A circular arc structure (CAS) is a mesh of a surface with edges realized by circular arcs, such that all edges incident to a vertex are tangent to a common plane, and form a repeatable pattern of angles about interior vertices so that all vertices are congruent. The goal is to approximate a free-form architectural design with repetitions of the costly parts, primarily the node and arc structures.

The CAS is computed by a nonlinear optimization of a “functional” that penalizes deviations from the multiple goals. A good initialization is crucial; a conformal mapping from a regular network (say, hexagonal) with the prescribed vertex angles provides it. We can avoid the torsion at nodes by following principal lines of curvature. We can augment the functional so that circular-arc edges all have the same radius; this is another cost savings.

A Möbius transformation preserves smoothness, angles, and circular arc-ness, and so provides geometric guidance for CASs. In particular, applying Möbius transformations to tori results in the so-called Dupin cyclides, and leads to cyclidic CASs with attractive properties. For example, the offset of a cyclidic patch is again cyclidic, which enables construction of structural elements with parallel glass panels.

A cyclidic CAS model may also be naturally converted to a developable strip model, which has manufacturing implications: we can form such strips by bending flat strips.

This paper is rich with ideas, emerging from the fruitful interaction of classical differential geometry, modern discrete geometry, numerical optimization, and architectural pragmatics. The 25 color figures are nothing short of stunning.