Paper accepted in the journal Computer Aided Geometric Design

We got our paper The architectural application of shells whose boundaries subtend a constant solid angle accepted in the journal Computer Aided Geometric Design. The paper is freely available here: https://www.sciencedirect.com/science/article/pii/S0167839624000657

Our recent work shows that it is possible to generate a consistent principal curvature net for a structural grid – that is, a grid composed of planar panels meeting at right angles – by using a special class of surfaces known as surfaces of constant solid angle. A key geometric restriction in our approach is that the surface boundaries lie in a single plane. This construction guarantees both the planarity of the panels and the orthogonality of the grid directions, while also ensuring that the panels do not need to be cut at the boundary. As a result, the proposed strategy is particularly well suited to structurally rational and fabrication-friendly grid designs.

Abstract: Surface geometry plays a central role in the design of bridges, vaults and shells, using various techniques for generating a geometry which aims to balance structural, spatial, aesthetic and construction requirements.

In this paper we propose the use of surfaces defined such that given closed curves subtend a constant solid angle at all points on the surface and form its boundary. Constant solid angle surfaces enable one to control the boundary slope and hence achieve an approximately constant span-to-height ratio as the span varies, making them structurally viable for shell structures. In addition, when the entire surface boundary is in the same plane, the slope of the surface around the boundary is constant and thus follows a principal curvature direction. Such surfaces are suitable for surface grids where planar quadrilaterals meet the surface boundaries. They can also be used as the Airy stress function in the form finding of shells having forces concentrated at the corners.

Our technique employs the Gauss-Bonnet theorem to calculate the solid angle of a point in space and Newton's method to move the point onto the constant solid angle surface. We use the Biot-Savart law to find the gradient of the solid angle. The technique can be applied in parallel to each surface point without an initial mesh, opening up for future studies and other applications when boundary curves are known but the initial topology is unknown.

We show the geometrical properties, possibilities and limitations of surfaces of constant solid angle using examples in three dimensions.