DETC2004/CIE VOLUME-BASED CUT-AND-PASTE EDITING FOR EARLY DESIGN PHASES

Proceedings of DETC 04 ASME 004 Design Engineering Technical Conferences and Compuers and Informaion in Engineering Conference Sepember 8-Ocober, 004, Sal Lake Ciy, Uah USA DETC004/CIE-57676 VOLUME-BASED CUT-AND-PASTE EDITING FOR EARLY DESIGN PHASES Hiroshi Masuda Graduae School of Engineering The Universiy of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan e-mail: masuda@nakl..u-okyo.ac.jp Yoshiyuki Furukawa Digial Manufacuring Research Cener Naional Insiue of Advanced Indusrial Science and Technology 1- Namiki, Tsukuba-shi, Ibaraki 305-8561, Japan e-mail: y-furukawa@ais.go.jp Yasuhiro Yoshioka Graduae School of Engineering The Universiy of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan e-mail: yeshero@nakl..u-okyo.ac.jp ABSTRACT Since produc conceps are frequenly changed in he early sage of design, he creaion of is rough models is useful for communicaion among he design eam. In his paper, we propose modeling operaions based on a volume-based cu-andpase mehod for generaing rough 3D models using exising 3D models. Cu-and-pase ediing exracs a characerisic feaure from a source model and copies i o a arge model. Our mehod allows pasing a wide variey of feaures ha may have overhangs and handles, while mos exising mehods canno manage such shapes. To realize such a volume-based cu-andpase echnique, his paper inroduces a new mehod ha consiss of mesh segmenaion and surface/volume fiing. In our mehod, a feaure region and is surrounding area are separaed from he user-specified area, and hey are used for generaing a parameric volume ha involves he feaure region. In feaure pasing, parameric volumes are deformed and pased o he arge model and hen feaures inside he volume are adapively deformed and pased o he arge. On he basis of his cu-and-pase mehod, we propose modeling operaions ha enable feaure regisraion, feaure removal, feaure pasing, and exure pasing. These operaions were implemened and demonsraed using example models. The resul shows our operaions can effecively generae new models by removing and reusing parial shapes in exising models. Keywords: CAD, 3D Modeling, Cu-and-Pase, Concepual Design, Volume 1. INTRODUCTION Afer he produc specificaion is deermined based on cusomer needs, rough models are ofen creaed by exper designers in he early sage of produc developmen. Since a lo of uncerainies exis in his sage, he designer creaes only rough models and frequenly refines hem. Such rough models are usually represened in concep skeches or hreedimensional models, and hey are used for he design evaluaion among he developmen eam. This early evaluaion is essenially imporan o produce compeiive producs, because mos of he values including cos, weigh, qualiy, manufacurabiliy, assemblabiliy, mainenancabiliy, ec., are srongly consrained by decisions in his phase. The early design review is helpful o preven inconsisency in design, and o shoren he oal developmen ime. Since he design conceps mus be undersood by a wide variey of specialiss from differen perspecives, rough models, such as D skeches or 3D models, are useful o enrich communicaion wih op managemen, vendors, eam members, cusomers, and invesors [1]. In his phase, however, i may no be reasonable o spend a lo of ime for creaing deail 3D models, because he design conceps are very ofen changed or discarded. One way o rapidly creae 3D models is o prepare design emplaes ha produce design variaions. This capabiliy is provided by some commercial CAD sysems [,3]. However, hese emplaes canno deal wih variaions ha are no previously defined. In addiion, i is a very expensive process o 1 Copyrigh 004 by ASME

reveal aci design knowledge and embed i ino a emplae model. Anoher way is o produce produc variaions on he base of exising 3D models of similar producs or 3D models scanned from real producs [4-16]. In his paper, we discuss modeling operaions for his approach. Since sric surface represenaion is no necessarily required o convey he design concep in he early sage, we confine our discussion o mesh models, which are suiable for 3D models ha are boh approximaed from precise CAD models and scanned from real producs. Several echniques have been developed o modify exising models [4-16]. Shape deformaion [4-6], cu-and-pase ediing [7-13], and 3D paining [14-16] seem o be promising echniques for his purpose. While he shape deformaion echnique modifies coordinaes of 3D models wihou changing Figure 1. Cu-and-pase ediing he conneciviy of meshes, he cu-and-pase ediing removes or adds deail feaures by changing he conneciviy. On he oher hand, 3D painings, which draw D figures on a 3D model, change colors on he model wihou changing he coordinaes and he mesh conneciviy. We mainly discuss cu-and-pase ediing in his paper because shape deformaion is a relaively maure echnique. In his paper, we propose a new volume-based cu-and-pase ediing mehod and hen apply i o modeling operaions for modifying exising models. Our modeling operaions include feaure removal, feaure pasing, feaure regisraion, and 3D paining. Cu-and-pase ediing exracs a characerisic feaure from a source model and copies i o a arge model. We simply call such a characerisic feaure a feaure. As shown in Figure 1, he region seleced by he user is separaed ino he base surface and he deail surface, and only he deail surface is used as a feaure o be pased. Suppose he deail feaure is hin and does no involve overhangs; he base surface can be calculaed as he smooh approximaion of he original surface by applying surface fiing or polygonal simplificaion. The deail surface, on he oher hand, is obained as he difference of geomery beween he original surface and he base surface. The difference is measured in he normal direcion a each verex and is sored as a heigh-field, which is a.5d shape. To pase he deail surface o a arge model, he corresponding verices of he arge model are moved o he normal direcions using he heigh values of he deail surface. However, i is difficul o apply hese cu-and-pase operaions o general feaures, which may be hick or have overhangs and handles, as shown in Figure, because a feaure canno be easily separaed by he approximaion of he original surface; a hick feaure may cause self-inersecions; and a feaure ha produces handles or holes canno be mapped on a planer surface. To pase feaures wih overhangs, Kanai e al. [11] proposed a mesh fusion echnique based on he morphing of mesh models. This approach is very ineresing, bu i may cause unnaural disorions because of he parameerizaion based on harmonic mapping. Forsey and Barels [8] proposed hierarchical B-spline surfaces ha can be used o pase feaures in parameric space, bu heir mehod resrics he pasing wihin surface paches. No self-inersecions Figure. Examples of volume-based pasing. Our mehod is based on a volume-pasing approach, which pases a parameric volume insead of a heigh-field. Since we pase a volumeric region on he arge model, we can deform and pase a feaure in he volume even if he feaure has overhangs or handles. In addiion, our opimizaion algorihm for volume pasing avoids self-inersecions of hick feaures, as shown in Figure. A parameric volume maps coordinaes from parameric space o Euclidean space using a funcion V :(u, v, w) (x, y,z). A ypical mapping V ( u, v, w) is defined by he ensor produc of hree B-spline curves [17]. A regular hree-dimensional laice of conrol poins P }, each { ijk Copyrigh 004 by ASME

of which has a { x, y, z} coordinae, deermines such a parameric volume V ( u, v, w). To realize a volume-based cu-and-pase echnique, we propose he following mehods: 1. To calculae he base and he deail surfaces, we divide he user-specified region ino a pure feaure region and is surrounding region. The base surface is calculaed using only he surrounding region. We inroduce a local mesh segmenaion mehod o exrac pure feaure regions auomaically.. Parameric volumes ha fi o he source or arge models need o be calculaed. We solve his problem using B-spline volume fiing. In his fiing mehod, he parameerizaion, which assigns a (u, v) parameer o each verex in a mesh, is gradually updaed o improve he fiing errors. In he objecive funcion for volume fiing, he weighs of he fiing error merics and he smoohness merics are conrolled o avoid selfinersecions. 3. On he basis of our cu-and-pase mehod, we inroduce modeling operaions ha allow feaure regisraion, feaure removal, feaure pasing, and exure pasing. The res of he paper proceeds as follows. The nex secion provides he overview of our mehod. In Secion 3, we describe how o separae a region ino he base surface and he deail surface. In Secion 4, our volume fiing echnique is explained, segmenaion separaion feaure geomery conex geomery base surface parameerizaion base volume and in Secion 5, modeling operaions based on volume fiing are described. In Secion 6, we demonsrae our modeling operaions, and finally, Secion 7 concludes he paper.. OVERVIEW Our objecive is o develop modeling operaions suiable for he reuse of exising 3D models o suppor communicaion in he early sage of design. We suppose mesh models generaed by scanning real producs or essellaing CAD models. On he basis of our volume fiing mehod, we inroduce he following operaions: Feaure regisraion: Feaures are exraced from mesh models and sored in a feaure library. Coordinaes of a feaure are normalized in a parameric volume for reuse. Feaure removal: Unnecessary feaures are removed, and replaced by heir base surfaces. Feaure pasing: An applicable feaure is adaped o and pased on he arge model. Texure pasing: If users canno find appropriae feaures, hey can ineracively skech on he arge model. Figure 3 shows he proposed modeling process: (a) Firs, he user selecs a region including a feaure and is surrounding area. Then, he sysem calculaes he border of hese wo regions. Here, we call he pure feaure area a feaure region, and he surrounding area a conex region. (b) The region is separaed ino wo regions a he border. Then, he base surface is calculaed by approximaing he conex region. (c) The iniial parameric volume is defined so ha i involves he enire feaure region. Then, he conrol poins of he volume are opimized so ha he boom surface corresponds o he base surface and he feaure region is no largely modified. We call his parameric volume he base volume. (d) The feaure region is parameerized in he base volume. (e) To remove a feaure, he feaure region is replaced by he base surface. (e) To pase a feaure, he base volume of a feaure is deformed o fi o he arge model. Then, he feaure region is deformed. (f) When he user canno find suiable feaures in a feaure library, he/she can draw srokes on he arge model afer unnecessary feaures are removed. The drawings are represened in images and pased on he model using exure mapping. feaure removal arge surface feaure pasing 3. SEPARATION INTO FEATURE AND CONTEXT To calculae a base surface for a cu-and-pase operaion, we have o separae he user-specified region ino a feaure region and a conex region. exure pasing For he segmenaion of regions, several mehods have been repored [18-]. They are roughly caegorized ino local search mehods [18,19] and global opimizaion mehods [0- ]. Figure 3. Modeling process. In local search mehods, a region is ypically separaed ino several small regions by invesigaing he angency or he curvaure of adjacen faces. However, hese mehods may no 3 Copyrigh 004 by ASME

separae a region ino exacly wo regions and may no manage nearly fla areas correcly. In addiion, local searches are no robus for noisy daa, such as scanned models. Therefore, we will use a global opimizaion mehod. Alhough several algorihms have been repored so far, we use he formalism proposed by Kas e al. [], which reduces he segmenaion problem o a well-known maximum flow, minimum cu problem [3]. This mehod is sable and divides a region ino exacly wo regions. In our experimens, i produces correc and non-jaggy boundaries for mos 3D models. However, his mehod is very ime-consuming for large models. Therefore, we use he mehod only for he ineracive selecion of feaures. (a) Original model (b) Seleced region Figure 4 shows a segmenaion process in our sysem. Firs, he user selecs a region ha includes a feaure region and a conex region, as shown in Figure 4(b). The region mus be seleced so ha he mesh model is separaed ino exacly wo regions by eliminaing faces in he region; in oher words, he region includes cu-se edges of he mesh. Then, he opimal cu-se is calculaed and he feaure region is correcly separaed from he user-specified region, as shown in Figure 4(c). Since he seleced region conains he relaively small number of faces, i can be segmened in a very shor calculaion ime. The segmenaion algorihm consiss of he following seps: (1) In Figure 5, he region roughly specified by he user is shown as medium region C, and faces in he region are shown in dashed lines. () The mesh in he seleced region is convered o is dual graph, which is generaed by mapping every face in he mesh o a node, and by connecing wo dual nodes by an edge if he corresponding faces are adjacen, as shown in solid lines in Figure 5. (3) Source S and sink T are locaed in each of wo regions A and B. Then, S and T are conneced o nodes of he dual graph. (4) A capaciy is assigned o each dual edge so ha he capaciy becomes smaller when he angle of wo adjacen faces is larger. We use he capaciy funcion proposed by Kas e al. [], as follows. Given wo adjacen faces, f i and f j, which correspond o nodes v i and v j in a dual graph, he angle beween heir normals is referred o as α. Then, he capaciy C(i,j) of a dual edge i, j beween nodes v i and v j is defined as: d i, j) = η(1 cosα ), ( i, j 1 (i, j s, ) d( i, j) C( i, j) = 1+ (1), average( d( R)) (i, j = s, ) where η is a consan value. The appropriae value of η varies depending on wheher a feaure is convex or concave. We used η = 0. 1 for convex feaures, and η = 1 for concave ones. One of hese values is seleced afer a convexiy check is applied o a feaure. (5) The minimum cu of his undireced flow nework graph is calculaed. We solve his maximum flow, minimum cu problem using he Ford-Fulkerson algorihm [3]. Since his minimum cu ends o flow hrough edges wih small capaciies, (c) Exraced feaure geomery. Figure 4. Segmenaion of a seleced region. Figure 5. Flow nework of a dual graph. his mehod correcly deecs feaure boundaries even when fla faces exis on he boundary. 4. VOLUME FITTING The user-specified region is divided ino a feaure region and a conex region by he segmenaion algorihm described in he previous secion. Then, we calculae he base surface using he conex region, and he base volume using he feaure region. 4.1. Noaions Le S denoe a B-spline surface and V a B-spline volume. They are defined as follows: n 1 m 1 S ( u, v) = N ( u) N ( v) P () i= 0 j= 0 n 1 m 1 l 1 i= 0 j= 0 k = 0 p i V ( u, v, w) = N ( u) N ( v) N ( w) P (3) p i where P ij and P ijk are conrol poins; N i p is a B-spline basis funcion; p, q, and r are orders; l, m, and n are he numbers of conrol poins. q j q j ij r k ijk 4 Copyrigh 004 by ASME

4.. Surface Fiing We denoe poins of verices in he conex region by {Q = 0,, s-1}, and heir parameers in he fied surface by {(u, v )}. Then, we define he fiing error merics as: F S p = s 1 = 0 Q S( u, v ) (4) We also inroduce he smoohness merics: S Fs = ( S uu + Suv + Svv )dudv. (5) These merics are imporan for smoohly exrapolaing holes in he conex region as well as penalizing he disorion and he folding of faces. Then, he surface-fiing problem is formalized as he opimizaion of: S ( S ) S ( S ) Fp ( P ) + β Fs ( P ) min (6) where P (S) is conrol poins of he base surface, and β is a non-negaive weigh. To calculae conrol poins of surface S(u,v) by opimizing Eq. (6), we have o assign a (u, v) parameer o each poin in {Q }. In our implemenaion, he iniial parameer seings are roughly deermined by projecing poins {Q } on a plane ha minimizes he sum of he leas square disance errors. Then, conrol poins ha minimize Eq. (6) are calculaed by solving a sparse linear sysem. When he calculaed surface causes selfinersecions or he folding of faces, he value of β is changed o be large so ha he disorion is more largely penalized han he fiing error. In cu-and-pase ediing, he parameers mus be assigned very carefully, because he poor parameerizaion causes disored unnaural feaure pasing [13,4-6]. To improve he iniial seings of parameers, we use a similar approach o he one used by Weiss e al. [7], which was originally developed for reverse engineering. In heir mehod, (u, v) parameers are gradually refined by calculaing conrol poins and parameerizing {Q } alernaely. Suppose parameer (u, v ) is assigned o Q, and i is refined o ( u ˆ, ˆ v ), which is defined as he parameer of he neares poin o Q in he fied B-spline surface. The parameer ( u ˆ, ˆ v ) can be calculaed by solving he following equaions [8]: T S ( ˆ ˆ u ( S u, v ) Q ) = 0, (7) T S ( ( ˆ, ˆ v S u v ) Q ) = 0 where S u and S v are parial differenials of S. When he surface has large fiing-errors even afer parameers are refined, he sysem auomaically insers knos o increase he degrees of freedom for fiing. See [7] for he deail kno-inserion algorihm. The above seps, which consis of he opimizaion, parameerizaion, and kno-inserion, are ieraed unil he fiing error saisfies a predefined hreshold value. In our experience, his mehod successfully generaes naural and smooh parameerizaion by a few ieraions. 4.3. Volume Fiing We exend he surface fiing mehod o B-spline volume fiing. In volume fiing, parameers and conrol poins are refined alernaely in he same way as surface fiing. Le { f = 0,, r 1} Q be he coordinaes of verices in a f feaure region. We generae a recangular bounding box ha involves hese poins and roughly assign heir parameers (u,v,w ) in he bounding box. Then, we define he fiing error merics F V d and he smoohness merics F V s, as follows: F F V p V s = v 1 = 0 Q S( u, v, w ) (8) ( uu vv ww = V + V + V. (9) + Vuv + Vvw + Vwu )dudvdw Since he boom surface V(u,v,0) corresponds wih he base surface S(u,v), conrol poins of he base volume can be calculaed by minimizing he following objecive funcion: V ( V ) V V ( V ) Fp ( P ) + β Fs ( P ) min (10) ( V ) ( S ) subjec o P ij0 = Pij where P (V) is conrol poins of he base volume. The parameer (u,v,w ) of poin Q is refined by solving he following equaion: T V ( ˆ ˆ ˆ u ( V u, v, w ) Q ) = 0 T V ( ( ˆ, ˆ, ˆ v V u v w ) Q ) = 0 (11) T V ( ( ˆ, ˆ, ˆ w V u v w ) Q ) = 0 Then, a B-spline volume is calculaed again using Eq. (10) wih updaed parameers. In a pasing operaion, a volume is deformed on he arge model using his volume fiing mehod and hen he feaure region inside he volume is also adapively deformed. 5. MODELING OPERATIONS 5.1. Regisraion o Feaure Library Afer he base surface and volume are calculaed using he fiing algorihm, he feaure region is refined for regisraions o a feaure library. Firs, he border of he feaure region is recalculaed so ha i approximaely rides on he base surface. I is refined by remeshing riangles near he border and invesigaing differences beween he mesh and he base surface. If a verex is conneced o he verices on he border and is difference is large, he verex is added o he feaure region. Second, coordinaes in a feaure region are normalized as (u,v,w) ( 0 u, v, w 1) in parameer space of he parameric volume. The normalized mesh models of feaures are sored in a feaure library. In he curren implemenaion, he feaure daa are sored in he STL forma or he VRML forma. The size of he bounding box of a feaure is also sored. I is used o deermine he iniial defaul size of a volume o be pased on a arge model. The heigh of a volume, which is no 5 Copyrigh 004 by ASME

specified by he user in a pasing operaion, is se o have a similar raio o he one of he sored size. 5.. Feaure removal A feaure removal operaion removes a feaure region from a mesh model and fills he region wih a smooh surface. Figure 6 shows a process of feaure removal: (a) (c) (b) (d) Figure 6. Feaure removal. (a) The user roughly specifies a region by drawing srokes on a 3D model and he sysem calculaes he border of he feaure region using he segmenaion mehod described in Secion 3. (b) Faces in he feaure region are removed from he mesh model. Then, he sysem applies he B-spline surface fiing o he remaining verices, as described in Secion 4.. Since a UV parameer of each verex is also calculaed in his fiing process, he border of holes can be mapped in UV parameer space. (c) The holes are essellaed in UV parameer space and filled by riangles. (d) Coordinaes in UV space are mapped in 3D Euclidean space. Finally, he feaure region is replaced by he smooh surface ha exrapolaes he conex region. 5.3. Feaure Pasing In feaure pasing, he user specifies a region by drawing a recangular region on a arge model. Srokes specified on a screen are projeced ono a 3D model, as shown in Figure 8, and he specified region is regarded as he boundary of he base surface. Then, he base surface and volume are fied o his region using he surface and volume fiing mehods described in Secion 4. The defaul size of he volume is deermined by values sored in he feaure library. Verices {(u i,v i,w i )} in he feaure are embedded in parameer space of he base volume, and he shape of he feaure is deformed. In his phase, he user can edi he feaure by changing he heigh or each conrol poin of he B-spline volume, or by adding new consrains o he parameric volume. Then, he sysem remeshes he arge model so ha he border of he feaure region is included in he mesh. Then, he feaure pased model arge model Figure 7. Feaure pasing. (a) Exraced feaures. 3D model camera drawing plane (b) A feaure in he volume. Figure 8. Drawing on a 3D model. Figure 9. Exraced feaures. 6 Copyrigh 004 by ASME

sysem removes faces on which he feaure is pased, as shown in Figure 7. Finally, he feaure is conneced o he arge model. 5.4. Texure pasing In new produc developmen, he designer may no be able o find appropriae feaures in he feaure library. To suppor such cases, we define operaions ha pase D drawings on 3D models. This operaion is mainly used in combinaion wih feaure removal operaions. The user firs removes feaures on a 3D model o creae a drawing region on he model. Then, he user draws figures on a drawing plane, as shown in Figure 8. In exure pasing, srokes on he drawing plane are sored as a exure image, and are mapped on a 3D model. Each verex in he 3D model is relaed o a exure coordinae (i, j), which represens a posiion in he image. For real-ime mapping, we only calculae exure coordinaes near srokes drawn on he 3D model. This operaion is very simple and canno define a correc shape, bu i will be complemenarily used wih feaure pasing. The careful use of exure pasing can suppor he communicaion in he iniial design phase. The source model (a) Feaure removal The arge model 6. RESULTS We implemened our operaions and applied o several models. We used a PC wih a Penium 4.15GHz CPU and 1.0GB memory. Figure 9 shows exraced fender mirrors o sore in a feaure library. Their feaure regions were ineracively specified by srokes. The calculaion ime for each segmenaion process was less han 0.0sec. Calculaion ime of volume-fiing using various numbers of conrol poins is shown in Figure 10. In our experimens, volumes wih less han 500 conrol poins can be calculaed ineracively. The number of conrol poins is deermined by he fiing error, which can be conrolled by he user in our implemenaion. We observe ha his calculaion ime is (b) Feaure pasing Figure 11. Examples of feaure removal and pasing. Figure 10. Elapsed ime of volume-fiing. Figure 1. A combinaion of feaure removal and pasing. 7 Copyrigh 004 by ASME

and-pase ediing. Our volume-based approach can successfully enhance he capabiliy of cu-and-pase mehods. On he basis of he mehod, we developed modeling operaions, which involve feaure regisraion, feaure removal, feaure pasing, and exure pasing. These operaions were implemened and evaluaed using sample models. The resul shows hey can effecively generae new models by regisering, removing, and reusing parial shapes in exising models. Figure 13. An example of exure pasing. sufficien for our modeling operaions, because precise surface fiing, such as a 10-5 order, is no required in feaure pasing. In addiion, volumes need a small number of conrol poins in he heigh direcion. Figure 11 shows feaure removal and pasing operaions. In he feaure removal, he specified feaure could be smoohly replaced by he base surface. In feaure pasing, he fender mirror in Figure 11(a) could be also smoohly pased on anoher model in Figure 11(b). Figure 1 shows a combinaion of feaure removal and pasing operaions. Since boh operaions use he common volume fiing process, hey can be applied simulaneously. Figure 13 shows an example of exure pasing. In he lef figure, he user specifies unnecessary feaures in he par of buons, and he sysem generaes he base surface ha approximaes he specified region. Afer he feaure removal in he middle figure, he user can draw a new concep on he region. 7. CONCLUSIONS AND FUTURE WORK Our research aims o develop a modeling environmen ha suppors he early phases of produc developmen. In he early sage of design, rough models are useful for represening, evaluaing, and narrowing down a number of design conceps. We discussed modeling operaions suiable for creaing rough 3D models by modifying exising 3D models and models scanned from real producs. We do no inend o creae precise deail 3D models by our modeling operaions, because such models should be creaed using exising commercial CAD sysems afer conceps are narrowed down o a few candidaes. In his paper, we proposed a volume-based cu-and-pase mehod. Our mehod allows pasing shapes ha may be hick or have overhangs and handles. A B-spline volume is fied o a region in a 3D model by he opimizaion of he fiing error merics and he smoohness merics. This mehod enables users o smoohly pase a wide variey of feaures and o efficienly avoid self-inersecions. In addiion, mesh segmenaions based on a maximum-flow minimum-cu algorihm can correcly separae feaure regions and conex regions. These regions are used o calculae he base surfaces and base volumes for cu- In fuure work, we need o develop easy-o-use inerface for 3D modeling, because our modeling operaions would be used in an ineracive environmen. We need o sudy 3D skeches and oher ineracive modeling echniques. In addiion, for pracical use, our modeling operaions will be used in combinaion wih deformable modeling, which allows locally deforming shapes. Our curren sysem is jus a prooype, bu we would like o add more modeling operaions for realizing a pracical sysem for he early design phases. ACKNOWLEDGMENTS The auhors graefully acknowledge he suppor provided by Saoshi Kaneda of he Universiy of Tokyo. REFERENCES 1. Ulrich, K. T. and Eppinger, S. D. (1995). Produc design and developmen, McGraw-Hill, New York, NY.. hp://www.caia.com 3. hp://www.unigraphics.com 4. Sederberg, T. and Parry, S. R. (1986). Free-form deformaion of solid geomeric models, Proceedings of SIGGRAPH 86, pp. 151-160. 5. Coquillar, S. (1990). Exended free-form deformaion: a sculpuring ool for 3D geomeric modeling, Proceedings of SIGGRAPH 90, pp. 187-196. 6. MacCracken, R. and Joy, K. I. (1996). Free-Form Deformaion wih Laices of Arbirary Topology, Proceedings of SIGGRAPH 96, pp.181-188 7. Eck, M. DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M. and Suezle, W. (1995). Muliresoluion analysis of arbirary meshes, Proceedings of SIGGRAPH 95, pp. 173 18. 8. Forsey, D. R. and Barels, R. H. (1995), Surface fiing wih hierarchical splines, ACM Transacions on Graphics, 14(), pp.134 161. 9. Barghiel, C., Barels, R., and Forsey, D. (1994). Pasing spline surfaces, Mahemaical Mehods for Curves and Surfaces, edied by M. Daehlen, T. Lyche, L. L. Schumaker, Vanderbil Universiy Press, Nashville, TN., pp. 31-40. 10. Chan, L.K.Y., Mann, S., and Barrels, R. (1997). World space surface pasing, Proceedings of Graphics Inerface, pp. 146-154. 11. Kanai, T., Suzuki, H., Miani, J., and Kimura, F (1999). Ineracive mesh fusion based on local 3D meamorphosis, Proceedings of Graphics Inerface, pp. 148-156. 1. Suzuki, H., Sakurai, Y., Kanai, T., and Kimura, F (000). Ineracive mesh dragging wih an adapive remeshing echnique, The Visual Compuer, 16(3-4), pp. 159-176. 8 Copyrigh 004 by ASME

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