Functional design of flexible surfaces

Any object whose shape or function can be changed due to geometric, mechanic or physical properties will be called a flexible surface. These kinds of surfaces have many applications in architecture, industrial manufacturing processes, and computer aided design. Within this project we consider data structures and algorithms that facilitate the intuitive design and efficient manipulation of those surfaces and their transformations. Transformations between different shape configurations always have to be reversible which prohibits the use of plastic deformations. Because of their practical relevance we are primarily interested in elastic deformations that are isometric or close to being isometric. Special cases are the so called developable surfaces. Despite their popularity in many applications those surfaces are still treated with techniques that were devised for a totally different type of surfaces. A big part of this project is concerned with the representation, the design and manipulation of developable surfaces. To this end we introduce new ways of representation and interaction that are adapted to the case of developable surfaces. This results in a design scheme that is easily accessible for non-expert users. The insight gained from the special case of developable surfaces is used to devise algorithms for the general case of (almost) isometric free- form deformations. A crucial part to the solution of this problem is the formulation of the deformations process as a constrained optimization problem and the application of adequate optimization techniques. Nonetheless a purely numerical treatment cannot yield optimal results. We use surface discretizations that are adapted to a given transformation in order to reduce the dimensionality of the optimization problem, hence speeding up computation and improving robustness. Adapted surface discretizations can be determined by analyzing differential geometric surface properties. The type of deformation and the mechanism chosen for actual realization determines the number and type of side conditions of the optimization problem.

Transformable structures have received attention in industrial design and architecture. Since production processes almost always start from a planar material configuration this project focuses on developable surfaces. Restricting the set of deformations in order to preserve length, angles, and area, i.e., isometric deformations, economic manufacturing techniques can be employed at several stages of the production pipeline. Folding is the most basic and cost efficient approach to transform a planar sheet into a three dimensional shape. While folding has been practiced for centuries its application on a larger scale remains challenging. This is mainly due to the complex mechanisms that are required to drive the folding process without human interaction. This complexity limits the number of folds in a design and led to the consideration of more complicated fold types. Remarkably, one can fold along general curves, not just along a straight line. Folding along a general curve facilitates the creation of both, intricate and aesthetically pleasing shapes with a minimal number of folds. While the design of such curved folding patterns is an open problem there have been successful attempts to use industrial robots in the fabrication process. In contrast to this manufacturing driven approach, we consider non permanent string/cable based actuation that allows to repeat the folding/unfolding process indefinitely. To this end we devised a surface deformation model that is tuned towards developable surfaces with curved creases. Given a crease pattern we demonstrate how to compute a sequence of shapes along the folding motion. We use this deformation sequence to compute a set of strings attached to certain points on the surface that can reproduce the folding motion when pulled. Applications include dynamic structures in architecture and design. We verified our results by building small scale models using paper, plastic and thin aluminum sheets.