Analysis for Balloon Modeling Structure based on Graph Theory

Transcription

1 Anlysis for lloon Moling Strutur bs on Grph Thory Abstrt Mshiro Ur* Msshi Ym** Mmoru no** Shiny Miyzki** Tkmi Ysu* *Grut Shool of Informtion Sin, Ngoy Univrsity **Shool of Informtion Sin n Thnology, hukyo Univrsity -mil: ur@ngoy-u.jp, ysu@is.ngoy-u..jp, {mym noh miyzki}@sist.hukyo-u..jp lloon moling is n rt thniqu to mk shps suh s nimls from blloons. In this stuy, w propos strutur nlysis n iffiulty rting of blloon moling by using grph thory. In ition, w vlop blloon rt mking support tool bs on propos mtho. A rough shp of th blloon moling is onvrt into ulrin grph n fin th ulrin pth, whih rprsnts th prour of th blloon moling. Kywor lloon Moling, Strutur Anlysis, Grph Thory, Mking Support 1. Introution Whn somthing is m, gomtri rstritions my our from tools mtrils n tools uring th ours of mking. For xmpl, it is nssry to mk 3 shp by trnsforming ppr into ppr-rfting. Thus, in this s it is iffiult to know how to mk by tht rstrition, so mk support is ppli by using omputr [1][2][3]. lloon moling is n rt thniqu to mk shps suh s nimls from blloons. Shps r m of on or mor blloons. Vrious shps n b m by twisting blloons; howvr it is not sy to fin how to mk blloons into ny givn mol. In this ppr, w propos n nlysis of blloon moling strutur by using grph thory, n iffiulty rting of blloon moling. In ition, w vlop mking support tool for blloon moling (MSM) bs on th propos mtho. ginnrs will b bl to mk blloon moling by mking rough shps with MSM. W trgt only blloon moling gnrt by on blloon in this stuy. 2. About lloon Moling In mking blloon moling, vrious shps suh s nimls s shown in Fig.2-1 is m by 1) mking smll prt by twisting blloon, 2) fixing it oring to th position n th ir prssur of th blloon. Rbbit Prrot () r () Girff Figur 2-1 xmpl of lloon Moling 2.1. Ovrviw Th most ommon siz of blloons us in lloon moling is 5m in imtr n 160m in lngth t mximum swlling. A prt gnrt by twisting blloon is ll "ubbl". A bubbl is not fix only by twisting; thrfor it is nssry to fix th bubbl by spil twist thniqu. This thniqu is ll Lok Twist Thniqu Thr r svrl thniqus to Lok in lloon Moling [4]. Hr w xplin bsi twist thniqu whn on bubbl is us. In h figur, n lphbt of lrg lttr mns bubbl, n n lphbt of smll lttr mns twist point.

2 (1) Rok Twist Rok Twist is thniqu for loking h bubbl by four bubbls. It loks by foling "b", n twisting "" n "" s shown in Fig.2-2. It is th most bsi twist thniqu. A b () Figur 2-2 Lok Twist (2) Loop Twist Loop Twist is thniqu for loking h bubbl by thr bubbls. It loks by looping th bubbl of "", n twisting "" n "b" s shown in Fig.2-3. A b A Figur 2-3 Loop Twist (3) Pinh Twist Pinh Twist is Loop Twist whn "" is shortn s shown in Fig.2-3. iffiulty in rting is mor thn Loop Twist bus th twist bubbl is smllr thn th Loop Twist. A b A b A b A b b Figur 2-4 Pinh Twist (4) Tulip Twist Th tulip twist is thniqu for fixing th bubbl by pushing th blloon knot into blloon s shown in Fig.2-4, n twisting th blloon inluing th knot s shown Fig.2-4 (). A A A () Figur 2-5 Tulip Twist 3. Anlysis for lloon Moling Strutur Th strutur of h mol n b rprsnt by grph, whr bubbls n twist positions r trt s gs n vrtis. Whn mol is m of on blloon, th grph hs n ulrin pth whih visits h g xtly. An grphs involving ulrin pths r ll ulrin grphs [5]. Thrfor, numbr of bubbls n th twist points tht r nssry to mk th mol r lrifi by grphing th mol, n rqusting th ulrin pth Strutur Anlysis by using ulrin grphs A nssry onition for th xistn of ulrin pth is ithr ll vrtis in th grph hv n vn gr, or ll but two vrtis hv n vn gr. An it is possibl to istinguish whthr th grph is ulrin grph or not by using jny mtrix. Our strutur nlysis, first, fin n ulrin grph of givn mol. Th grph in Fig.3-1 is n obtin ulrin grph of th mol in Fig.3-1. In Fig.3-1, blk ots mns vrtx, n lins mns n g of grph. As for this grph, "" n "b", "b" n "", "" n "", "" n "b", "" n "b" r onnt rsptivly. Tbl 3-1 shows th jny mtrix of th grph. W n onfirm tht th grph stisfis th onition for ulrin grphs, bus this grph hs only two vrtis "" n "" gr of whih is o numbrs. A b Figur 3-1 Mk grph of lloon Moling Tbl 3-1 Ajny mtrix of Fig.3-1 j = b i = b

3 Thn, ll ulrin pths in th grph r gnrt by pth-first srh. h pth obtin by this srh rprsnts th orr of rrngmnt of th bubbl prts. Th grph in Fig.3-1 hs two ulrin pths shown in Fig.3-2. Whn th sm vrtx in th pth pprs two or mor tims, thy bom th twist positions. Vrtx "b" in th grph shown in Fig. 3-1 pprs twi, s shown in Fig.3-2. Thrfor, th givn mol is obtin by twisting "b 1 " n "b 2 " s shown in Fig.3-3 or. A b2 b iffiulty Rting Whn n ulrin grph hs two or mor ulrin pths, blloon moling hs two or mor mking prours. Thrfor, w n mtho for lulting h pth s iffiulty. Whn bubbl is not lok with Twist, it is nssry to lok thos bubbls by hn; thrfor it is iffiult to mk lloon Moling by inrsing bubbl tht is not lok. Thn, by using sum totl numbr of bubbls not fix, w provi th iffiulty of lloon Moling xmpl of Strutur Anlysis W try to nlyz blloon moling strutur by using n tul mol. Fig.3-4 is prrot whih ws m of blloon. A b b A b b Figur 3-2 ulrin pths of Fig.3-1 A b1 b2 Figur 3-3 Mking prours of Fig.3-1 A b1 b2 First, w onvrt th mol into grph, n onfirm whthr th grph is n ulrin grph or not by using n jny mtrix.tbl.3-2 shows n jny mtrix of th grph. Thn, if th grph is ulrin grph, w fin n ulrin pth of th grph Tbl 3-2 Ajny mtrix of Fig.3-4 j = b i = b Nxt, w obtin th mking prour from th rqust ulrin pth. Fig.3-5 shows th mking prours of th prrot. In figurs, ott lins r twist points. Figur 3-4 Prrot of blloon moling n prrot of grph b b A b F G b A b F b G G F b Ab b () G b F b A b () Figur 3-5 Four mking prours of prrot A b G F

4 Finlly, if th mol hs two or mor mking prours, w rt iffiulty of th mking prours. In Fig.3-6, 3-7, 3-8 n 3-9, figurs show th stt of blloons whn bubbls r lok in th mking prours. In h figur, bubbls of th sm olor xprss tht thos bubbls r fix s group. Our iffiulty rting uss th totl numbr of th group s n lmnt for omprison. For xmpl in th s of Fig.3-6, bus lulting formul is , th iffiulty rting of Fig.3-5 is 8. y th sm tokn, is 7 by , () is 8 by , n () is 12 by As rsult, Fig.3-5 is th sist of ths mking prours. Figur 3-8 Mking prours of Fig.3-5 () Figur 3-6 Mking prours of Fig.3-5 Figur 3-9 Mking prours of Fig.3-5 () 4. Mking Support Tool W vlop MSM bs on th bov strutur nlysis. MSM hs two mos, sign mo n Ltur mo. Fig.4-1 is ptur srn imgs in h mo. In sign mo, usrs n rrng bubbl prts. In ltur mo, th mking prour tht is lrifi by th strutur nlysis is squntilly isply. Figur 3-7 Mking prours of Fig.3-5 sign Mo Ltur Mo Figur 4-1 Mking Support Tool

5 5. xprimnttion To onfirm th fftivnss of MSM, w us MSM to rt si Animl whih is th bsi mol of blloon moling. First, w onstrut mol shown in Fig.5-1 in sign mo. Thn, MSM intrnlly gnrts th grph lik Fig.5-1. Nxt, it ollts th vrtis within onstnt istn lik prt surroun in ott lins of. As rsult, is onvrt into th grph lik (). Lok Twist A b G J h F H f I g () Figur 5-1 Pross of mking grph of si niml If th grph is n ulrin grph, MSA fins ulrin pths of th grph. Tbl 5-2 shows th jny mtrix of th grph. Tbl 5-2 Ajny mtrix of Fig.5-1 () j= b f g h i= b f g h Lok Twist us th ulrin pth of th grph foun only on, MSA prsnts mking prours s shown in Fig.5-2 bs on th pth. Lok Twist A b b F G f H g U f J Figur 5-2 Mking prours of Fig.5-1 h In Ltur Mo, w n s th orrsponing mking prours. Figur 5-3 shows th mking prours by Ltur Mo. Figur 5-3 xmpl of isply of Ltur Mo

6 W wr bl to rt th mol shown in Fig. 5-4 oring to th mking prour shown in Fig Figur 5-4 si niml m oring to th prsnt prours 6. onlusion In this ppr, w propos strutur nlysis n iffiulty rting of blloon moling by using grph thory. W fous on grph tht n b writtn by strok whn w mk blloon moling tht is m of on blloon; thrfor w tri to trt bubbls s gs n twist positions s vrtis in th grph. As rsult, mking prours of blloon moling ws obtin by using ulrin grph. In ition, w vlop MSM bs on strutur nlysis; n w onfirm th fftivnss of MSM by rting bsi mol of blloon moling with MSM. Aknowlgmnt I woul lik to thnk Mrvin Lnor for his onstnt support. Rfrns [1] Miyzki S, Ysu T, Yokoi S, Toriwki J: "An Intrtiv Simultion Systm of ORIGAMI s on Virtul Sp Mnipultion", Pro. I Intrntionl Workshop on Robot n Humn ommunition '92, pp ( ) [2] Jun Mitni n Hiroms Suzuki: "Mking Pprrft Toys from Mshs using Strip-bs Approximt Unfoling", AM Trnstions on Grphis, Vol. 23, No. 3, pp (2004) [3] Yuki Igrshi, Tko Igrshi: "Pillow: Intrtiv Flttning of 3 Mol for Plush Toy sign", Ltur Nots in omputr Sin 5166, pp.1-7 (2008) [4] ru Fif, r. ropo: lloon sulpting, Piilly ooks (1994) [5] Aln M. Gibbons: Algorithmi grph thory, mbrig Univrsity Prss (1985)