Abstract and Figures

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Abstract and Figures We introduce the foldabilization problem for space-saving furniture design. Namely, given a 3D object representing a piece of furniture, our goal is to apply a minimum amount of modification to the object so that it can be folded to save space --- the object is thus foldabilized. We focus on one instance of the problem where folding is with respect to a prescribed folding direction and allowed object modifications include hinge insertion and part shrinking. We develop an automatic algorithm for foldabilization by formulating and solving a nested optimization problem operating at two granularity levels of the input shape. Specifically, the input shape is first partitioned into a set of integral folding units. For each unit, we construct a graph which encodes conflict relations, e.g., collisions, between foldings implied by various patch foldabilizations within the unit. Finding a minimum-cost foldabilization with a conflict-free folding is an instance of the maximum-weight independent set problem. In the outer loop of the optimization, we process the folding units in an optimized ordering where the units are sorted based on estimated foldabilization costs. We show numerous foldabilization results computed at interactive speed and 3D-print physical prototypes of these results to demonstrate manufacturabilit Foldable scaffold. A folding configuration is flat if all of itspatches are co-planar. A scaffold Sis said to be foldable if, start-ing with its current configuration (i.e., the hinge angles), it can befolded into a flat configuration via a valid folding transform.Figure 1: Two automatic foldabilizations of a chair with respect totwo folding directions. Shown on the right are fabrication resultsproduced by a MakerBot Replicator II 3D printer. Folding is pos-sible by adding hinges, shrinking parts (chair back in the top row),or allowing slanting or shearing, leading to less hinges and betterstructural soundness (bottom right). The foldable chair in the toprow resembles the Stitch Chair by the designer Adam Goodrum.in a rich variety of ways, offering an abundant source of appealingand challenging geometry problems.An interesting geometry question about folding is: what makessome 3D objects more amenable to folding than others? Sincerigid parts cannot be folded, hinges need to be inserted to make theparts foldable. Moreover, folding involves constrained movementsof object parts and such movements often require necessary clear-ing space to avoid collision. Hence reducing the size or extent offurniture parts to make space is beneficial to folding, e.g., the backof the chair in Figure 1 (top) is shrunk to allow folding of the seat.Taking the two factors to the extreme, we arrive at structures formedby thin frames with many hinges; famous examples of such objectsare scissoring structures such as the Hoberman spheres. However,due to structural strength and functionality considerations, furniturefoldabilization can hardly go to that extreme.In this paper, we pose the novel foldabilization problem for space-saving furniture design. Given a 3D object Orepresenting a pieceof furniture, our goal is to apply a minimum amount of modifica-tion to Oto allow its parts to be folded flatly onto themselves oreach other. We focus on one particular instance of the foldabi-lization problem where folding is allowed only with respect to aprescribed folding direction and the allowed modifications includeadding (line)hinges onto furniture parts and shrinking the parts.Figure 1 shows two automatic foldabilization results for a chair,and the resulting folding sequences on fabricated 3D prototypes.Foldabilization is not an easy task for humans. It resembles a 3Dpuzzle with a large search space and a multitude of constraints.Solving the problem requires delicate spatial reasoning and a keenforesight to adapt to the dynamic changes to the shape configura-tion as folding sequences proceed. While humans are highly aptat pattern recognition, they are not as skilled at precise 3D manip-ulation while relying solely on visual guidance. In particular, inhuman perception, lengths in 3D are systematically distorted dueto perspective viewing [Baird 1970; Norman et al. 1996]. Thus, by

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