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Surface geometry plays a central role in the design of bridges, vaults andshells, using various techniques for generating a geometry which aims tobalance structural, spatial, aesthetic and construction requirements.  In this paper we propose the use of surfaces defined such that given closedcurves subtend a constant solid angle at all points on the surface and form itsboundary. Constant solid angle surfaces enable one to control the boundaryslope and hence achieve an approximately constant span-to-height ratio as thespan varies, making them structurally viable for shell structures. In addition,when the entire surface boundary is in the same plane, the slope of the surfacearound the boundary is constant and thus follows a principal curvaturedirection. Such surfaces are suitable for surface grids where planarquadrilaterals meet the surface boundaries. They can also be used as the Airystress function in the form finding of shells having forces concentrated at thecorners.  Our technique employs the Gauss-Bonnet theorem to calculate the solid angleof a point in space and Newton's method to move the point onto the constantsolid angle surface. We use the Biot-Savart law to find the gradient of thesolid angle. The technique can be applied in parallel to each surface pointwithout an initial mesh, opening up for future studies and other applicationswhen boundary curves are known but the initial topology is unknown.  We show the geometrical properties, possibilities and limitations of surfacesof constant solid angle using examples in three dimensions.

The architectural application of shells whose boundaries subtend a constant solid angle