Geometric and area parameterization of N-edged hyperbolic paraboloidal umbrellas

The legacy of master builder Félix Candela is cemented by his widespread adoption of thin hyperbolic paraboloid (hypar) umbrellas within architecture across the Americas. Exhibiting any number of edges, with or without parabolic bisectors, these inverted umbrellas perfectly embody the amalgamation of efficiency, economy, and elegance as emblematic of structural art. Yet, a consistent mathematical description of their unique form does not presently exist in the literature. As such, Candela's umbrellas have rarely been the subject of rigorous structural analysis, nor frequently featured in contemporary architectural planning. This paper introduces equations to parametrize the geometry of hypar umbrellas with any arbitrary number of edges and parabolic bisectors for analysis and design applications. A simplified method computing the surface area of Candela's umbrellas based on regular pyramids is also presented. As such, this work provides an exact geometrical description of N-sided hypar umbrellas, thus removing the need for manual surface generation via computer-aided design (CAD) prior to numerical analyses.