We discretize isometric mappings between surfaces as correspondences between checkerboard patterns derived from quad meshes. This method captures the degrees of freedom inherent in smooth isometries and enables a natural definition of discrete developable surfaces. This definition, which is remarkably simple, leads to a class of discrete developables which is much more flexible in applications than previous concepts of discrete developables. In this paper, we employ optimization to efficiently compute isometric mappings, conformal mappings and isometric bending of surfaces. We perform geometric modeling of developables, including cutting, gluing and folding. The discrete mappings presented here have applications in both theory and practice: We propose a theory of curvatures derived from a discrete Gauss map as well as a construction of watertight CAD models consisting of developable spline surfaces.

Fig. 1. Le : A statically sound space structure designed and optimized with our framework, motivated by the real architectural project shown in Figure 2. Right: The space structure is constructed with six types of customized beams to minimize the total volume of the material used for beams while maintaining moderate manufacturing complexity. Here, a ho er color indicates a larger beam cross-section area. Our framework automatically determines the optimal cross-section areas of the six types of beams as well as the assignment of beam types.We study the design and optimization of statically sound and materially e cient space structures constructed by connected beams. We propose a systematic computational framework for the design of space structures that incorporates static soundness, approximation of reference surfaces, boundary alignment, and geometric regularity. To tackle this challenging problem, we rst jointly optimize node positions and connectivity through a nonlinear continuous optimization algorithm. Next, with xed nodes and connectivity, we formulate the assignment of beam cross sections as a mixed-integer programming problem with a bilinear objective function and quadratic constraints. We solve this problem with a novel and practical alternating direction method based on linear programming relaxation. The capability and e ciency of the algorithms and the computational framework are validated by a variety of examples and comparisons.

Figure 1: We approximate the Cour Visconti roof in the Louvre, Paris, by a quad mesh with planar faces which caps a honeycomb (consisting of hexagonal cells whose walls intersect at 120• ). This structure exhibits several features important in freeform architectural design: planar faces, low valence of nodes, a torsion-free support structure, and repetitive node geometry. AbstractMotivated by requirements of freeform architecture, and inspired by the geometry of hexagonal combs in beehives, this paper addresses torsion-free structures aligned with hexagonal meshes. Since repetitive geometry is a very important contribution to the reduction of production costs, we study in detail "honeycomb structures", which are defined as torsion-free structures where the walls of cells meet at 120 degrees. Interestingly, the Gauss-Bonnet theorem is useful in deriving information on the global distribution of node axes in such honeycombs. This paper discusses the computation and modeling of honeycomb structures as well as applications, e.g. for shading systems, or for quad meshing. We consider this paper as a contribution to the wider topic of freeform patterns, polyhedral or otherwise. Such patterns require new approaches on the technical level, e.g. in the treatment of smoothness, but they also extend our view of what constitutes aesthetic freeform geometry.