Shape Compaction via Stacking and Folding

Space-saving, or collapsible, objects are ubiquitous in our living and working space. They can adjust configurations to either perform their intended functionality or save space, for example, while storing and shipping. This additional space-saving characteristic of changing forms makes collapsible objects more preferable than their non-collapsible counterparts, especially in environments where space is costly. Shape compaction is an important real-world design problem, where a 3D object is geometrically modified, such that it can be more compactly stored by changing to a different configuration, while preserving its functionality and aesthetic. This thesis argues the need for computational tools to support shape compaction of 3D objects and proposes novel algorithms to support the compaction via stacking and folding.The first problem is stackabilization --- making objects more amenable to stacking. As a group collapsing principle, a collection of shapes can cooperatively occupy less space when stacked than they do individually. Given a 3D object and a stacking direction, a measure of stackability is defined to reflect the space-saving ratio of stacking the given object along the given stacking direction. The stackabilization algorithm deforms the input 3D object to meet a target stackability score using energy minimization. This energy accounts for the scales of the deformation parameters as well as the preservation of per-existing geometric and structural properties in the objects. The second problem is foldabilization --- modifying the input 3D object such that it can be folded into a flat configuration along a prescribed direction. Folding an object involves rearranging its parts via hinging; the folded part configuration usually occupies less space than the unfolded one. The input 3D object is first abstracted into a scaffold, which consists of a collection of connected planar patches. The foldabilization algorithm minimizes the amount of modifications, e.g. shrinking and split, on these patches such that the modified scaffold can be folded into a flat configuration. Structure soundness is considered by allowing slanted folding and patch disconnection, which usually result in fewer splits on the input scaffold. The fully automatic foldabilization results can be computed at interactive speed. The prototypes can be fabricated while folded for cost-effective printing, and unfolded to show their usage configurations.