting argument for graphic research that has revealed an unknown property regarding the intersection of rotational quadratic surfaces which they share one of their foci. This article shows the heuristic-geometric reasoning carried out stemming from Archimedes’ text transcriptions and a conjecture that can be deduced when the initial property is generalised for the rest of the quadratic surfaces. Moreover, an explanation is offered for the possibilities of this property to be used for the discretisation of architectural surfaces through the use of parametric design and digital fabrication.

The property discovered in this research is summarised as follows: “If two rotational quadratic surfaces share the position of one of their foci at the same point, then the intersection curves between the two surfaces are always planar” (The oblate ellipsoid and one-sheeted hyperboloid are excluded.).

This new property, which currently remains only a conjecture, has been formulated from purely graphic thinking. However, its validity has been fully tested through a heuristic method which involves checking the planarity on all possible combinations of quadric intersections in a necessary and sufficient number of cases. For this purpose, the power of CAD tools has been used as a true geometric research laboratory where the validity of the theoretical approaches is subject to trial and error.

Keywords

Quadric theorems Descriptive geometry Treatises Architectural geometry Confocal surfaces Rotational quadratic surfaces Graphic thinking 

Notes

Acknowledgements

We would like to thank prof. Dr. José María Gentil Baldrich for introducing us to this line of research. We would like to thank prof. Daniel Hernández Macías for his participation in this research, without which the results achieved would not have been possible. We also want to emphasize his contribution as a co-author of the conjecture. Our thanks also go to prof. Dr. María-José Chávez for her corrections and mathematical advice. We would like to thank the sponsors of the Archimedean Pavilion: University of Seville, through ETSIE, Department of Graphic Engineering and Vicerrectorado de Estudiantes, and Dow Building Solutions [a division of The Dow Chemical Company]. Thanks also to Shuming Wang Zhu for his collaboration and to the group of students, lecturers, FabLab Sevilla, and Ehcofab for their support and collaboration throughout the development, fabrication and assembly of the Archimedean Pavilion (http://archimedeanpavilion.blogspot.com.es).

References

  1. Archimedes-Gechauff, Thomas (Alias Venatorius). 1544. Archimedis Syracusani… opera, quae quidem extant, omnia, multis iam seculis desiderata, atque a quam paucissimis hactenus visa, nuncque primum & Graece & Latine in lucem edita, Basel: Herwagen, Johannes, d.Ä.  https://doi.org/10.3931/e-rara-8997
  2. Archimedes-Commandino. 1558. Archimedis opera non nulla a Federico Commandino. Venice: P. Manutium. Aldus. [Two parts, the second: Comenterii in opera non nulla Archimedis]. http://gallica.bnf.fr/ark:/12148/bpt6k58156t
  3. Archimedes-Maurolico, Francesco. 1685. Monumenta omnia mathematica, quae extant, quorumque catalogum inversa pagina demonstrat: opus praeclarissimum, non prius typis commissum, a matheseos vero studiosis enixe desideratum, tandemque e fulgine temporum accurate excussum/ex traditione Francisci Maurolici: Panormi [Palermo]: Apud Cyllenium Hesperium, 1685. [De Conoidibus et Sphaeroibidus in two books: 1\(^{\underline {\rm o}}\) pp. 226–246; 2\(^{\underline {\rm o}}\) pp. 247–275].  https://doi.org/10.3931/e-rara-12132
  4. Archimedes-Heiberg, Johan Ludvig. 1880. Archimedis opera omnia cum commentariis Eutocii. Vol 1. Lipsiae [Leipzig]: In aedibus B. G. Teubner. http://archive.org/details/archimedisoperao01arch
  5. Bobenko, A. I., Schief, W. K., Suris, Y. B. and Techter, J. 2015. On a discretization of confocal quadrics. I. An integrable systems approach. Journal of Integrable Systems, 1(1), 1–34.  https://doi.org/10.1093/integr/xyw005
  6. Bobenko, A. I., Schief, W. K., Suris, Y. B. and Techter, J. 2017. On a discretization of confocal quadrics. II. A geometric approach to general parametrizations. arXiv:1708.06800 [math.DG]
  7. Brianchon, Charles-Julie. 1817. Mémoire sur les lignes du second ordre. Paris: Bachelier.Google Scholar
  8. Chasles, Michel. 1837. Aperçu Historique sur l’origine et le développement des méthodes en géométrie, particulièrement de celles qui se rapportent a la géométrie moderne, Bruxelles: M. Hayez.Google Scholar
  9. Chasles, Michel. 1852. Traité de géométrie supérieure, París: Bachelier.Google Scholar
  10. Chasles, Michel. 1870. Rapport sur les progrès de la géométrie, par M. Chasles, Paris: Impr. Nationale.Google Scholar
  11. Chasles, Michel and Graves, Charles. 1841. Two Geometrical Memoirs on the General Properties of Cones of the Second Degree. (Trad. additions and appendix by Graves, C.). Dublin: Graisberry and Gill.Google Scholar
  12. Darboux, Gaston. 1896. Leçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal, Vol 2-3, Paris: Gauthier-Villars.Google Scholar
  13. Dinca, Ion. 2014. Thread configurations for ellipsoids. Rev. Roumaine Math. Pures Appl., (3), 371–398.Google Scholar
  14. Dupin, Charles. 1819. Essai historique sur les services et les travauxs cientifiques de Gaspard Monge, par M. Ch. Dupin, Paris: Bachelier.Google Scholar
  15. Dupin, Charles. 1822. Applications de géométrie et de méchanique à la marine, aux ponts-et-chaussées, etc., pour faire suite aux “Développements de géométrie” par Charles Dupin,…, Paris: Bachelier.Google Scholar
  16. Frèzier, Amédée François. 1737-1739. La théorie et la pratique de la coupe de pierres et de bois pour la construction des voûtes, et autre parties des bâtiments civils et militaire, ou traité de stéréotomie à l’usage de l’architecture. Strasbourg-Paris: Jean Daniel Doulsseker-L. H. Guerin.Google Scholar
  17. Gentil Baldrich, José María. 1997. Sobre la intersección de las cuádricas de revolución de ejes paralelos. Sevilla: Departamento de Expresión Gráfica Arquitectónica.Google Scholar
  18. Gentil Baldrich, José María. 2016. Teorema de la esfera intrusa. In: ACCA 015. Sevilla/Málaga: dEGA/RU Books.Google Scholar
  19. Hachette, Jean Nicolas Pierre. 1817. Éléments de Géométrie à trois dimensions (partie synthétique), de M. Hachette. Paris: L’Auteur.Google Scholar
  20. Izquierdo-Asensi, Fernando. 1985. Geometría Descriptiva Superior y Aplicada. Madrid: Dossat [1º edición 1978].Google Scholar
  21. Ivory, James. 1809. On the attraction of homogeneous ellipsoids. Phil. Trans. Royal Soc. London, 99, 345–372.Google Scholar
  22. La Gournerie, Jules M. de 1860–64. Traité de Géométrie descriptive, 3 vols. Paris: Mallet-BachelierGoogle Scholar
  23. Lazard, S., Peñaranda, L. M. and Petitjean, S. 2006. Intersecting Quadrics : An Efficient and Exact Implementation. Computational Geometry, 35, 74–99.  https://doi.org/10.1016/j.comgeo.2005.10.004
  24. Martín-Pastor, A. and Granado-Castro, G. 2017. Some controversies on representation of the solar shadow in 17th century. The manuscript ‘Artes Excelençias de la Perspectiba’ in context. Disegnare idee immagini, 54, 12–23.Google Scholar
  25. Miller, J. R. and Goldman, R. N. 1995. Geometric algorithms for detecting and calculating all conic sections in the intersection of any two natural quadric surfaces. Graphical Models and Image Processing, 57(1), 55–66.  https://doi.org/10.1006/gmip.1995.1006
  26. Monge, Gaspard. 1798. Géométrie Descriptive - Lecons données aux Écoles normales, l’An 3 de la République; Par Gaspard Monge. Paris: Impr. de Baudouin.Google Scholar
  27. Monge, Gaspard. 1809. Application de l’analyse à la géométrie, à l’usage de l’Ecole impériale polytechnique; par M. Monge. Paris: Ve Bernard [1ª ed. in-fol, 1795, Feuilles d’Analyses appliquée à la Géométrie, Ecole polytechnique]Google Scholar
  28. Narvaez-Rodriguez, R. and Barrera-Vera, J. A. 2016. Lightweight Conical Components for Rotational Parabolic Domes: Geometric Definition, Structural Behaviour, Optimisation and Digital Fabrication. In S. Adriaenssens, F. Gramazio, M. Kholer, A. Menges, & M. Pauly (Eds.), Advances in Architectural Geometry 2016 pp. 378–397. Zurich: vdf Hochschulverlag AG an der ETH Zürich.  https://doi.org/10.3218/3778-4_25
  29. Poncelet, Jean-Victor. 1862-1864. Applications d’analyse et de géométrie qui ont servi de principal fondement au Traité des propriétés projectives des figures/par J.-V. Poncelet; et accompagnés de divers autres écrits… par MM. Mannheim et Moutard,… Paris: Mallet-Bachelier.Google Scholar
  30. Raynaud, Dominique. 2018. Sociologie des controverses scientifiques. Paris: Édition Matériologiques [1st edition 2003].Google Scholar
  31. Salvatore, Marta. 2011. Prodromes of Descriptive Geometry in the Traité de stéréotomie by Amédée François Frézier. Nexus Network Journal, 13(3), 671–699.  https://doi.org/10.1007/s00004-011-0086-0
  32. Shene, C. K. and Johnstone, J. K. 1994. On the Lower Degree Intersections of Two Natural Quadrics. ACM Trans. Graph., 13(4), 400–424.  https://doi.org/10.1145/195826.197316
  33. Staude, Otto. 1883. Geometrische Deutung der Additionstheoreme der hyperelliptischen Integrale und Functionen. Mathematische Annalen, XXII (1), 1–69, 146–176.Google Scholar
  34. Taibo-Fernández. 1983. A. Geometría Descriptiva y sus aplicaciones. Tomo II. Madrid: Tébar Flores.Google Scholar

Copyright information

© Kim Williams Books, Turin 2018

Authors and Affiliations

  1. 1.Universidad de SevillaSevilleSpain

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