1 research outputs found
The architectural application of shells whose boundaries subtend a constant solid angle
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