Construction of 3-D Paper-made Objects from Crease Patterns

This paper proposes an approach to constructing a 3-D paper-m ad e ob ject from a crease pattern, a set of line segm ents (creases) on a sheet of paper w hich is usually prod uced w hen a paper-m ad e ob ject such as origam i is unfold ed . F irst, a coord inate transform ation m ethod from 2 -D crease patterns to 3-D space is proposed . S ince faces (form ed b y the creases) m ore than tw o are lik ely transform ed onto the sam e plane in 3-D space, it is necessary to d ispose these faces consistently . Therefore, second , a m ethod for analy z ing positional relationships of the ov erlapping faces is proposed . F inally , w e show som e ex am ples of 3-D ob jects constructed b y our m ethod . The proposed m ethod is useful for pack aging and architectural m od eling. 1 I n tro d u ctio n O r ig am i is a f o r m o f v is u al and s c u lp tu ral rep res entatio n th at is d efi ned p r im arily by th e fo ld ing o f th e m ed iu m (u s u ally p ap er) . T rad itio nal o r ig am i m o d els h av e been p res ented by d r ill bo o k s , in wh ic h th e fo ld ing p r o c es s es c o ns is ting o f s ev eral s im p le fo ld ing o p eratio ns (e.g . m o u ntain-f o ld ing ) are intellig ibly ins tr u c ted s tep -by s tep th r o u g h a s eq u enc e o f illu s tratio ns . M iy az ak i et al. [ 4 ] h av e d ev elo p ed a v ir tu al interac tiv e m anip u latio n s y s tem f o r eac h s im p le f o ld ing o p eratio n. H o wev er, rec ently , we fi nd th at th ere are a lo t o f realis tic , m o d ern and c o m p lex o r ig am i m o d els d es ig ned no t by th e trad itio nal fo ld ing o p eratio ns bu t by g eo m etric o r ig am i d es ig n m eth o d s [ 3 ] . T h ere is no s p ec ifi c p r o c es s to f o ld th es e m o d els . T h e o r ig am i c reato r s d raw line s eg m ents (c reas es ) o nto a s h eet o f p ap er to g enerate a c reas e p attern, th e u nfo ld ed s tate o f an o rig am i m o d el. In o th er wo rd s , th ey d es ig n a new m o d el befo re th ey f o ld it. D es ig ning s u c h m o d els c o ns u m es m u c h tim e and energ y o f th e c reato r s bec au s e th ey h av e to im ag ine th e c o m p leted m o d el f r o m a c reas e p attern with o u t ac tu ally f o ld ing it. In th is p ap er, we p ro p o s e an ap p r o ac h th at rep res ents 3 D m o d els c o ns tr u c ted f r o m c reas e p atterns to h elp o r ig am i c reato r s eas ily s ee d es ig ned m o d els as o c c as io n d em and s . W e no tic e th e fo llo wing th ree p r o blem s : 1 . T h e c reas e p atterns g enerated by g eo m etric o r ig am i d es ig n m eth o d s d o no t inc lu d e any info r m atio n abo u t f o ld ing p r o c es s o r f o ld ing o p eratio ns . 2 . T h e fac es m o re th an two fo r m ed by c reas es are lik ely o v er lap p ed o n th e s am e p lane in 3 D s p ac e. 3 . S o m e c reas e p atterns allo w s ev eral way s o f f o ld ing . T o d eal with th es e p r o blem s , we fi rs t p r o p o s e a c o o r d inate trans f o r m atio n m eth o d th at is able to trans f o r m all fac es in a 2 D c reas e p attern into th e fac es in 3 D s p ac e s im u ltaneo u s ly , in s ec tio n 3 . T h en, in s ec tio n 4 , we d es c r ibe h o w to d is p o s e th e fac es o v er lap p ed o n th e s am e p lane in 3 D s p ac e s o th at th e p o s itio nal relatio ns am o ng th em are c o ns is tent. A lth o u g h th ere m ay be m u ltip le s o lu tio ns f o r way s o f f o ld ing , we are tr y ing to ex trac t o ne o f th e feas ible s o lu tio ns . A s related wo rk , U c h id a et al. [ 6 ] h av e p r o p o s ed an ap p r o ac h to d ed u c ing a fo ld ing p r o c es s f r o m a c reas e p attern o f o r ig am i m o d els , bu t it is p rem is ed th at th e o rig am i m o d els are d es ig ned by trad itio nal f o ld ing o p eratio ns . E is enberg et al. [ 2 , 1 ] h av e d is c u s s ed a fo ld ing net p r o blem , f o r th e p u rp o s e o f trans f o r m ing 3 D v ir tu al o bjec ts s u c h as p o ly h ed r o ns into a p ap er rep res entatio n o f th e m o d el as a fo r m o f h ar d c o p y f r o m v ir tu al env ir o nm ents , as th e o p p o s ite o f o u r ap p r o ac h . H o wev er, fac es o f th e 3 D p o ly g o n m o d els d o no t o v er lap eac h o th er. S o , th e s ec o nd p r o blem d es c r ibed abo v e d o es no t ex is t. 2 F ro m C re ase P atte rn s to 3 D M o d e ls In o r d er to c o ns tr u c t 3 D v ir tu al p ap erm ad e o bjec ts f r o m c reas e p atterns , th e ro tatio nal trans f o r m atio n bas ed o n c reas es u s ing th e ad jac ent relatio ns h ip am o ng fac es is need ed . W e g enerate a g rap h o f th e c reas e p attern in wh ic h no d es rep res ent fac es and ed g es rep res ent c reas es is c o ns titu ted , wh ic h c o m es to o btain th e p o s itio nal relatio ns h ip s am o ng fac es eas ily (F ig . 1 ) . W e c all th e g rap h crease pattern graph (C P -graph f o r s h o r t) . F ig u re 1 . A c reas e p attern (left) , and fac e-c reas e g rap h (rig h t) . 2 .1 C re ase P atte rn T h e c reas e p attern is g iv en and s atis fi es th e fo llo wing p rec o nd itio ns ; • f o ld -able, • h as a s et o f fac es d iv id ed by c reas es , and • all th e fac es are p o ly g o ns arrang ed o n xy p lane in th e rig h th and ed c o o r d inate s y s tem , and o n th e o bv ers e s id e MVA2005 IAPR Conference on Machine VIsion Applications, May 16-18, 2005 Tsukuba Science City, Japan