Computers in Industry 19 (1992) 185-191 Elsevier

185

Modeling in Computer Graphics

Three-dimensional garment design and animation A new deo gn tool for the garment industry

Ying Yang, Nadia Magnenat Thalmann MIRALab, CUI, Universitd de Gen~ve, 12 rue du Lac, CH 1207 Gen~ve, Switzerland

Daniel Thalmann Computer Graphics Lab~.,awry, Swiss Federal Institute of Technology, Lausanne, Switzerland

Garment design is traditionally carried out in two dimensions, and some software has been developed and applied in the garment industry in the design of garment panels. In this paper, a new tool for the interactive design of garments in three dimensions is introduced. Making use of an elastic surface model, animation allows us to examine the garment design in three dimensions dynamically. The designer can use this tool to visualize his original ideas and changes interac- tively, and to see the garment vividly portrayed, including texture mapping on the final design, before the real cloth panels are cut. Application of this tool in the garment indus- try could reduce design time and costs substantially.

Keywords: Cloth animation, Garment panels, Deformable models

1. Introduction

As in many other industries, computers are being considered for use in the garment industry for both design and manufacturing. The tradi- tional approach to garment making is first to take measurements of the human body, second to draw panel patterns on rectangular fabrics in two di- mensions according to the style and fashion de- sired, then to cut the panels out, and finally to sew them together by hand or by sewing ma- chines. Before the dress is sewn the tailor cannot

know for sure what the dress will look like, and what the effect will be of wearing it on the human body. For a new fashion design, the tailor can only imagine the results, depending on his experi- ence and talent.

In recent years, computer technologies have begun to be used in the garment industry. Soft- ware has been developed and applied to the interactive design of 2-D garment panels and to optimizing the layout of garment panels on the fabric. In Hinds and McCartney's work [1], a static trunk of a mannequin's body is represented by bicubic B-spline surfaces. Garment panels are considered to be surfaces of complex shapes in 3-D. The garment panels are designed around the static mannequin body, and then are reduced to 2-D cutting patterns. This approach is contrary to the traditional approach to garment design. The garment is modeled by geometric methods. To visualize the folds and drapes, harmonic func- tions and sinusoidal functions are superimposed on the garment panels. Mangen and Lasudry's work [2] propose an algorithm for finding the intersection polygon of any two polygons. This is applied to the automatic optimization of the lay- out of polygonal garment panels in 2-D rectangu- lar fabrics. Both of these projects concern stages of garment design and manufacturing in real in- dustrial contexts.

0166-3615/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

186 Modeling in Computer Graphics Computer in Industry

Techniques of computer graphics offer many other possibilities for the development of high- tech tools for garment design and manufacturing. Not only can the interactive design of 2-D gar- ment panels be achieved by general computer graphics, but the sewing of garment panels and the examination of garment movement on the human body can also be visualized through cloth animation based on dynamic surface models. Ter- zopoulos et al. [3] and Aono [4] both proposed

Ying Yang is a PhD student at MI- RALab, University of Geneva. He re- ceived his MSc in CAD/CAM from Beijing University of Aeronautics and Astronautics. His research interests include three-dimensional computer animation and geometric modeling.

- Nadia Magnenat Thalmann is cur- rently full Professor of Computer Sci- ence at the University of Geneva, Switzerland and Adjunct Professor at HEC Montreal, Canada. She has served on a variety of government advisory boards and program commit- tees in Canada. She has received sev- eral awards, including the 1985 Com- munications Award from the Govern- ment of Quebec. In May 1987, she was nominated woman of the year in sciences by the Montreal community.

Dr. Magnenat Thalmann received a BS in psychology, an MS in biochemistry, and a PhD in quantum chemistry and com- puter graphics from the University of Geneva, She has written and edited several books and research papers in image syn- thesis and computer animation and was codirector of the computer-generated films Dream Flight, Eglantine, Rendez- t,ous a Montreal, Galaxy Sweetheart, lAD and Flashback. She served as chairperson of Graphics Interface '85, CGI '88, Computer Animation '89 and Computer Animation '90.

Daniel Thalmann is currently full Professor and Director of the Com- puter Graphics Laboratory at the Swiss Federal Institute of Technology in Lausanne, Switzerland. Since 1977, he was Professor at the University of Montreal and codirector of the MI- RALab research laboratory. He re- ceived his diploma in nuclear physics and PhD in computer science from the University of Geneva. He is coed- itor-in-chief of the Journal of Visual. ization and Computer Animation,

member of the editorial board of the Visual Computer and cochairs the EUROGRAPHICS Working Group on Com- puter Simulation and Animation. Daniel Thalmann's research interests include 3-D computer animation, image synthesis, and scientific visualization. He has published more than 100 papers in these areas and is coauthor of several books includ- mg: Computer Animation: Theory and Practice and linage Synthesis: Theory and Practice. He is also codirector of several computer-generated films.

elastically deformable surface models to simulate and animate the movement of cloth in various physical environments. Another interesting ap- proach by Kunii and Gotoda [5] incorporates both the kinetic and geometric properties for generating garment wrinkles. Magnenat Thai- mann et al. used a modified elastic model to create and animate various articles of clothing, such as a skirt, underwear, T-shirt and trousers, on a synthesized actor's body [6,7]. Based on the above techniques, we are developing a new de- sign tool for use in the garment industry. This tool interactively designs the garment panels in 2-D by computer, sews the garment panels in 3-D on the computer screen, and dynamically simu- lates the garment's shape on the moving body of a synthesized actor. Texture patterns of various fabrics can be mapped onto the garment to make it look more realistic. The designer can modify the 2-D panels if the 3-D garment is not satisfac- tory. After all the examinations and changes, the final design is drawn by a plotter or is directly sent to a cutting machine which cuts the garment panels out of the fabric. A pattern library of garment templates can be connected to this tool. Adding artificial intelligence techniques, it would be possible for the tool to automatically design garments for the public.

In the following sections, the strategy and tac- tics of the tool are sketched.

2. A system for interactive garment design

The system for the interactive garment design tool consists of the following five parts: (1) Interactive graphic interface for the 2-D de-

sign of panels; (2) deformable cloth model; (3) pattern library of garment templates; (4) movable human body model; (5) output interface. The structure and the relationships of the system are shown in Fig. 1.

The interactive graphic design of the garment panels is carried out within the 2-D design inter- face. With cursor movements of the mouse, the designer can draw and modify the patterns for the garment panels on the coordinate grid on the computer screen in two dimensions. The garment templates in the library can be loaded into the

Computers in Industry Y. ,fang et al. / 3-D garment design and animation 187

[ / / /11/ / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /1] ~.2D Design Interface z~] y~HH.~HH'H'H~

~/// / / / / / / / / / / / / / / /) /#A [~. Garment Template ~]

lihrary

L 3 D Examination

[ 31) Deformable,~\ ,ql__llp ~ 3D Human I~ Cloth Model ~ I~ Body Model ,~.'~

Fig. I. Structure and relationships of the system.

2-D design interface, so that they can be used or modified. The 3-D human body model provides the movable mannequin bodies and the motion sequences. Different sequences of human move- ments, such as walking, running, dancing, fashion modeling, and so on can be generated. The 3-D deformable cloth model is used to create the garment from the fabric panels in three dimen- sions, and to simulate the changing shape of the garment on the mannequin as the body moves. Various properties of the cloth fabric, such as its mass, stretching and bending factor coefficients, damping density as well as characteristics of the physical environment, such as gravity and wind forces, are used to simulate the movements of the garment. In the template library, there are pat- terns of many different ruled or traditional gar- ments. These templates can be used directly in the design or modified for the particular individu- als. After the design is finally decided upon, the patterns for the panels in the final design are saved in the library. The patterns can be drawn on papers by a plotter or sent to the cutting machine to produce garment templates and the cloth panels.

3. Two.dimensional panel design

The functions of the 2-D panel design inter- face include mainly interactive drawing of the

panel polygons, digitizing existing templates, and optimal placement of garment panels on rectan- gular fabrics of various sizes. Button positions, seam lines, and the sizes of the panels and gar- ment, are also indicated on the patterns.

3.1. Digitization of templates

Many templates already exist for various fash- ion styles of different people, for different body shapes, in different countries. They are the most valuable resources for the garment designer. Putting them in the template library is helpful in that the designer can easily access them and modify them slightly to make new garments. This requires digitization of the existing templates. Only the tablet and mouse need to be used to digitize the polygonal templates. Sometimes the templates have also some curvilinear arc edges. The arc edges can be simplified to several termi- nal lines, so that all templates can be regarded simply as polygonals. With the tablet and mouse, starting at one vertex of the polygon, the shape of the garment can be digitized into the library vertex by vertex. For curvilinear edges, additional points are chosen to be vertices.

3.2. Interactive panel design

Less complicated than other CAD systems in mechanics or architecture, the interactive gar- ment panel design is carried out in only two dimensions. Because all the panels can be simpli- fied to polygons, the designer can easily create and modify their shape using general 2-D interac- tive graphics. With the mouse and keyboard, the designer fixes vertex positions and inputs sizes, thus creating the polygon. Button positions and seam lines are indicated within the panel poly- gons. If the designer is not satisfied with his work, he can modify his design in the same way.

3.3. Layout of garment panels

In the garment industry, most garment panels are not ruled polygons, the cloth fabric usually is rectangular and it comes in certain sizes only. It is important that the panels be laid out correctly on the rectangular fabric, otherwise much fabric will be wasted in the large-batch manufacturing of the garment. To optimize the layout of the

188 Modeling in Computer Graphics Computer in lndrt;t:y

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Fig. 2. Geometric design of T-shirt.

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panels, a simulated annealing algorithm [2] is used. First, all the panels are placed on the rectangular fabric arbitrarily, and the intersec- tions of panel polygons are tested. If some poly- gons overlap, they are moved apart; if the gaps between polygons are too large, they are moved closer. The testing and moving continues until the necessary length of fabric is obtained, without any superposition of panels. At this point, we also decide which panel edges will be seamed to- gether, and which edges will be attached to the actor's body.

For example, consider the geometric design of a T-shirt (Fig. 2) and pants (Fig. 3). The T-shirt and pants are very simple so each of them could be regarded as a single panel. As shown by Fig. 2, t h e T-shirt is designed in a 2-D rectangular mesh cloth ABCD by specifying the polygon's vertices vt, v2, v3,. . . ,v2s. It is also specified that the edge v~v2s will be attached to the waist of the actor, and the edge vsv 6 will be seamed with the edge v24v23, the edge vttv12 will be seamed to v~8v,7, and the edge vtv 2 seamed to U28U27. All

20

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A 0 10 15 20 25 30 35 40 45

Fig, 3. Geometric design of pants.

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the information, such as edge numbers of the polygon, the coordinates of vertices, which edge will be seamed and attached, etc. are stored in the data structure of the panel. The 2-D polygo- nal cloth panel will be transferred into a 3-D polygonal panel in ruled surfaces.

4. T h e d e f o r m a b l e c l o t h m o d e l a n d i t s p a r a m e -

t e r s

To simulate the sewing and animation of the garment, deformable cloth models must be used. Physically based models are preferable in the hope of increased realism. This should take into account such physical properties as mass, stiff- ness, damping factors, inhomogeneity, anisotropy and viscoelasticity. The model should be de- formable under external forces and its own inter- nal elastic energy, should detect collisions of the cloth with itself and with external objects, and should be able to create constraint forces when collisions occur. With this model, diverse kinds of clothing can be created and animated by defining and adjusting the geometric sizes, the physical properties of cloth and the external forces ap- plied to it. After some comparisons [7], Terzopou- los' elastic surface model [3] was chosen for our system. In this model, the main parameters are as follows: - mass density of the nodal point of the fabric; - damping density of the fabric; - stretching coefficients of the fabric; - bending coefficients of the fabric; - gravity; - external forces, including wind force, collision

forces, etc.; - time step for calculating the deformation; - number of relaxation steps. These parameters are determined by the physical properties of the fabric. Different fabrics have different physical properties. For example, silk is hard to stretch but easy to bend, woolen cloth is more massive and rigid. When worn on the hu- man body, garments with the same polygonal panels but different fabrics will take on different shapes. The environmental factors, such as grav- ity, wind forces, and collision forces also affect the shapes of the garments, and they can be changed dynamically to examine the garment un- der various environments.

Computers in Industry

The most useful parameters for modifying the appearance of the motion are the density, the damping factor, resistance coefficients, the wind and the time interval.

5. Sewing garment panels in three dimensions

Once the desired panels are designed, they are sewn along the indicated seam lines around the mannequin's body in 3-D making use of the de- formable cloth model. At this stage, the man- nequin's body is static in a standing posture, and gravity is the single environmental factor. The garment panels are first placed around the man- nequin's body, then external sewing forces are applied to the indicated seam lines shown on the panels. These sewing forces gradually deform ac- cording to a relaxation step and a time step. Collisions among the different parts of the gar- ment are detected and repulsive forces are ap- plied between any two parts of the garment in contact. When the panels are close enough to the mannequin's body, a collision between the gar- ment and the body will occur. The body creates the repulsive forces to beep the garment outside the body [8]. Spring forces are used to simulate the repulsive force. When the seam lines are all sewn up and the deformation of the garment is complete, the 3-D garment has been created (Fig. 4) ~. Some special features of garment, such as wrinkles (Fig. 5), folds, and drapes, are automati- cally calculated and formed by the deformable cloth model. Texture mapping can also be ap- plied to the garment, so that it will look more realistic as shown in Fig. 6.

For example, as shown in Fig. 7, we create a T-shirt and pants in the 2-D plane and transfer them into 3-D space around Elvis' body. During the seaming and attachment procedure, the edges of both the T-shirt and the pants near the body's waist are attached to the waist, the four edges of the T-shirt near the shoulders are seamed to- gether, and the two bottom edges of the pants are seamed to each other. As the result, a suit of clothes including the T-shirt and the pants has been designed and fabricated. Figures 8 and 9 show examples.

i Figures 4-12 may be found on pages 195 and 196.

Y. Yang et al. / 3-D garment design and animation 189

In the same way, Fig. 10 shows a dress and Fig. 11 shows a view of Marilyn wearing this dress.

6. Garment animation with human body move- ment

Garment animation during human body move- ment is performed by the deformable cloth model and the human body model. First, a series of sequences of human body movement, such as walking (Fig. 12), running, dancing, jumping, fashion modeling etc. are generated by the hu- man body model. When the mannequin is mov- ing, collisions between different parts of the gar- ment itself, and between the garment and the body are tested and repulsive forces are automat- ically calculated and applied. Environmental fac- tors, such as gravity, variable wind forces and air viscosity, are added to the deformable cloth model. The sewing forces assure that the panels remain joined together. With the movement of the body, the shape of the garment, including wrinkles, folds and drapes, is changed automati- cally. The parameters of both the fabric and the environment can be adjusted flexibly.

7. Garment examination and change

During the procedures of sewing and anima- tion of the garment, the designer checks the appearance of the garment in three dimensions. If the result is not satisfactory, he can interac- tively modify the shapes of panels, or the parame- ters corresponding to the fabric's properties, as well as the factors of the environment. The ani- mation is repeated and the whole process is iter- ated until the desired effect is obtained.

8. Implementation

8.1. Data structures

At the present stage of development of the garment design tool system, the human body model and the deformable cloth model have both been completed. The 2-D interactive design in- terface for the garment panel and the template library are still under development.

190 Modeling in Computer Graphics Computer in Industry

A suit of clothes

panel I I ... panel n l panel 2 panel 2 panel 1 ... panel n3

/1\ point I ... point k l

point 2

Fig. 13. The hierarchical data structure of clothes.

The clothes on the actor's body may include several articles, such as T-shirts, pants, jackets and trousers, and each article may consist of several cloth panels, so the data structure of clothes in the software is hierarchical, as shown in Fig. 13. In this cloth data ztructure, the seam- ing information between panels or within the panel itself, and the information about attaching each article to the actor's body are also included.

The panel is the elementary unit treated by the elastic surface model. In a panel data struc- ture, there are geometric data and physical data, seaming information and attaching information, as shown in Fig. 14. The geometric data about a panel include the polygons' edge numbers, the polygons' vertices, the number of points in the mesh panel, the center of the panel ~nd its rotat- ing angle, etc. From the geometric data on a

panel

Fig, 14. The data structure of a panel.

article

rJ//llll/~I/ll/l#I/ll/I//4llll~fllllllllll/lll/llllll/lll~ [ ~ panel data: number of panels, V~ series of panels "//A VI/ /# / / / / / / / / /~ / / / /# / / / / / / /# / / / f f / IY lA

~/JJT////////////////////////////////~//////////////////////////////////////~ seaming:, edge x l o f y l s e a m e d with edge ~//~]

x2 of panel y2, edge x3 of panel y3 seamed with edge x4 of anei y4, ...

Fig. 15. The data structure of an article.

panel, we can derive its shape, size, position, normal, and so on. The physical data on a panel include its mass, the damping factor, speed, forces on it, the stretch factor, the curvature factor, elastic energies (the stretch energy and curvature energy), etc. The seaming information for one panel concerns which edges of the panel should be seamed together. It includes the number of nodes on the edges and the coordinates of the nodes in the 2-D mesh plane, and indicates which node is seamed to which. The attachment infor- mation indicates which edges of the panel are attached to specific points on the actor's body. It includes the number of nodes on the edges and the coordinates of the nodes in the 2-D mesh plane, and an indication of which node is seamed with which point on the actor's body.

An article of clothing consists of several panels seamed together, so its data structure contains the panel data and the information about seam- ing the panels together, shown in Fig. 15.

The texture mapping approach is also being worked out. For the moment, WAVEFRONT soft- ware is used to put the texture pattern on the garment.

8,2. Collision detection

When we consider collisions between the cloth and the body, we have a situation of an actor-en- vironment interface using a physical motion con- trol method. Collision detection adds extra con- straints and requires a specific algorithm. For very flexible objects like clothes, it is necessary to introduce self-detection. In our method [8], colli- sion avoidance consists of creating a very thin force field around the obstacle surface to avoid collisions. This force field acts like a shield reject- ing the points. The collision detection process is

Computers in Industry Y. Yang et al. / 3-D garment design and animation 191

almost automatic. The animator has only to pro- vide the list of obstacles to the system and indi- cate whether they are moving or not. For a walk- ing synthetic actor, moving legs are of course considered as a moving obstacle. A number of parameters have been planned in order to modify the behavior of the collision detection method: shield depth, shield force and damping factor.

As the algorithm speed depends on the num- ber of obstacle polygons, it is prudent to take into account only polygons which are likely to inter- sect the cloth. For the example of Marilyn's skirt, only the pelvis and the legs are considered (Fig. 16) 2. With this method, we created animated flags in the wind and a skirt in the computer-gen- erated film Flashback (Fig. 17).

8.3. Methodology of use

To use this new tool in garment design, the procedure consists of the following steps: (1) Take the measurements of the human body. (2) Interactively draw the polygonal patterns of

the garment panels or select the templates from the garment template library.

(3) Modify the shapes of the garment panels. (4) With the deformable cloth model and the

human body model, create the garment on the mannequin body in three dimensions.

(5) Simulate and animate the changing shape of the garment with moving sequences of the mannequin's body.

(6) Examine the changing shape of the garment to see if it is satisfactory or not.

(7) If the garment is not satisfactory, repeat (3) to (6).

(8) Save the patterns in the garment template library.

(9) Draw the patterns of the garment panels on paper.

The above steps illustrate the superiority of this tool over the traditional design approach. The designer can dynamically visualize his design be- fore the garment is actually made. Much time and cloth can therefore be saved.

2 Figures 16 and 17 may be found on p. 196.

9. Conclusion

Using new animation techniques, we are devel- oping a high-tech CAD tool for garment design. This tool not only designs garment panels in 2-D, but it also allows the visual examination of the garment in 3-D on a moving human body with cloth animation, before the garment is actually manufactured. This improves on traditional gar- ment design, which is only carried out in two dimensions, and makes the design proces~ more convenient and economical.

Acknowledgements

The research is partly supported by the Fonds National Suisse de la Recherche Scientifique, le fonds FCAR du Qu6bec and the Natural Sci- ences and Engineering Research of Canada. The authors would like to thank Arghyro Paouri for the design of several pictures.

References

[1] B.K. Hinds and J. McCartney, "Interactive garment de- sign", Visual Comput., Vol. 6, 1990, pp. 53-61.

[2] A. Mangen and N. Lasudry, "Search for the intersection polygon of any two polygons: Application to the garment industry", Comput. Graph. Forum, Vol. 10, 1991, pp. 19- 208.

[3] D. Terzopoulos, J. Platt, A. Barr and K. Fleischer, "Elastically deformation models", Proc. SIGGRAPH'87, Comput. Graph., Vol. 21, No. 4, 1987, pp. 205-214.

[4] M. Aono, "A winkle propagation model for cloth", Com- puter Graphics huerface, Springer, Singapore, 1990, pp. 96-115.

[5] T.L. Kunii and H. Gotoda, "Modeling and animation of garment wrinkle formation processes", Proc. Computer Animation'90, Springer, Tokyo, 1990, pp. 131-147.

[6] N. Magnenat Thalmann and Y. Yang, "Techniques for cloth animation", in: N. Thalmann and D. Thalmann (eds.), New Trends in Animation and Visualization, Wiley, New York, 1991, pp. 243-256.

[7] N. Magnenat Thalmann, Y. Yang and D. Thaimann, "The problematics of cloth animation", Proc. 2nd Conf. on CAD/CG, International Academic Publishers, Beijing, China, 1991, pp. 1-7.

[8] B. Lafleur, N. Magnenat Thalmann and D. Thalmann, "Cloth animation with self-collision detection", in: T.L. Kunii (ed.), Modehng in Computer Graphics, Springer, Tokyo, 1991, pp. 179-188.

Computers in Industry X. Zhou, W. Straper / NURBS approach to cyclides 193

Fig. 4. NURBS representation of a cyclide with its control net.

:Y

Fig. 5. A horned cyclide with its control net.

Fig. 6. A cyclide patch with the knot vectors 7' = (uo, uo, u o, 0, O, ul, ul, ul), 8 - ( v o , vo, vo, O, 0, vl, vt, vt).

Fig. 7. A cyclide patch with the knot vectors T = (Uo, Uo, uo, 0, 0, ul, ul, ul), S - ( V o , Vo, Co, vl, vl, vl).

194 B. Falcidieno et al. / Feature representation in different contexts Computers in Industry

Fig. 13. SFOG representation of the bench-mark object after shape feature extraction.

Fig. 14. Functional recognition of the T-slot features in the SFOG representation.

Computers in Industry Y. Yang et al. / 3-D garment design and animation 195

. . . . ~ i:

2

Fig. 4. An example of 3-D cloth.

~ii~iii ~ ................................ !

Fig. 5. Cloth with wrinkles.

Fig. 6. Texture mapping. Fig. 7. Seaming clothes and putting them on Elvis.

,Fig. 8. Putting pants on Marilyn. Fig. 9. Marilyn wearing a T-shirt and pants.

196 Modeling in Computer Graphics Compu, o m Industry

Fig. 10. A dress for Marilyn.

Fig. 12. Clothes animation in Marilyn's walking sequence.

Fig. 11. Marilyn wearing a dress.

Fig. 16. Skirt animation.

!i~!~i!ii~ii~!ii~i~!!~iiii I

Fig. 17. Scenes from the animation film Flashback.

Computers in Industry R. van Kleij / Display of quadric CSG models 197

Fig. 14. CSG models with quadrics and display resolutions: (a) simple pipe (512x512); (b) block with holes (512 x512); (c) Bing carburettor (512 x 512); (d) open Bing carburettor (990 × 990); (e) molecule (9990 × 990); (f) Mig-29 fighter (990 x 990).

198 P. Astheimer et ai. / Scientific visualization Computers in Industry

Fig. 5. Visualization of the air flow in the human nose using polylines in combination with volume rendering of CT data.

w

Fig. 6. Visualization of the air flow in the human nose using lighted triangle stripes in combination with volume rendering

of CT data.

Fig. 7. Visualization of the air flow in the human nose showing the results of merging volume rendering of CT data

with the rendering of geometric objects.

Fig. 9. Different presentations of finite element data. To the left: wi're-frame. To the right: transparent display with bound-

ing box (upper) and solid (lower).

Fig. 10. Display of vector field. Fig. 11. Data probing as example for semantic interaction.

Computers in Industry H. Yamashita et al. / Interactive data input 199

Fig. 11. Editing window of: (a) R i and (b) G i coordinate system.

Fig. 12. Editing window: (a) input of dimensions of R2; (b) input of subdivision numbers of G 2.

Fig. 13. Input window of: (a) R i and (b) G i coordinate system.

200 H. Yamashita et al. / Interactive data input Computers in Industry

Fig. 14, Model of the first example (a) and subdivision maps: (b) whole analysis region; (c) magnification of the region near the core; (d) coil) core and aluminum plate.