We propose a novel type of planar-to-spatial deployable structures that we call elastic geodesic grids. Our approach aims at the approximation of freeform surfaces with spatial grids of bent lamellas which can be deployed from a planar configuration using a simple kinematic mechanism. Such elastic structures are easy-to-fabricate and easy-to-deploy and approximate shapes which combine physics and aesthetics. We propose a solution based on networks of geodesic curves on target surfaces and we introduce a set of conditions and assumptions which can be closely met in practice. Our formulation allows for a purely geometric approach which avoids the necessity of numerical shape optimization by building on top of theoretical insights from differential geometry. We propose a solution for the design, computation, and physical simulation of elastic geodesic grids, and present several fabricated small-scale examples with varying complexity. Moreover, we provide an empirical proof of our method by comparing the results to laser-scans of the fabricated models. Our method is intended as a form-finding tool for elastic gridshells in architecture and other creative disciplines and should give the designer an easy-to-handle way for the exploration of such structures.

Figure 1: We present a medium sized structure that was built to investigate the scalability of the elastic geodesic grids approach. The desktop model to the left was scaled by a factor of four to receive a structure of 3.1 × 2.1 × 0.9 meters in size.

Given a designer created free-form surface in 3d space, our method computes a grid composed of elastic elements which are completely planar and straight. Only by fixing the ends of the planar elements to appropriate locations, the 2d grid bends and approximates the given 3d surface. Our method is based purely on the notions from differential geometry of curves and surfaces and avoids any physical simulations. In particular, we introduce a well-defined elastic grid energy functional that allows identifying networks of curves that minimize the bending energy and at the same time nestle to the provided input surface well. Further, we generalize the concept of such grids to cases where the surface boundary does not need to be convex, which allows for the creation of sophisticated and visually pleasing shapes. The algorithm finally ensures that the 2d grid is perfectly planar, making the resulting gridshells inexpensive, easy to fabricate, transport, assemble, and finally also to deploy. Additionally, since the whole structure is pre-strained, it also comes with load-bearing capabilities. We evaluate our method using physical simulation and we also provide a full fabrication pipeline for desktop-size models and present multiple examples of surfaces with elliptic and hyperbolic curvature regions. Our method is meant as a tool for quick prototyping for designers, architects, and engineers since it is very fast and results can be obtained in a matter of seconds.

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