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

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.