Tensioned principle curvature cable nets on minimal surfaces
Paper in proceeding, 2021
Cable nets are efficient and elegant structures that are prestressed to limit their deflection under loading. The best known cable net structure is the Munich Olympiastadion built for the Olympic games in 1972 by Frei Otto and Jörg Schlaich. Frei Otto believed that the ideal shape for such structures is a minimal surface with uniform surface tension under prestress. However, a minimal surface can only be approximated by the equal mesh nets used by Frei Otto, but it is possible to produce a true pretensioned minimal surface with a fine net of cables forming a pattern of curvilinear squares, which includes a net following the principal curvatures of the surface. A analytical and a numerical approach for the formfinding of minimal surfaces with a principal curvature net are described. The analytic approach uses the fact that every minimal surface with principal curvature coordinates can be expressed by a single function of a complex variable. This is a special case of the Weierstrass–Enneper parameterization which uses two functions, but one of them effectively only controls the pattern of coordinates on the surface. The numerical approach automatically produces a minimal surface and the principal curvature coordinates at the same time and can be applied to any minimal surface whose boundaries are either principal curvature or asymptotic directions, or a combination of the two. Straight lines and cable boundaries form asymptotic lines and a surface which is normal to a sphere has a principal curvature direction as its boundary.