Planar Structures with Automatically Generated Bevel Joints

Planar structures with automatically generated bevel joints

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As Published 10.1016/j.cag.2018.02.005

Publisher Elsevier

Version Author's final manuscript

Citable link https://hdl.handle.net/1721.1/124169

Detailed Terms http://creativecommons.org/licenses/by-nc-nd/4.0/ ARTICLE IN PRESS JID: CAG [m5G; February 27, 2018;2:49 ]

Computers & Graphics xxx (2018) xxx–xxx

Contents lists available at ScienceDirect

Computers & Graphics

journal homepage: www.elsevier.com/locate/cag

Technical Section

Planar structures with automatically generated bevel joints

∗ ∗∗ Q1 Zhilong Su a, Lujie Chen b, , Xiaoyuan He a, , Fujun Yang a, Lawrence Sass c

a Southeast University, Nanjing, 2 Si Pai Lou, Nanjing, China

b Singapore University of Technology and Design, 8 Somapah Road, Singapore

c Massachusetts Institute of Technology, MA, USA

a r t i c l e i n f o a b s t r a c t

Article history: A generative method based on computer algorithms is proposed to automatically produce parts of planar Received 31 August 2017 structures ready for fabrication. The parts resemble surface patches of the digital model of a structure.

Revised 29 January 2018 Each part is generated with bevel joints on the edges so that part-to-part connection is enabled by fric- Accepted 13 February 2018 tion of the joints. The shape of a bevel joint is determined by the interior angle between two parts, and Available online xxx is modelled by a number of parameters, including the thickness of a planar material in use. The bevel Keywords: joints consist of slanted planes, and in principle when they are assembled, no gap exists on the surface Planar structures of the physical structure. Due to the slanted planes, the joints cannot be fabricated by a laser cutter Bevel joint that can only produce vertical cuts. We experimented the fabrication with a three-axis CNC router and a Digital fabrication 3D printer; both produced accurate and robust parts; however, there is limitation in using a CNC cutter,

1 1. Introduction For large-scale prototyping [8] , i.e. practically any dimension 23 that is beyond one meter, planar structures are one of the most 24 2 Creative design is inevitably influenced by the capability of adopted approaches [9] . They have long been used in architecture, 25 3 production. Many years of research has led designers to recognize construction, and ship building. A typical work process starts from 26 4 that physical artefacts, either in the form of low-fidelity prototypes a digital 3D model; create a representation of the model by 2D 27 5 or high-quality products, give valuable feedback on a conceptual planes in Computer-Aided Design (CAD) software; materialize 28 6 idea and are critical to the success of an iterative design pro- the planes by assigning a thickness; generate slots and joints for 29 7 cess [1] . Methods of rapid physical production are therefore connecting the planar parts; fabricate and assemble the parts to 30 8 valuable to designers of various industries. Automobile, marine produce a physical artefact. Although the process involves a lot of 31 9 and air plane designers make mockups to study the aesthetics, manual work such as drafting in CAD software, it offers a designer 32 10 dynamics, and mechanics of their products [2–4] . Architects make great flexibility in materialize his design. Designers also see oppor- 33 11 scaled models to understand the interaction between a building tunities to automate part of the process and come up with creative 34 12 and an environment [5] . From a designer’s point of view, physical solutions. Sass [10,11] proposed a shape grammar to streamline 35 13 prototypes are best to be created with low cost, in a short time, the generation of interlocking planar parts to produce house 36 14 while having good fidelity and strength because they are supposed models. He used computer-numerically controlled (CNC) machines 37 15 to be used for just-in-time evaluation [6] . The two-sided expec- to fabricate the parts. This work demonstrated preliminary idea on 38 16 tation on prototypes has motivated research in rapid prototyping automation of planar structures, while a shape grammar is not a 39 17 across multi-disciplines. 3D printing as a highly automated rapid computer program but rule sets to guide manual drafting. 40 18 prototyping method has been widely used [7] . A printed artefact Researchers in computer graphics started to look into direct 41 19 obtains a solid surface that closely resembles the original digital physical production in 2006 [12] , where direct means that the 42 20 model. However, 3D printing has limitations, such as low speed parts of a physical artefact are directly generated from a digital 43 21 and limited build volume. In areas where these limitations are model with little human intervention in modeling (drafting). This 44 22 unacceptable, other methods are applied instead. approach also relies on CNC machining but its impact is sub- 45 stantial: Computer algorithms can independently extract relevant 46 shape information to produce the drawings of parts, which are 47 later fabricated using CNC machines. As a result, human error is 48 Q2 ∗ Corresponding author . ∗∗ 49 Corresponding author . largely eliminated from modeling and fabrication. Since then, new

E-mail address: [email protected] (L. Chen). ideas in automating the production of planar structures have been 50

Please cite this article as: Z. Su et al., Planar structures with automatically generated bevel joints, Computers & Graphics (2018), https://doi.org/10.1016/j.cag.2018.02.005 ARTICLE IN PRESS JID: CAG [m5G; February 27, 2018;2:49 ]

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51 ﬂourished. They address various questions such as improving slots 52 and joints for complex models [13] , differentiating importance 53 of planar parts [14,15] , chair-design system [16] , parametric de- 54 sign system [17] , unequally spaced slots [18] , non-perpendicular 55 parts [19] , and bendable parts [20] .

56 2. Related work

Fig. 1. (a) Finger joints of a watertight structured achieved in [25] . (b) Bevel joints 57 This paper is concerned with a type of planar structures of a planar structure. (c) An exploded view of the bevel joints in (b). 58 that is a watertight representation of a digital model. Methods 59 aforementioned produce openly sliced models that are not suit- 60 able for certain applications; for instance, be used as mold for 61 casting. Watertight planar structures are physical models as a 62 complete volume, as opposed to sets of overlapping slices. There 63 is sparse exploration of this method of model production. Chen 64 et al. [21] developed a multiplanar modeler that subdivided a 65 digital model into several planar surfaces. Interior connectors were 66 used to join the planar parts. Due to the connectors, fabrication 67 of a model requires two separate processes, one for the planar 68 parts and the other for the connectors. Song et al. [22] described a 69 coarse-to-fine prototyping system, where the interior of a model is 70 made of low-fidelity planar structures and the exterior is made of 71 high-fidelity 3D printed parts. The complete model is an assembly 72 of 3D printed parts onto the planar structures. 73 While the above ideas from the computer graphics field have 74 accomplished watertight representation of a model, explorations

75 in architecture have found real-world applications of these struc- Fig. 2. Overview of the production system for planar structures bevel joints. 76 tures. Robeller et al. [23] created interlocking planar structures 77 with dovetail joints. The structures were strong and could be used 78 as pavilion in an architecture scale. Dovetail joints are widely We have made an interesting observation based on the work 115 79 used in conventional woodwork, the intellectual contribution of in [25] : Adding one more constraint to a watertight structure, we 116 80 [23] lies in a computational method to calculate the angle of a would get a planar structure without surface gaps. Fig. 1 shows 117 81 dovetail joint from the geometry of adjacent planar parts. The the difference in finger joints between the former achieved in 118 82 authors also applied robotic machining to fabricate the parts using [25] and a type of the latter. Clearly, the latter is more elegant 119 83 bevel cutting to accurately produce the joints along the edges. This and structurally robust. Note that the dovetail joints in [23] are 120 84 process greatly simplifies the production of self-interlocking planar different from those in Fig. 1 (b), which hold an object by friction, 121 85 structures, in which no additional connectors are needed; all parts just like those in Fig. 1 (a). Dovetail joints require much more 122 86 are held together by an interlocking mechanism and friction. complicated fabrication, assembly, and disassembly processes 123 87 Howe [24] applied watertight structures made of finger joints because they rely more on intricate shape for interlocking than 124 88 on concrete casting. He showed feasibility of casting large-scale friction between joints. 125 89 concrete structures that have curves, and demonstrated that form- This paper extends the algorithms developed in [25] to a new 126 90 work (mold) production can be a potential industry application of production system that generates structures without surface gaps 127 91 watertight structures. This work was focused on architecture de- ready for fabrication. The system not only deals with bevel joints 128 92 sign but technical details on how finger joints were created were along the edges of adjacent planar faces but creates accurate shape 129 93 not described. It is clear that to rapidly generate formwork of a around a vertex shared by multiple faces. 130 94 customized shape, it is important to have a fully automatic algo- 95 rithm. Only then, variations in design can be quickly prototyped 3. Principle 131 96 and evaluated. The algorithm must generate a watertight struc- 97 ture that has an interior volume equal to a digital model, which An overview of the proposed production system for planar 132 98 makes methods that focus on exterior representation unsuit- structures based on bevel joints is illustrated in Fig. 2 . The system 133 99 able [21,22] because the interior of an object made by these meth- accepts a triangle mesh model as input, converts it to a polygon 134 100 ods is either occupied by connectors or by supporting structures. mesh, assigns a thickness to each surface patch, and generates 135 101 Chen and Sass [25] developed an intelligent modeling system bevel joints on all edges. 136 102 that is able to process a 2-manifold triangle mesh model and The data-processing pipeline consists of four steps: (1) con- 137 103 generate a corresponding watertight structure made of finger version from a triangle to a polygon mesh, (2) interior angle 138 104 joints. They described the work flow of finger joint generation and determination, (3) bevel joint generation, and (4) vertex formation. 139 105 showed that the principle is applicable not only to edge-to-edge The first step has been described in detail in [25] . Steps 2 to 4 are 140 106 finger joints but also to a vertex shared by several faces. If an described in the following subsections. 141 107 input model has all planar faces perpendicular to each other, the 108 generated physical model will not have gaps on the exterior and 3.1. Interior angle determination 142 109 interior surfaces. Through many experiments, they demonstrated 110 the work steps of the system and produced a solid spiral staircase The interior angle between any two adjacent faces should 143 111 by casting it from plywood mold. The original digital model of the be determined for accurate bevel joint generation. This angle is 144 112 spiral staircase did not have all faces perpendicular to each other; indicated as α in Fig. 3 , where two faces are shown in thickened 145 113 hence, the interior volume has gaps between parts, which subject solid lines. Generally, the angle between two faces is often rep- 146 114 the cast model to undesired surface artefacts. resented by β ∈ [0, π], which does not contain information about 147

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Fig. 3. Side view of two faces. The interior angle α is (a) between 0 and π, and (b) between π and 2 π. The angle β does not indicate interior and is always between

0 and π. The normal vectors, n 1 and n 2 , of the faces point to exterior.

Fig. 4. Perspective view of two faces. n c is the cross product of n 1 and n 2 . v i and v i +1 are the i th and ( i +1) th vertices of face 1.

148 which part of the space is considered as the interior volume; α ∈ π α ∈ π π α ∈ π π 149 nevertheless, β can be calculated directly from the normal vector Fig. 5. Bevel joint generation. (a) (0, /2), (b) ( /2, ), (c) ( , 3 /2), and (d) α ∈ (3 π/2, 2 π). The original polygon surface is indicated by the thickened 150 of the faces, n and n : 1 2 lines. The ﬁrst column shows a mold-making scheme where the interior surface is n · n the original. In the second column, the exterior surface is the original. The third β = 1 2 . arccos (1) column shows the bevel joints in perspective view. | n 1 || n 2 | 151 Then, an empirical procedure described in [25] may be applied to 152 obtain α from β. 153 Alternatively, the relationship between α and β can be deter- 154 mined from n , n , and the intersection of the two faces. 1 2 π − β n (v + − v ) ≥ 0 α = c i 1 i (2) π + β n c (v i +1 − v i ) < 0

155 where nc is the cross product of n1 and n2 , and the intersection is Fig. 6. A type of bevel joint that deals with degeneracy when α = π. 156 denoted by v and v + , the mutual edge of faces 1 and 2. Basically, i i 1 −−−→ , α π − β 157 if nc is in the same direction as vi vi +1 equals to and

158 α ∈ [0, π], as shown in Fig. 4 (a); otherwise, α equals to π + β and distance to withdraw from the extended face 1 so as to produce 180

159 α ∈ [ π, 2 π), as shown in Fig. 4 (b). an indented tooth. Fig. 5 illustrates the offset distances against 181 α 160 Note that the above relationship is valid regardless of which several values of . The mold-making scheme is shown in the 182

161 face is deﬁned as face 1. If we interchange faces 1 and 2 in ﬁrst column, which leads to a physical model larger than that pro- 183 duced by the second scheme (the second column); nevertheless, 184 162 Fig. 4 (i.e. reverse the sign of nc ), vi and vi +1 will be interchanged

163 as well because the direction of a face’s normal vector is dictated the shape of the bevel joints generated by both schemes is the 185

164 by the sequence of the vertices. same, as shown in the third column. 186 Eqs. (3) and (4) represent, respectively l and s in terms of α, T , 187

165 3.2. Bevel joint generation and m, a parameter denoting the scheme. m is 1 for mold making, 188 and −1 otherwise. Both l and s are signed values. They directly 189

166 The shape of a bevel joint is related to three parameters at the inform how an algorithm should generate the thickened face 1. 190

167 vicinity of a mutual edge between two faces. The ﬁrst parameter is Since the bevel joints on faces 1 and 2 are complement to each 191

168 the thickness of the material in use, T . The other two parameters other, the offset distances of face 2 can be deduced from l and 192

169 are the offset distances, l and s , for creating the teeth of a bevel s. Where face 1 has a protruding tooth, face 2 should have an 193

170 joint. Due to the thickness of the material, the original polygon indenting tooth, and vice versa. 194 171 surface is either the interior or exterior of a physical model; mT l = , α ∈ (0 , π ) AND α ∈ (π, 2 π ) (3) 172 it cannot be both. Hence, there are two production schemes in sin α 173 generating bevel joints. The ﬁrst scheme is mold making, in which π 195 174 the interior of a physical model is set as the original surface. The s = −mT tan α − , α ∈ (0 , π ) AND α ∈ (π, 2 π ) (4)

175 model can be used as mold for casting. In the second scheme, the 2 176 exterior is set as the original surface. When α takes on value 0 or π, degeneracy is found in the above 196 177 The offset l deﬁnes the distance that face 1 extends in the equations. While not much can be done for α = 0 , one may force 197

178 direction perpendicular to n1 and the mutual edge so as to pro- an algorithm to generate a type of bevel joint shown in Fig. 6 for 198 179 duce a thickened face 1 of thickness T . The offset s deﬁnes the α = π. 199

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Fig. 7. Vertex formation between two adjacent edges. (a) An initially processed edge. (b) A subsequently processed, adjacent edge. (c) Intersection of the bevel joints on the two edges. (d) Shape of the vertex on complete.

200 3.3. Vertex formation

201 A polygon mesh model has many mutual edges between ad- 202 jacent faces. They are processed sequentially to generate bevel 203 joints. The order of processing is not ﬁxed, and in general, differ- 204 ent sequences cause some variation in the bevel joints on certain 205 faces but they do not result in any change in the overall shape of 206 a model. 207 It is crucial to keep track of processed edges and to record the 208 parameters, such as α, l , and s , used to generate the bevel joints on 209 each edge because the shape of a vertex is mutually determined 210 by the bevel joints on adjacent edges. Fig. 7 shows the formation 211 of a vertex on a particular face. An edge is processed initially to 212 generate some bevel joints in (a). The shape of the vertex is not 213 fully determined at this stage except for a slanted plane over the 214 vertex. In (b), an adjacent edge is processed, which also generates Fig. 8. Vertex formation among three faces. (a) The front vertex of a tetrahedron is 215 a slanted plane as part of a bevel joint. In (c), an intersection common to three faces. (b) Face 1: bottom face; faces 2 and 3: side faces. (c) and

216 of the slanted planes produced by the two edges is calculated. (d) Results based on a particular processing sequence. (e) and (f) Results based on 217 Finally, the shape of the vertex is fully determined as shown in (d). a different configuration of protruding and indenting bevel joints. 218 Vertex formation may produce different-shaped parts at a 219 mutual vertex based on different processing sequences and con- 220 figurations of protruding and indenting bevel joints. Fig. 8 (a) and 4. Results and discussions 250 221 (b) show a tetrahedron whose front vertex is common to faces 222 1, 2, and 3. Figs. 8 (c) and (d) show the results of a particular 4.1. Convex shape 251 223 processing sequence. Edges 1–2 and 1–3 are processed first. Face 224 1 obtains a protruding bevel joint at the mutual vertex on both The production system of planar structures based on bevel 252 225 edges. Edge 2–3 is processed later and face 3 obtains a protruding joints has been tested to generate various objects. Fig. 9 shows 253 226 bevel joint near the vertex. At the vicinity of the thickened vertex, an example of a soccer model. The original digital model, (a) and 254 227 face 1 occupies most of the space and face 2 occupies the least (b), consists of several pentagons and hexagons. Bevel joints were 255 228 space. However, if the protruding and indenting bevel joints are generated on all edges shown in (c). Fig. 9 (d) shows 32 parts of 256 229 generated differently on the faces, the shape of the parts would the model, 12 pentagons and 20 hexagons. As this was an initial 257 230 vary at the vertex. Fig. 8 (e) and (f) show an example, in which test of the system, we did two ad hoc treatments to the parts 258 231 each face obtains a protruding and an indenting bevel joint. In this to facilitate assembly. Firstly, a hole was created on each face by 259 232 particular case, different processing sequences of edges 1–2, 1–3, manual drafting so that we could reach inside the model if needed 260 233 and 2–3 would end up in the same results. The three faces occupy during assembly. Secondly, at the vicinity of all vertices an extra 261 234 equal amount of space at the vertex. amount of material was removed to increase relief between parts. 262 235 From a pure geometry point of view, the overall shape of Ideally, to fabricate a slanted bevel joint a five-axis CNC cutter 263 236 the model is unchanged regardless of the shape variation of the with bevel cutting capability should be used; however, we were 264 237 parts; however, in practice if a model has many sharp vertices, constrained by availability of equipment and used a three-axis 265 238 we recommend a vertex-sharing strategy illustrated in Fig. 8 (e) CNC router as shown in Fig. 9 (e). We applied an incremental 266 239 and (f). This strategy does not produce necks at vertices that are tool path to mimic bevel cutting. The red-highlighted path is an 267 240 present in Figs. 7 (d) and 8 (c). A thin neck may be formed when approximate to the slanted surface. It was achieved by several 268 241 the indent of a joint is relatively large compared to the angle of cuts at incremental z levels. At each level, the tool path was offset 269 242 a vertex. Such a neck can be easily broken during assembly due slightly with respect to the previous level. Consequently, the red 270 243 to friction force, while vertex sharing eliminates thin necks. More path was produced by a number of steps. The more the steps, i.e. 271 244 importantly, the vertices of parts thus produced, e.g. Fig. 8 (e), are the more the levels, the better is the approximation to a bevel cut, 272 245 not subject to large friction force because at the mutual vertex while the more time would be spent in cutting. Fig. 9 (f) shows 273 246 the bevel joint of each part is relieved by a neighbouring part and some fabricated parts ready for assembly, and (g) is a picture of 274 247 is shaped into a small tetrahedron. When several tetrahedra form the assembled structure. A digit model can be scaled easily; the 275 248 the mutual vertex, they do not bite into each other, e.g. Fig. 8 (f), physical structure is 55 cm in diameter, around 2.5 times larger 276 249 therefore they experience little friction. than a real soccer. 277

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Fig. 10. Comparison of a bevel joint structure with a ﬁnger joint structure. (a) 3D printed parts of the pyramid model shown Fig. 2 . (b) Partly assembled pyramid. (c) Fully assembled pyramid. (d) A laser-cut pyramid made of ﬁnger joints, generated by methods described in [25] .

should be used; however, this also suggests that a planar structure 290 fabricated by a CNC cutter is likely to have gaps on the surface. 291 3D printing does not have the limitation of CNC cutting caused 292 by the radius of a drill bit. The resolution of 3D printing can 293 reach sub-millimeter even on low-cost, consumer 3D printers. 294 We printed the parts of the pyramid model shown in Fig. 2 , and 295

obtained a 65 × 65 × 60 mm 3 tightly assembled physical model. 296 Fig. 10 (a–c) show the assembly process. Fig. 10 (d) shows a laser- 297 cut pyramid made of finger joints, generated by methods described 298 in [25] . Comparison of (c) and (d) demonstrates the advantage of 299 bevel joints over finger joints when the angle between adjacent 300 parts is not 90 °. In (d), the connection between the base and the 301 side parts exhibits visible gaps, while in (c) the connection has no 302 gap. 303 The top vertex is generated properly in (c) but not in (d) as 304 finger joints are not suitable in dealing with a vertex of acute 305 angles. Such vertex is problematic to bevel joint generation as well 306 because of the neck issue discussed in Section 3.3 . In practice, we 307 implemented a vertex-sharing strategy: if a vertex is of an acute 308 angle on all associated faces, vertex sharing will be applied on 309 this vertex; otherwise, a regular bevel joint will be generated. For 310 example, the top vertex of the pyramid is associated with four 311 side faces, and each face has an acute angle on the vertex; hence, 312 vertex sharing is applied on the top vertex. In contrast, the front 313 vertex shared by the base and two side faces in (c) is of a right 314 angle on the base; hence, vertex sharing is not applied on the front 315

Fig. 9. Production of a soccer model. (a, b) Digital model. (c) Processed model with vertex, and a regular bevel joint is generated. In our implemen- 316 bevel joints. (d) Model parts. (e) Fabrication of a part using a three-axis CNC router. tation, we used 90 ° as a threshold to determine whether vertex 317 (f) Parts ready for assembly. (g) Assembled model. The arrows indicate arcs at the sharing should be applied on a particular vertex but this value can 318

intersection of two linear toolpaths produced by a drill bit. be changed. If parts of a structure are relatively thick comparing 319 to their sizes, female bevel joints tend to extend inward by a large 320 278 On close inspection of Fig. 9 (g), we see that there are gaps distance and produce thin necks; then increasing the threshold 321 279 between the bevel joints. This is partly due to the fact that the will force vertex sharing to be applied on more vertices, and sub- 322 280 friction between the joints were too high and the teeth of adjacent sequently reduce the number of thin necks. A suitable threshold 323 281 faces could not fully contact each other. To reduce the friction, can be determined by observing the generated parts. If too many 324 282 one can loosen the tolerance between the joints but this will thin necks are produced, increase the threshold. Based on our ex- 325 283 affect the strength of the structure, which is further discussed in perience, it is rarely needed to increase the threshold above 120 °. 326 284 Section 4.4 . Another factor is that the drill bit of the CNC cutter 285 is cylindrical; so at the intersection of two linear toolpaths an arc 286 is produced. The radius of the arc is the radius of a drill bit. The 4.2. Concave shape 327 287 arrows in Fig. 9 (g) indicate some of these arcs. The arc prevents 288 a male and a female bevel joint from fully contacting each other, If given a concave model, the algorithm may produce parts 328 289 leaving a small gap in between. To reduce the gap, a small drill bit that are diﬃcult to fabricate by a CNC router. Two examples are 329

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Fig. 11. Concave surface. (a) Face 1 is concave as its angle with faces 2 and 3 crosses π/2. (b) Thickened face 1 in (a). (c) An edge of face 1 is common to faces 2 and 3. (d) Thicken face 1 in (c).

Fig. 13. Bunny produced by 3D printing. (a) Simpliﬁed bunny model that has 35 planar faces. (b) Simulated model assembly with bevel joints; each face is rendered in a different color. (c) Partly assembled model. (d) Fully assembled model.

generated structures with bevel joints. The parts were produced 344 by 3D printing shown (c) and (d). Fig. 12 (e) is the assembled 345 model viewed in two different angles. As can be seen, the physical 346 structure closely resembles the input model, and there is no gap 347 at the intersection between any two parts. 348 Fig. 13 shows another model that has concave surface patches. 349 The model is a simplified bunny with 35 planar faces. Our method 350 cannot generate a desktop-sized bunny based on the original 351 bunny model that has smoothly varying surface patches because 352 of a number of limitations discussed in Section 4.5 . Parts of the 353 simplified model were 3D printed and were assembled in 30 354 minutes. One does not have to follow a specific assembly sequence 355 as each part can be attached to the structure without being 356 blocked by other parts. For example, in Fig. 13 (c) a front part on 357 the lower body has not been attached, while its top, left, and 358 right neighbours do not obstruct its assembly path; hence, it can 359 be attached by pressing into the structure. The fully assembled 360 model, Fig. 13 (d), exhibits small gaps at some joints. They are 361 caused by warping of the parts during 3D printing. When printing 362 a flat plate, the warping artefact is a common problem in most 363

Fig. 12. Stairs produced by 3D printing. (a) Original model. (b) Processed model. (c) consumer 3D printers. Despite of the issue of fabrication precision, 364 3D print of a part. (d) Several printed parts. (e) Assembled stairs. the structure is fully closed on exterior and interior surfaces. 365

330 shown in Fig. 11 . In (a), the interior angle between faces 1 and 4.3. Model assembly 366 331 2 is larger than π, while that between faces 1 and 3 is smaller 332 than π. As a result, face 1 has some bevel joints faced upwards A physical structure produced by the proposed method is an as- 367 333 and others faced downwards as shown in (b). In (c), face 1 has an sembly of planar parts. The interaction between any two adjacent 368 334 edge partly common to both faces 2 and 3. The angle of faces 1–2 parts is through bevel joints along their mutual edge. The parts 369 335 and 1–3 crosses 3 π/2, which produces an edge of face 1 difficult are connected to each other by pressing the bevel joints into one 370 336 to fabricate, as shown in (d). Face 1 in (b) and (d) can only be another. To assemble each part to the structure, there is a direct 371 337 CNC cut by two separate passes. Prior to the second pass, face 1 line of motion, coherent to the orientation of a bevel joint; hence, 372 338 must be flipped so that an inversely slanted bevel joint can be no specific assembly sequence is needed in principle. This is dif- 373 339 produced; however, this would involve manual realignment and ferent from structures built with an interlocking mechanism [22] , 374 340 would affect the accuracy of fabrication. Hence, we did not use in which some parts can block the assembly motion direction 375 341 the three-axis CNC router to fabricate concave shapes. of other parts. Without an interlocking mechanism, the proposed 376 342 We used 3D printing to avoid the aforementioned fabrication structure holds itself by friction between parts. Friction also affects 377 343 problems. Fig. 12 (a) and (b) show a digital model of stairs and the the model strength ( Section 4.4 ). If the friction between parts is 378

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◦ ◦ Due to this factor, sharp corners within [ −15 , 15 ] on a model 427 may be generated with an inaccurate shape by our method. 428

5. Conclusion 429

This paper describes a computational method for automatic 430 generation of bevel joint planar structures. The method is incor- 431 porated in a production system that can process triangle mesh 432 data, the most common format used to represent a 3D object. 433 Planar surfaces are extracted from the mesh data, and the angle 434 between adjacent faces is calculated in a range of 2 π. Slanted 435 Fig. 14. Three factors that affect model strength. bevel joints are generated based on the angle, and their shape is 436 affected by the processing sequence of the edges and the settings 437 of protrusion and indent. The shape of a vertex of a particular face 438

379 too large caused by inappropriate fabrication tolerance, a structure is mutually determined by the bevel joints of the edges connected 439

380 cannot be assembled as the bevel joints do not ﬁt into each other. to the vertex. A vertex common to several faces can be processed 440

381 Pragmatically, we ﬁnd it ease to start with large parts, and then without any problem; when assembled the vertex is formed by 441

382 gradually add parts that share an edge to the partly assembled the parts at its vicinity. 442

383 structure. However, that is only a heuristic approach, and it does Part fabrication of a convex model can be achieved by a ﬁve- 443

384 not imply that other sequences are not feasible in practice. axis CNC cutter with bevel cutting capability, while we have also 4 4 4 demonstrated that it is feasible to use a three-axis CNC router by 445 385 4.4. Model strength generating multiple passes to fabricate the slanted bevel joints. 446 However, if a model has concave surfaces, some parts will obtain 447 386 Model strength is affected by three factors, as shown in Fig. 14 ; bevel joints facing reverse directions, which poses a challenge in 448 387 all of them have some bearing on the friction between joints. Large using a CNC cutting machine to produce the parts. In this study, 449 388 friction lead to a structurally strong model, while small friction we resort to additive manufacturing, i.e. 3D printing, to fabricate 450 389 leads to a weak model. Increasing the part thickness increases the parts with aforementioned features. 451 390 friction and the weight of the model but a thick part may induce Bevel joint planar structures have already found potential 452 391 faulty bevel joints, which is discussed in Section 4.5 . The width of applications in architecture, model production, and building con- 453 392 joints determines the number of joints along an edge. The bigger struction. With the proposed method, people can rapidly produce 454 393 the number, the larger the contact area of two adjacent parts, and accurate and high-quality planar structures with little human 455 394 the larger the friction; therefore, a smaller width (more joints) intervention in part modeling. This may lead the technology into 456 395 leads to a stronger model. Fabrication tolerance directly affects even more impactful application areas. 457 396 the friction between joints. We found that a tolerance between 397 0.1 and 0.2 mm could produce reasonably strong structures using Acknowledg ments 458 398 plywood (CNC router fabrication) or polylactic acid (3D printing) 399 as the part material. A suitable tolerance should be calibrated for This work is funded by the International Design Center re- 459 400 different materials and fabrication methods when ﬁxing the other search grant IDG21500107 at Singapore University of Technology 460

Q3 401 two factors. In general, the part thickness and width of joints and Design. 461 402 are user speciﬁed, whereas the fabrication tolerance is inherently

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Please cite this article as: Z. Su et al., Planar structures with automatically generated bevel joints, Computers & Graphics (2018), https://doi.org/10.1016/j.cag.2018.02.005

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