Band Decomposition of 2-Manifold Meshes For...

Band Decomposition of 2-Manifold MeshesFor Physical Construction of Large Structures

Qing Xing, Gabriel Esquivel, Ergun Akleman & Jianer ChenTexas A&M University

Jonathan GrossColumbia University

Figure 1: This large sculpture of Bunny is constructed with laser cut poster-board papers assembled with brass fasteners.

With the design and construction of more and more unusu-ally shaped buildings, the computer graphics community hasstarted to explore new methods to reduce the cost of thephysical construction for large shapes. Most of currentlysuggested methods focus on reduction of the number of dif-ferently shaped components to reduce fabrication cost. Inthis work, we focus on physical construction using devel-opable components such as thin metals or thick papers. Inpractice, for developable surfaces fabrication is economicaleven if each component is different. Such developable com-ponents can be manufactured fairly inexpensively by cuttinglarge sheets of thin metals or thin paper using laser-cutters,which are now widely available.

(a) (b) (c)

Figure 2: Construction elements for bunny. (a) is an exam-ple of vertex component that is cut with laser cutter, (b)shows elements of vertex component and (c) shows the pro-cess of assembling vertex components with fasteners.

We observe that one of the biggest expenses for construc-tion of large shapes comes from handling and assemblingthe large number components. This problem is like puttingpieces of a large puzzle together. However, unlike puzzleswe do not want construction process to be challenging. In-stead, we want to simplify the construction process in such away that the components can be assembled with a minimuminstruction by the construction workers who may not haveextensive experience.

In this work, we introduce an approach to automatically cre-ate such easily assembled developable components from anygiven manifold mesh. Our approach is based on classicalGraph Rotation Systems (GRS). Each developable compo-

nent, which we call vertex component, is a physical equiva-lent of a rotation at the vertex v of a graph G. Each vertexcomponent is a star shaped polygon that physically corre-sponds to the cyclic permutation of the edge-ends incidenton v (See Figure 2(a)). We engrave edge-numbers with laser-cutters directly on edge-ends of vertex components to sim-plify finding corresponding edge ends. When we print edge-numbers, we actually define a collection of rotations, one foreach vertex in G. This is formally called a pure rotationsystem of a graph.

The fundamental Heffter-Edmunds theorem of GRS assertsthat there is a bijective correspondence between the set ofpure rotation systems of a graph and the set of equivalenceclasses of embeddings of the graph in the orientable surfaces.As a direct consequence of the theorem, to assemble thestructure all construction workers have to do is to attachthe corresponding edge-ends of vertex components. Onceall the components are attached to each other, the wholestructure will correctly be assembled.

Gauss-Bonnet theorem, moreover, asserts that the totalGaussian curvature of a surface is the Euler characteristicstimes 2π. Since the structure is made up only developablecomponents, Gaussian curvature is zero everywhere on thesolid parts. The Gaussian curvature happens only in emptyregions and that are determined uniquely. Since, we cor-rectly form Gaussian curvature of holes, the structures willalways be raised and formed 3-space.

We also develop strategies to simplify finding correspondingpieces among a large number of vertex components. Us-ing this approach, Architecture students have constructed alarge version of Stanford Bunny (see Figure 1) in a designand fabrication course in College of Architecture. The costsof poster-board papers and fasteners were very minimal, lessthan $100. We are currently working on to construct evenlarger shapes using stronger materials. We are also planningto use the structures obtained by this approach as molds tocast large plaster or cement sculptures.

This work partially supported by the National Science Foun-dation under Grant No. NSF-CCF-0917288.