Boxelization: folding 3D objects into boxes

We present a method for transforming a 3D object into a cube or a box using a continuous folding sequence. Our method produces a single, connected object that can be physically fabricated and folded from one shape to the other. We segment the object into voxels and search for a voxel-tree that can fold from the input shape to the target shape. This involves three major steps: finding a good voxelization, finding the tree structure that can form the input and target shapes' configurations, and finding a non-intersecting folding sequence. We demonstrate our results on several input 3D objects and also physically fabricate some using a 3D printer

Shape Design and Optimization for 3D Printing

In recent years, the 3D printing technology has become increasingly popular, with wide-spread uses in rapid prototyping, design, art, education, medical applications, food and fashion industries. It enables distributed manufacturing, allowing users to easily produce customized 3D objects in office or at home. The investment in 3D printing technology continues to drive down the cost of 3D printers, making them more affordable to consumers. As 3D printing becomes more available, it also demands better computer algorithms to assist users in quickly and easily generating 3D content for printing. Creating 3D content often requires considerably more efforts and skills than creating 2D content. In this work, I will study several aspects of 3D shape design and optimization for 3D printing. I start by discussing my work in geometric puzzle design, which is a popular application of 3D printing in recreational math and art. Given user-provided input figures, the goal is to compute the minimum (or best) set of geometric shapes that can satisfy the given constraints (such as dissection constraints). The puzzle design also has to consider feasibility, such as avoiding interlocking pieces. I present two optimization-based algorithms to automatically generate customized 3D geometric puzzles, which can be directly printed for users to enjoy. They are also great tools for geometry education. Next, I discuss shape optimization for printing functional tools and parts. Although current 3D modeling software allows a novice user to easily design 3D shapes, the resulting shapes are not guaranteed to meet required physical strength. For example, a poorly designed stool may easily collapse when a person sits on the stool; a poorly designed wrench may easily break under force. I study new algorithms to help users strengthen functional shapes in order to meet specific physical properties. The algorithm uses an optimization-based framework — it performs geometric shape deformation and structural optimization iteratively to minimize mechanical stresses in the presence of forces assuming typical use scenarios. Physically-based simulation is performed at run-time to evaluate the functional properties of the shape (e.g., mechanical stresses based on finite element methods), and the optimizer makes use of this information to improve the shape. Experimental results show that my algorithm can successfully optimize various 3D shapes, such as chairs, tables, utility tools, to withstand higher forces, while preserving the original shape as much as possible. To improve the efficiency of physics simulation for general shapes, I also introduce a novel, SPH-based sampling algorithm, which can provide better tetrahedralization for use in the physics simulator. My new modeling algorithm can greatly reduce the design time, allowing users to quickly generate functional shapes that meet required physical standards

Stackabilization

We introduce the geometric problem of stackabilization: how to geometrically modify a 3D object so that it is more amenable to stacking. Given a 3D object and a stacking direction, we define a measure of stackability, which is derived from the gap between the lower and upper envelopes of the object in a stacking configuration along the stacking direction. The main challenge in stackabilization lies in the desire to modify the object’s geometry only subtly so that the intended functionality and aesthetic appearance of the original object are not significantly affected. We present an automatic algorithm to deform a 3D object to meet a target stackability score using energy minimization. The optimized energy accounts for both the scales of the deformation parameters as well as the preservation of pre-existing geometric and structural properties in the object, e.g., symmetry, as a means of maintaining its functionality. We also present an intelligent editing tool that assists a modeler when modifying a given 3D object to improve its stackability. Finally, we explore a few fun variations of the stackabilization problem

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