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Research Papers

# Design Patterns of Soft Products Using Surface FlatteningOPEN ACCESS

[+] Author and Article Information
Dongliang Zhang

Zhejiang University,
Hangzhou 310058, China
e-mail: dzhang@zju.edu.cn

Jituo Li

Mechanical Engineering Department,
Zhejiang University,
Hangzhou 310027, China
e-mail: jituo_li@zju.edu.cn

Jin Wang

State Key Laboratory of Fluid Power and
Mechatronic Systems,
Zhejiang University,
Hangzhou 310027, China
e-mail: dwjcom@zju.edu.cn

1Corresponding author.

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received April 24, 2017; final manuscript received February 8, 2018; published online April 26, 2018. Assoc. Editor: Yan Wang.

J. Comput. Inf. Sci. Eng 18(2), 021011 (Apr 26, 2018) (8 pages) Paper No: JCISE-17-1084; doi: 10.1115/1.4039476 History: Received April 24, 2017; Revised February 08, 2018

## Abstract

In this paper, we present a pattern development method for soft product design. We utilize a surface fattening method based on a mass-spring model to create 2D patterns unfolding from a three-dimensional (3D) model. Multilevel meshes are proposed to expedite the flattening process, and a boundary optimization method is employed to guarantee 2D patterns can be sewn well. We apply the proposed method to the design of real soft products. Experimental results show that it can deal with complex surfaces efficiently and robustly, and manufactured products are satisfactory.

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## Introduction

In the process of designing 2D patterns from a three-dimensional (3D) model, the most important technology is surface flattening. Surface flattening has wide applications in soft product design, including plush toys, garments, and shoes and upholstery. Even in mechanical engineering, pattern development of sheet metal components is a prerequisite for manufacturing sheet metal with a free-form surface [1,2]. The flattening process of sheet metal, which is the inverse process for sheet metal forming, improves the efficiency of design, enhances the quality of sheet metal, and provides the useful information of the optimum blank including deformation and stress concentration. In practice, to fabricate 2D patterns into a product with the expected 3D shape, physical properties of materials should be taken into consideration. Therefore, physically based methods, or energy-based flattening methods, are preferred in the product design industry. Although much research has been conducted with regard to solving the flattening problem, existing physically based flattening methods [35] have some limitations for the complex surface, which has a large and irregular mesh with many triangles or holes. First, the computational time increases dramatically when the number of triangles increases. Second, large distortion and overlapping tend to occur easily for a large and irregular mesh.

To produce a soft product, first the material is cut into pieces with shapes of 2D patterns, and then the 2D pieces are sewn together to form a 3D shape. In the sewing process, usually a pair of boundary lines to be sewn together should be of the same length to guarantee that two patterns are sewn well and the resulting shape is satisfactory. Some length preserved flattening algorithms [69] have been studied for product design and surface parameterization, but they usually make the boundary length of a 2D pattern the same as that of a 3D surface. For a soft product, although we require that the boundary lines to be sewn together should have the same length, the boundary length of a 2D pattern is not necessarily the same as that of a 3D surface, since the material will be deformed when the real product is produced. For example, for a plush toy, the surface will be deformed after the filling material is stuffed. Therefore, optimization is needed to adjust the boundary lengths of 2D patterns.

To solve above problems, we present a new surface flattening algorithm using a hierarchical mesh structure. In the method, a basic flattening method based on a mass-spring model is employed, and a hierarchical mesh structure is proposed to deal with complex surfaces with large and irregular meshes. Then, a boundary optimization technique is introduced to automatically adjust the boundary lengths of 2D patterns to facilitate the sewing process and guarantee the quality of the resulting product.

The main contributions of the study are as follows:

1. (1)A surface flattening method using a hierarchical mesh structure is proposed to expedite the flattening process and improve the quality of the flattening result.
2. (2)A boundary optimization technique is proposed to adjust the boundary lengths of 2D patterns to make them suitable for sewing, so as to enhance the manufacturability of the product.

The structure of the paper is as follows: Sec. 2 reviews related work. In Sec. 3, we present a basic flattening model using a mass-spring model. In Sec. 4, a hierarchical mesh structure is employed to expedite the process of flattening complex surfaces. Section 5 presents a boundary optimization method to make two boundary lines to be sewn together having the same length. In Sec. 6, experimental results are presented, and finally we summarize the paper in Sec. 7.

## Related Work

Surface flattening is a subject widely studied in the field of computer-aided design and computer graphics. In the computer-aided design community, physically based flattening methods are usually used in order to take into account of material properties. In the computer graphics community, surface flattening is mainly used for texture mapping, which requires converting a 3D surface into a 2D mesh. The flattening procedure used for texture mapping is also known as surface parameterization [1016]. In the methods, the preservation of angles during mapping is of major concern, but the lengths of mesh edges are not preserved and the material properties are not considered. Therefore, the surface parameterization methods are seldom used for product design in the industry.

Most physically based flattening methods use a variety of an energy model. McCartney et al. [3] and Wang et al. [4] presented surface flattening algorithms based on the energy model to design patterns of 3D garments. Li et al. [17] proposed a planar mesh flattening algorithm introducing an inverse forming approach based on physical plastic deformation of metal materials. The algorithm flattens a mesh onto a plane with lower area distortion and minimizes the angular distortion. Zhong and Xu [5] introduced a method of generating 2D flat patterns from a 3D triangulated surface by opening the bending configuration of pairs of adjacent triangles. In the method, a 3D triangulated surface is modeled with a mass-spring system that simulates the surface deformation during flattening, and an unwrapping force field is built to drive the mass-spring system to a developable configuration through the numerical integration. Skouras et al. [18] proposed a computational approach for interactive design of inflatable structures. They presented an automatic physically based pattern generation method, combining fast simulation based on the tension field theory and constraint optimization. Liu et al. [1] proposed an energy-based surface flattening method using a simplified mass-spring model for flat pattern development of sheet metal components. Liu et al. [2] developed a flattening system for sheet metal with free-form surfaces, which is also based on a planar mass-spring model. These existing energy-based flattening algorithms used for product design are efficient and produce good results for relatively simple surfaces. However, when a surface becomes complex, these methods do not guarantee the preservation of the metric structure of the resulting 2D mesh or even its validity. In addition, the computational time increases dramatically.

In contrast to the methods of flattening a 3D surface into a 2D pattern, Aono et al. [6] presented an approach of fitting a 2D pattern to a 3D surface to design 3D broadcloth composite parts, in which a Tchebychev net model is employed to simulate the deformation of woven fabrics into a specific 3D shape. Wang [8] presented an approach, WireWarping, to compute a planar piece with length-preserved feature curves from a 3D piecewise linear surface patch. The method simulates warping a given 3D surface patch onto a plane with the feature curves as tendon wires to preserve the length of their edges. During warping, the surface-angle variations between edges on wires are minimized so that the shape of a planar piece is similar to its corresponding 3D patch. Zhang and Wang [19] extended the WireWarping method by introducing a new type of feature curve named elastic feature, which brings flexibility to shape control of the resultant 2D patterns. These methods try to preserve the boundary lengths of 2D patterns to make them the same as those of 3D surfaces. In our method, we allow the lengths of boundary lines to stretch by considering that the boundary lines will be deformed after wrapping a 2D pattern into a 3D surface for a soft product.

The hierarchical mesh method has been successfully used in computational sciences. Muller [20] proposed a multigrid-based method to significantly speed up the convergence of a position-based dynamics approach to process nonlinear constraints and constraints based on inequalities. Zhang and Yuen [21] presented a fast cloth simulation method using multilevel meshes. Some authors [10,20,22] proposed hierarchical parameterization techniques, which use mesh multiresolution structures to speed up the parameterization process for large models. Sheffer et al. [16] extended an angle-based flattening method [23] by combining sophisticated numerical tools with a multiresolution approach to achieve maximal speedup and generate angle preserving low stretch parameterizations of huge meshes. In these methods of surface flattening, the hierarchical mesh structure is mainly used for the methods of surface parameterization.

In this study, we apply the hierarchical mesh structure to a surface flattening method based on a mass-spring model. A basic flattening method is used to expand a simplified 3D surface into a mesh, and multilevel meshes are exploited for a coarse-to-fine procedure to obtain a fine flattening result. To make the flattened 2D patterns suitable for sewing, a boundary optimization method is proposed to automatically adjust the boundary lengths of 2D patterns.

## A Basic Flattening Method

Previously, we presented a flattening method based on a mass-spring model in Ref. [24]. In this study, the method is used as a basic method to flatten a simple triangular mesh, and we extend the method by using multilevel meshes to deal with a large mesh with many triangles and accelerate the flattening process. To keep the integrity of the paper, in this section, we briefly introduce the basic flattening method.

###### A Mass-Spring Model.

In the method, we use a planar triangular mass-spring model to flatten triangulated surfaces. The model consists of many virtual masses, each of which links to its neighbors, as shown in Fig. 1. The linkages between neighbors are achieved by two types of springs: tension spring and crossed spring. A tension spring links a mass to its directly connected masses. It causes the strongest internal force, which resists in-plane tension or compression. The stretching properties of the material can be reflected in the tension springs. A crossed spring crosses an edge and links two masses that are on two triangles that share the edge. The bending properties of the material can be reflected in the crossed springs.

Based on laws of motion, the equations that govern the mass-spring model can be set up. The acceleration of each mass is determined by the equation of force: F = ma. For each mass, the total force is the sum of the force due to each spring given by Hook's law (F = k·Δl, where k is the spring constant and Δl is the displacement of the spring from its original length), damping, and external forces. According to the standard equations for position and velocity, a set of differential equations are obtained [24]. To solve the differential equation, we use the Verlet method, which offers greater stability than the Euler method at no significant additional cost over the Euler method.

###### Flattening Procedure.

In the method, a concept of triangle strips is introduced [24,25]. Using triangle strips, the efficiency of flattening can be improved, and the possibility of overlapping can be reduced. The procedure of the flattening algorithm is as follows: Starting from a central triangle strip, a 3D mesh is expanded step by step using triangle strips. At each step, one or two adjacent triangle strips are flattened. A process of energy relaxation will be done in a triangle strip before it is added into the part that has been flattened. Energy relaxation is conducted using the differential equations mentioned in Sec. 3.1. The relaxation process will ease or disperse localized high strains by doing several flattening iterations to alter the positions of vertices. After the triangle strips are added into the flattened part, global energy relaxation will be done to make the whole mesh have an optimal shape.

## Flattening Using Multilevel Meshes

The basic flattening method works well for meshes with a relatively small number of triangles. For larger meshes, the flattening process becomes quite time-consuming and the resulting 2D mesh tends to be easily flipped. To efficiently expand large meshes with hundreds of thousands of triangles, we propose a flattening procedure using multilevel meshes.

The basic idea of the hierarchical approach is to reduce the problem size, then solve the smaller problem, and finally derive the solution to the original problem using multilevel meshes. In this work, we follow this approach to solve the flattening problem for the large mesh with many triangles. The procedure of the flattening method using multilevel meshes is divided into three successive steps:

• Step 1: Construct multilevel meshes using 3D mesh simplification.

• Step 2: Unwrap the simplified coarse mesh using the basic flattening algorithm.

• Step 3: Obtain the final flattened mesh using a coarse-to-fine procedure based on the flattened coarse mesh.

The details of step 1 will be described in Sec. 4.1, and the flattening of the coarsest mesh is conducted based on the method in Sec. 3.2. In step 3, the coarse mesh will be gradually recovered into the original dense mesh using a process of coarse-to-fine construction, which will be described in Sec. 4.2. Once a planar denser mesh is obtained, energy relaxation will be performed.

Figure 2 shows the process of flattening using multilevel meshes. In the method, an initial mesh M = M0 will be simplified into multilevel meshes (M0, M1,…, Mn) with different vertex numbers. A coarser mesh Mi is obtained by applying a sequence of successive edge collapse transformations from a denser mesh Mi−1. The number of levels depends on the number of vertices of the initial mesh. In our study, we set the number of vertices of the mesh Mi to be 30% of the number of vertices of the mesh Mi−1, and the minimal number of vertices of the mesh Mn is greater than 100.

In the last step, the flattened 2D mesh will be gradually transformed to a denser mesh until the final mesh has the same topology as the initial 3D mesh M0. In each step of transformation, energy relaxation will be conducted to ease the localized high strains. In our method, we use multilevel meshes to make the flattened mesh gradually transform to the final result. Theoretically, only two levels of meshes are needed to get the flattening result. However, for two-level meshes, the position of the vertex changes dramatically from the coarse mesh to the fine mesh. It will result in much longer time for energy relaxation, and the flattening mesh suffers large distortion.

###### Construct Multilevel Meshes.

Multilevel meshes are constructed using mesh simplification, which is performed through a sequence of edge collapse operations. In the mesh simplification method, we use the edge length as the criterion to select the edge to collapse at each iteration. That is, edge collapse starts from the shortest edge, and the sequence is in the order of edge lengths. The order of edge lengths will be updated after each edge collapse operation. In the study, we introduce three modifications to the typical edge selection procedure aimed at reducing the parameterization distortion and speeding up the parameterization process.

The first modification is aimed at avoiding extreme angles and flipped triangles during simplification. Extreme angles and flipped triangles may cause numerical problems during the coarse mesh flattening as well as the reconstruction process, slowing down the procedure. Therefore, during simplification, we disallow collapses that introduce extreme angles or flipped triangles. Based on experiments, we set the limit as 2 deg. Thus, after each edge collapse, all the angles in adjoining triangles have to be in the range between 2 deg and 178 deg.

The second modification is the method of calculating the length of newly created edges. When an edge is collapsed, one vertex will be removed, and its linked edges will be modified. Before computing the lengths of new edges, all triangles connecting to the vertex to be removed will be expanded into the 2D plane by keeping the lengths of edges linking the point and preserving the angle ratios of all triangles. Then, the actual length of the new edge is set as the distance of two points in the 2D plane. The actual length of the edge will be used in the process of flattening and energy relaxation. The method of computing the edge length can reduce the distortion of the flattened coarse mesh, and speed up the coarse-to-fine process. For a developable surface, this method will make the simplified surface “developable.”

The third modification is to build the mapping between the coarse mesh and the denser mesh. In the process of edge collapse, when one vertex is removed, the vertex will be mapped onto the coarse mesh. Since we unfold the triangles connecting the vertex into the 2D plane when we compute the length of a new edge, we can find the triangle in which the vertex is mapped, then, calculate its barycentric coordinate related to the triangle. For instance, in Fig. 3, P is the vertex to be removed. Since it is located on the triangle P0P3P4, it will be mapped onto the triangle. By saving the information of the barycentric coordinates of removed vertices and its mapped triangle, the original topology of the mesh will be kept once the coarsest mesh is refined to the original mesh.

###### Coarse-to-Fine Reconstruction.

After obtaining the coarsest mesh, the mesh is flattened using the basic flattening procedure. The final stage of the algorithm uses the flattening of the coarse mesh to compute the flattening of the original mesh using a coarse-to-fine reconstruction procedure.

The coarse-to-fine procedure gradually uses the denser mesh Mi−1 to replace the flattened coarser mesh Mi based on the mapping information until the planar original mesh is obtained. Once the planar denser mesh is obtained, iterations of smoothing using energy relaxation are performed. It updates the planar coordinates of the vertices of the mesh. The relaxation is constrained. Namely, if moving the vertex causes a triangle flip, the vertex will be left in place.

The multilevel flattening method quickly expands the 3D mesh into a two-dimensional (2D) mesh. It significantly reduces the time of flattening compared with the direct flattening method without using the hierarchical mesh structure. The vertices of the fine mesh are gradually relocated using global smoothing, and the accuracy of the flattening result is guaranteed.

## Boundary Optimization

In the section, we introduce the concept of sewing relation, which specifies pairs of boundary lines to be sewn together. Based on the sewing relation, a boundary optimization method is proposed to adjust the boundary lines of 2D patterns so that 2D patterns can be sewn well to produce a satisfactory product.

###### Sewing Relation of Two-Dimensional Patterns.

Since 2D patterns are flattened from 3D surface patches, the sewing relation between different patterns can be automatically obtained based on the adjacent relation of the 3D patches. Figure 4 shows the sewing relation of 2D patterns of a toy.

The sewing relation is important because the producer has to know how the patterns are sewn together before manufacturing a product. Normally, for two boundary lines, which will be sewn together, their boundary lengths should be equal. However, because the 3D surface patches of a product are not developable, the surface flattening algorithm does not guarantee the preservation of the boundary lengths. Furthermore, two boundary lines of a pair of patterns to be sewn together may have different lengths since there are deformations in the flattened meshes. To make sure that two boundary lines are sewn well and the resulting product has a satisfactory shape, it is necessary to adjust the boundary lengths to ensure they have the same length.

###### Compute Optimal Boundaries.

In our method, we propose an optimal method to adjust the boundary lines of flattened 2D patterns. We set an objective length for each boundary line, and then minimize the angle variations between boundary lines of 2D patterns. The shape of the resulting pattern will be similar to that of the original pattern. The objective length of a boundary line is the length of the line after it is adjusted. For two boundary lines to be sewn together, usually they have different lengths. In our method, we simply set the objective length as the average length of two lines. In practice, the objective length can be set as any reasonable value.

Before computing the optimal boundary lines, which are the resulting boundary lines of the 2D pattern after it is adjusted, the 2D pattern will be polygonized, that is, curved boundary lines will be replaced by straight lines, as shown in Fig. 5. After polygonizing a 2D pattern, the lengths of boundary lines and angles of boundary lines are calculated based on the 2D polygon, and the optimal boundary method will be applied to the polygon. Once we get the optimal boundary lines, we will recover the curved boundary lines accordingly by scaling the boundary lengths.

We compute the optimal boundary lines of a 2D pattern using a constrained optimization framework. The angle variation term is set as the soft constraint in the objective function, and the boundary length invariant term is set as the hard constraint. We employ the method [8] to formulate the problem in the angle space. The constrained optimization is as follows. Display Formula

(1)$minθi∑i=1n12(θi−αi)2$

The equation is subjected to the following conditions. Display Formula

(2)$nπ−∑i=1nθi≡2π$
Display Formula
(3)$∑i=1nli cos ϕi≡0$
Display Formula
(4)$∑i=1nli sin ϕi≡0$

where $θi$ is the targeting 2D angle associated with the corner point Pi of the 2D pattern, $αi$ represents the original angle of the corner point of Pi, li denotes the optimal length of a boundary line, and n is the number of corner points of the boundary, $ϕi$ is the angle with a relation $ϕi=iπ−∑b=1iθb$, as illustrated in Fig. 6.

The first constraint in Eq. (2) is obtained by the closed-path theorem, and the second and third constraints in Eqs. (3) and (4) are derived from the position coincidence requirement. Newton's method is adopted to solve this constrained optimization problem.

## Experimental Results

We have implemented our algorithm using C++ and OpenGL, and tested the performance of our algorithm with a variety of 3D models of triangulated surfaces. All the experiments are implemented on a PC with a central processing unit of Intel Core™ 2.5G. Experimental results are shown in Figs. 711.

At first, we test the performance of the surface flattening algorithm using multilevel meshes. Figure 7 shows the flattening results of four triangular surfaces, and Table 1 shows the performance of different flattening methods. In the table, the runtime of the multilevel flattening method includes the computational time of creating multilevel meshes. According to the experiments, the process of preparing multilevel is fast, and it only accounts for less than 10% of the total computational time.

In Table 1, we compare the proposed method with the hierarchical least squares conformal maps method [HLSCM, 13] and the basic flattening method [24] without using multilevel meshes, which is described in Sec. 3. In the table, EA is the area error, and EL is the perimeter error, which are defined as follows: Display Formula

(5)$EA=∑i=0n|Ai′−Ai|/∑i=0nAi$
Display Formula
(6)$EL=∑i=0m|Li′−Li|/∑i=0mLi$
where n and m are the number of faces and edges in a triangular mesh, $Ai′$ and $Ai$ are the areas of a face i in the flattened and original mesh, $Li′$ and $Li$ are the lengths of an edge i in the flattened and original mesh.

From the table, we can see that the multilevel flattening algorithm is significantly faster than the basic flattening algorithm without using multilevel flattening, and the deformation (in terms of both area and length) of the flattened mesh is reduced using multilevel meshes. Complex surfaces with large and irregular meshes or holes can also be handled. Figure 7(b) shows a flattening result of a surface of a car shell with holes, and Fig. 7(c) shows the flattening result of a metal sheet of an irregular triangular surface. In the flattened meshes, the deformation rate is reasonable, and there is no overlapping. The experiments prove that the multilevel method is efficient. The multilevel flattening method is slower than HLSCM. However, the deformation is large for HLSCM. In Fig. 8, we compare the flattening results of a half sphere using different methods. Supposing the surface is isotropic, the shape of the flattening mesh using the proposed multilevel method is a circle, but the other methods generate noncircular shapes. It demonstrates that the multilevel method generates better results for product design.

The multilevel flattening algorithm can significantly reduce the possibility of overlapping. We have tested some complex surfaces. For most cases, overlapping can be avoided using the proposed flattening algorithm. One thing should be noted that, for a surface with very low developability, the proposed method cannot guarantee that there are no flipped triangles in the flattened mesh.

Our algorithm has been applied in a program of soft product design. The program has been used by professional designers to design patterns for plush toys, garments, and sofas. Figures 911 show the flattening results of a plush toy, a garment, and a sofa. The process of pattern design is as follows: First, seam lines are drawn on the surface of a 3D model; then, the region surrounded by seam lines is selected, and finally the selected 3D surface patch is flattened into a 2D mesh to obtain a 2D pattern. After all 2D patterns are flattened, sewing relations are established, and the boundary optimization method is applied to all 2D patterns to automatically adjust the boundary lengths of 2D patterns. The speed of surface flattening is fast, and real-time pattern design can be achieved. If the number of triangles in a surface patch is less than 5000, the computational time of surface flattening can be done in 1 s.

## Conclusions

A new surface flattening algorithm for soft product design has been presented in this paper. A basic flattening method using a mass-spring model is employed, and the efficiency of flattening is significantly improved by the use of a hierarchal mesh structure. A boundary optimization method is proposed to adjust the boundary lengths of 2D patterns to guarantee that the patterns can be sewn well. Experimental results show that the flattening algorithm can deal with complex surfaces efficiently and robustly, and it is suitable for soft product design.

Further work remains to be done. The first work to do is to study real physical material properties of different products, and combine the material properties into the flattening method. The second work to do is the automation of drawing seam lines on the surface of a 3D model. For soft products, there are some special rules for draw seam lines, making it possible to draw seam lines automatically using artificial intelligence. The third work to do is to bilaterally edit seam lines and update the pattern in real time, that is, if a seam line is modified, the 2D pattern will be updated automatically, and if the 2D pattern is modified, the seam line will be updated automatically. The last work to do is to apply the proposed algorithm to the flattening of sheet metal by considering its physical properties and characteristics. Since our method is based on the mass-spring method, it is possible to be used to improve the efficiency of energy-based flattening methods [1,2] for development of sheet metal components.

## Funding Data

• National Natural Science Foundation of China (Grant Nos. 51775492, 61732015, 51575481, and 61472355).

• Science and Technology Department of Zhejiang Province (Grant No. 2018C01090).

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## References

Liu, Q. S. , Xi, J. T. , and Wu, Z. Q. , 2013, “ An Energy-Based Surface Flattening Method for Flat Pattern Development of Sheet Metal Components,” Int. J. Adv. Manuf. Technol., 68(5–8), pp. 1155–1166.
Liu, X. H. , Li, S. P. , Zheng, X. H. , and Lin, M. X. , 2016, “ Development of a Flattening System for Sheet Metal With Free-Form Surface,” Adv. Mech. Eng., 8(2), pp. 1–12.
McCartney, J. , Hinds, B. K. , and Seow, B. L. , 1999, “ The Flattening of Triangulated Surfaces Incorporating Darts and Gussets,” Comput.-Aided Des., 31(4), pp. 249–260.
Wang, C. L. , Smith, S. , and Yuen, M. F. , 2002, “ Surface Flattening Based on Energy Model,” Comput.-Aided Des., 34(11), pp. 823–833.
Zhong, Y. , and Xu, B. , 2006, “ A Physically Based Method for Triangulated Surface Flattening,” Comput.-Aided Des., 38(10), pp. 1062–1073.
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## Figures

Fig. 2

Process of flattening using multilevel meshes

Fig. 1

Mass-spring model

Fig. 3

Edge collapse: (a) 3D shape and (b) unfolded shape

Fig. 4

Sewing relation of 2D patterns: (a) 3D model and (b) 2D patterns and sewing relation

Fig. 7

Examples of surface flattening: (a) freeform surface, (b) car shell, (c) metal sheet, and (d) half sphere

Fig. 8

Comparison of flattening a half sphere using different methods: (a) half sphere, (b) MLF, (c) BF, and (d) HLSCM

Fig. 9

Two examples of plush toys: (a) 3D toy model with seaming lines, (b) 2D patterns, (c) produced plush toy, (d) 3D toy model with seaming lines, (e) 2D patterns, and (f) produced plush toy

Fig. 10

An example of garment design (only main patterns are displayed)

Fig. 11

An example of sofa design

Fig. 5

Create a polygon from a 2D pattern: (a) 2D pattern and (b) 2D polygon

Fig. 6

Compute optimal boundary

## Tables

Table 1 Timing comparison of different flattening methods
Note: MLF—multilevel flattening method, BF—basic flattening method without using multilevel flattening, and HLSCM—hierarchical least squares conformal maps.

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