EUROGRAPHICS 2017/ J. J. Bourdin and A. Shesh Education Paper

Fabrication of Simplified Shapes using Digital CNC Machines

A. Muntoni1

1University of Cagliari, Italy

Figure 1: Our fabrication pipeline: we simplify a given input shape in a 3D model composed of a small number of polygonal primitives andwith a restricted set of dihedral angles between adjacent faces. We then unfold the simplified model and we mill the unfolded shape on asheet of rigid material (e.g. wood) with a CNC Carving Machine using tips of well-known angles along the edges. We then easily assemblythe resulting simplified model, putting glue and folding along the edges.

AbstractOn my Ph.D. I am focusing my research on Fabrication of 3D shapes using subtractive techniques like 3-axis CNC milling orcarving machines. These techniques cannot manufacture free-form geometries, and they have very precise constraints on themanufacturable models. My research consist of algorithmically process free-form 3D models in order to obtain manufacturableshapes using these techniques. CNC Carving machines are able to work only on sheets of materials with a limited thickness, andfor 3-axis milling machines every millable piece needs to be an height-field. On this Research statement I’ll present a methodfor the simplification of 3D models in order to fabricate them with CNC Carving Machines, and I’ll introduce another projectfor the decomposition of 3D shapes in manufacturable blocks with 3-axis milling machines.

Context

My Ph.D. is being performed at the University of Cagliari, in thedepartment of Mathematics and Computer Science. In our lab twomain research fields are studied: Computer Graphics and HumanComputer Interaction. I am at the 3rd year and I’m working on theComputer Graphics field, supervised by Professor Riccardo Scateniand Dr. Marco Livesu, in the specific topic of Geometry Processingapplied to Fabrication. In this Research Statement I will focus on aproject for fabrication of 3D shapes with CNC Carving Machines,and I’ll briefly introduce another project focused on fabrication us-ing CNC Milling Machines.

Introduction and Objectives

Fabrication is a trending topic in Computer Graphics field. Themain goal is to get a physical and tangible representation of a 3D

digital shape, and it can be done with variegate techniques, that canbe manual, automatic or semi-automatic.

Automatic fabrication techniques can be distinguished in addi-tive and subtractive. Additive fabrication, or 3D printing, consistson adding, layer by layer, filament of material in order to obtainthe desired shape. However 3D printers have some cons given bythe limitation of usable materials, the limited dimensions of themanufacturable models (which sometimes lead to split the desiredmodel in different portions that need to be manually glued in post-processing) and the need to print supports given by the fact thatevery layer has to be based on a solid base.

On the contrary, subtractive techniques aim to remove portionsof material in order to obtain the desired shape. However, the fea-sibility of the fabrication is limited and depends on the model andon the used machine. 5-axis milling machines have a wide range

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A. Muntoni / Fabrication of Simplified Shapes using Digital CNC Machines

of manufacturability shapes, but they are very expansive and auto-matic generation of machine paths for the milling is still an openproblem ( [YDX99], [LZ07]). 3-axis milling machines are morecheaper, but they are able to produce only 2.5D models, e.g. shapeswith flat bases and with no undercuts. But, compared with addi-tive technologies, 3-axis milling allows the use of many differenttypes of materials (wood, stone, metals, etc) unavailable with 3Dprinting, and also allows manufacturing on bigger scales.

Also CNC laser-cutting and carving machines are widespreadautomatic subtractive technologies. These machines are able to cutor dig only sheets of rigid materials with limited thickness, butmanufacturing times are limited and they are very attractive forsemi-automatic fabrication methods.

Semi-automatic fabrication techniques aim to obtain simplifica-tions or decompositions of the desired model in order to make pos-sible and/or easier the automatic manufacture. Once the automaticmanufacture is ended, a manual process to assembly the model isrequired. Developing fabrication methods based on semi-automaticfabrication techniques introduces the challenge to make as easy aspossible and non-error prone the assembly process.

During the first two years of my Ph.D I’ve studied two differentapproaches, based on semi-automatic fabrication using CNC Carv-ing and Milling Machines.

The first approach, which I will discuss on this research state-ment, has the goal to propose a novel algorithm that, starting from ageneric input shape, generates a easy-to-assembly shape manufac-turable with CNC carving machines, avoiding errors given by theprecision of the user during the assembly process. The general ideaof the algorithm is to simplify the input shape in a coarse modelcomposed of a small number of flat polygonal faces, with the par-ticular property that every possible dihedral angle between adjacentfaces belongs to a restricted and well-known set of allowed angles,in order to facilitate the assembly process and avoid precision er-rors. The fabrication process will consist of cutting sheets of rigidmaterials (like wood sheets, but also rigid paper, glass or stone)with CNC carving machines. These machines are able to makefurrows produced by different tips of well-known angles (Figure2). After the fabrication process, a very simple assembly processwould be to put some glue on the furrows and to simply fold all theproduced faces.

The second approach I studied is a method for the decompositionof 3D models in blocks which can be manufactured with 3-axisCNC milling machines. I’ll briefly introduce it on Section 5.

Related Works

In a general speaking, mesh simplification has been studied in re-cent years. [CSAD04] proposes a method which produces an ap-proximation of the surface with a variational approach. This is ageneral purpose method, it can produce non-watertight meshes andit is not applied for fabrication. On the contrary, we aim to solvea simplification problem specifically for fabrication and producingalways watertight meshes.

Fabrication of 3D digital shapes using flat sheets of rigid materi-als has been studied in a lot of different applications. [RA15] repre-sent an input geometry using beams with rectangular cross-section.

Figure 2: Fabrication Process using a CNC Carving Machine.We restrict the possible angles between the polygonal faces of thesimplified models to a set based on the tips used for the carvingprocess. Then, the assembly process will be an easy folding alongthe carved edges.

The authors show some beams that can be made of wood and man-ufactured with laser cutter machines and assembled with particularjoints. [SP13] and [CPMS14] present methods for the fabricationof 3D models by the interlocking of planar pieces and strips. Thesethree papers propose a solution to problems which are substantiallydifferent to ours: they don’t aim to create a surface, but a set ofstrips or planar pieces that are joined each other (with interlocksor other type of joints). On the contrary, we want to produce a 3Dsurface composed of a small set of polygonal planar faces.

A more related problem is solved by [MS04]. Here they simplifythe input shape and their output is a set of triangle strips to cut,fold and glue together in the assembly process in order to obtain apaper-craft. However they limit the primitives which compose theiroutput to triangles, and the assembly process is quite difficult andit is suitable only for paper-craft lovers.

[CSaLM13] propose a method which solves a problem very sim-ilar to ours. They approximate the input surface in a 3D mesh witha small number of planar polygonal faces for fabrication with CNCcutter machines. However their assembly process is very complex:it needs special connectors and takes hours for a single model. Weinstead introduce a constraint on the possible angles between adja-cent faces that will simplify the assembly process.

[CS16] propose an alternative approach for the assembly pro-cess by creating a novel interlocking system with unique joiningfeatures. However they focus their work only on models producedwith CAD tools.

In [ZLAK14] the authors try to solve the problem to representan input surface with Zometool, a mathematical modeling systemused in various areas. Their problem introduces an angle constraint:every Zometool node has a small set of possible directions, andtherefore the angle formed between the edges belongs to a well-known restricted set. However they don’t need to simplify the inputmesh with the goal to obtain a small number of faces on their outputmesh.

Suggested Method, Partial Results and Remaining Work

Since the common tips used on CNC carving machines have com-mon degree angles (like 45◦, 60◦ and multiples), we are focusingon a method which is strictly related to the naive version of March-ing Cubes algorithm [LC87], that generates a surface starting froma scalar field. The generated surface is a triangle mesh, and theset of all possible triangle normals of that mesh is restricted and

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A. Muntoni / Fabrication of Simplified Shapes using Digital CNC Machines

well-known. Because of this, also all the possible dihedral anglesbetween triangles are limited an well-known.

We start generating a regular grid on the Bounding Box of theinput mesh, and for every grid point we check whether the pointis inside or outside the shape, assigning a sign accordingly. Apply-ing the Marching Cubes algorithm to the grid, we obtain a surfacemesh which is a possible solution of our problem (for the reasonswritten in the previous paragraph), but with a big number of polyg-onal faces (we consider adjacent triangles with the same normal asa unique polygonal face), as shown on Figure 3.

Figure 3: Input model (left) and a Triangle Mesh (right), gener-ated by the naive Marching Cubes Algorithm and composed of 408polygonal faces. Colors (RGB) are given by the triangle normals.

The idea of our method is to work on the signs of the regulargrid on which is applied the Marching Cubes algorithm. We aim tomodify, with an heuristic, the signs of some grid points in order toobtain, after re-applying Marching Cubes, a resulting mesh with asmaller number of polygonal faces and which still recalls the orig-inal input shape. We set the grid spacing to the average mesh edgelength multiplied by an user parameter, that will set the "granular-ity" of the final simplified mesh.

We make this putting all the cubes of the grid in a queueand looking if every cube and its neighbors match with aMask of cubes that belongs to a set of Masks. Every Maskof this set describes a combination of adjacent signs (andthen a combination of adjacent triangles generated by March-ing Cubes) that we don’t want to have on our output model,and it also describes how we should change this combinationof signs switching some Points of Interest inside the Mask.

An example of a Mask is shown onthe inset. The shown mask is composedof four adjacent cubes with specifiedsigns on its points, and is composed oftwo sets of Points of Interest (circled inorange and green, respectively). Onlyone of these sets will be selected andswitched on order to change the trian-gulation. Every switch of signs on thegrid points is a local modification on the

triangle mesh (a switch of a point sign reflects to a change of thetriangles generated by its 8 incident cubes of the grid) and thereforewe can locally update at every iteration the informations about the

areas of the polygonal faces of the current mesh. These informa-tions allow us to understand whether a switch is a good move andwhich signs should be switched after we validate a move.

Figure 4: The unfolding of the moai model (up-left of figure 5).

We show on Figure 5 some results using this method. These re-sults have some features that we don’t want on our final results butare inherent of the Marching Cubes Algorithm (like small spuriousfacets between big orthogonal faces). We deal with this problemby providing a Graphic User-Driven tool that allows the user to se-lect and remove some unwanted small polygonal facets keeping theshape manifold and watertight.

We also provide an edge-unfolding heuristic algorithm (Figure4) designed on purpose for our meshes that uses a normal orderingfor the faces and makes some post-processing in order to minimizethe final number of connected components of unfolded faces, andmaximize the average area of every connected component.

The results shown are improvable in different aspects. We needto improve the method on symmetric models, and we also needto focus on the post processing in order to transform the GraphicUser-Driven tool for the deletion of small faces in a full-automaticapproach. Another idea is to introduce a texturing-manager to themethod, in order to have a mapping from a texture attached to theinput model to a texture which could be printed and glued on thefinal simplified model. After dealing these problems, we will ableto finally fabricate our results.

Another idea is to link this work to the other work briefly intro-duced on section 5. The idea is to add a constraint on our simplifi-cation, making sure that our output model is inside the surface, andthen try to construct non-overlapping height-field blocks with flatbases relying on the faces of the simplified internal structure. Wethen try to cover the entire input surface with these blocks. Ensur-ing that every block is an height-field with a flat bases enables themanufacturability of these blocks with CNC milling machines.

3-Axis Milling Decomposition

I am also working on a project for the volumetric decomposition ofshapes in blocks that are manufacturable using 3-axis CNC millingmachines. These machines are cheap and they allow to use var-iegate materials. However, as shown on Figure 6, a model needsto observe some properties to be manufactured by a 3-axis CNC

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A. Muntoni / Fabrication of Simplified Shapes using Digital CNC Machines

Figure 5: Some results.

milling machine. Therefore, we propose a semi-automatic fabri-cation process where all the blocks that compose the input modelare manufactured separately with a milling machine, and then theassembly process requires to glue together the fabricated blocks.The algorithm therefore has the goal to find a consistent, non-overlapping decomposition of a free-form input shape, making surethat every single block of the decomposition observes the con-straints given by the manufacturability with a 3-axis milling ma-chine. An example is shown on Figure 7.

Figure 6: Some examples of manufacturable (a) and non manufac-turable (b and c) pieces using 3-axis CNC milling machines. Thepiece shown on (b) has undercuts that cannot be reached verticallyby the cutter without removing undesired portions, on (c) insteadthe piece has features on the back side of the piece while a 3-axismillable block requires flat-base models.

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[CSaLM13] CHEN D., SITTHI-AMORN P., LAN J. T., MATUSIK W.:Computing and fabricating multiplanar models. In Computer graphicsforum (2013), vol. 32, Wiley Online Library, pp. 305–315. 2

[LC87] LORENSEN W. E., CLINE H. E.: Marching cubes: A high res-olution 3d surface construction algorithm. In ACM siggraph computergraphics (1987), vol. 21, ACM, pp. 163–169. 2

Figure 7: The decomposition of the cube spike model in 6 blocksmanufacturable using 3-axis milling machine. The inset, if present,can be an internal bearing structure (a scaffold) or also empty inorder to save material.

[LZ07] LI L., ZHANG Y.: Optimal tool-path generation for 5-axis millingof sculptured surfaces. In Control and Automation, 2007. ICCA 2007.IEEE International Conference on (2007), IEEE, pp. 1207–1212. 2

[MS04] MITANI J., SUZUKI H.: Making papercraft toys from meshes us-ing strip-based approximate unfolding. In ACM transactions on graphics(TOG) (2004), vol. 23, ACM, pp. 259–263. 2

[RA15] RICHTER R., ALEXA M.: Beam meshes. Computers & Graphics53 (2015), 28–36. 2

[SP13] SCHWARTZBURG Y., PAULY M.: Fabrication-aware design withintersecting planar pieces. In Computer Graphics Forum (2013), vol. 32,Wiley Online Library, pp. 317–326. 2

[YDX99] YANG W., DING H., XIONG Y.: Manufacturability analy-sis for 5-axis sculptured surface machining. In Robotics and Automa-tion, 1999. Proceedings. 1999 IEEE International Conference on (1999),vol. 3, IEEE, pp. 2116–2121. 2

[ZLAK14] ZIMMER H., LAFARGE F., ALLIEZ P., KOBBELT L.: Zome-tool shape approximation. Graphical Models 76, 5 (2014), 390–401. 2

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