Grid Vertex-Unfolding Orthogonal Polyhedra

Transcription

1 Gid Vete-Unfolding Othogonal Polheda Miela Damian Robin Flatland Joseph O Rouke Septembe 25, 2006 bstact n edge-unfolding of a polhedon is poduced b cutting along edges and flattening the faces to a net, a connected plana piece with no ovelaps. gid unfolding allows additional cuts along gid edges induced b coodinate planes passing though eve vete. veteunfolding pemits faces in the net to be connected at single vetices, not necessail along edges. We show that an othogonal polheda of genus eo has a gid vete-unfolding. (Thee ae othogonal polheda that cannot be vete-unfolded, so some tpe of gidding of the faces is necessa.) Fo an othogonal polhedon P with n vetices, we descibe an algoithm that vete-unfolds P in O(n 2 ) time. Enoute to eplaining this algoithm, we pesent a simple vete-unfolding algoithm that equies a 3 1 efinement of the vete gid. Kewods: vete-unfolding, gid unfolding, othogonal polheda, genus-eo. 1 Intoduction Two unfolding poblems have emained unsolved fo man eas [DO05a]: (1) an eve conve polhedon be edge-unfolded? (2) an eve polhedon be unfolded? n unfolding of a 3D object is an isometic mapping of its suface to a single, connected plana piece, the net fo the object, that avoids ovelap. n edge-unfolding achieves the unfolding b cutting edges of a polhedon, wheeas a geneal unfolding places no estiction on the cuts. net epesentation of a polhedon finds use in a vaiet of applications [O R00] fom flattening monke bains [SSW89] to manufactuing fom sheet metal [Wan97] to low-distotion tetue mapping [THM04]. It is known that some nonconve polheda cannot be unfolded without ovelap with cuts along edges. Howeve, no eample is known of a nonconve polhedon that cannot be unfolded with unesticted cuts. dvances on these difficult poblems have been made b specialiing the class of polheda, o easing the stingenc of the unfolding citeia. On one hand, it was established in [DD + 98] that cetain subclasses of othogonal polheda those whose faces meet at angles that ae multiples of 90 have an unfolding. In paticula, the class of othostacks, stacks of etuded othogonal polgons, was poven to have an unfolding (but not an edge-unfolding). On the othe hand, loosening the citeia of what constitutes a net to pemit connection though points/vetices, the so-called vete-unfoldings, led to an algoithm to vete-unfold an tiangulated manifold [DEE + 03] (and indeed, an simplicial manifold in highe dimensions). vete This is a significant evision of the pelimina vesion that appeaed in [DFO06]. Dept. omput. Sci., Villanova Univ., Villanova, P 19085, US. miela.damian@villanova.edu. Dept. omput. Sci., Siena ollege, Loudonville, NY 12211, US. flatland@siena.edu. Dept. omput. Sci., Smith ollege, Nothampton, M 01063, US. oouke@cs.smith.edu. Suppoted b NSF awad DUE

2 unfolding maps the suface to a single, connected piece P in the plane, but P ma have cut vetices whose emoval disconnect P. second loosening of the citeia is the notion of gid unfoldings, which ae especiall natual fo othogonal polheda. gid unfolding adds edges to the suface b intesecting the polhedon with planes paallel to atesian coodinate planes though eve vete. The two appoaches wee ecentl maied in [DIL04], which established that an othostack ma be gid vete-unfolded. Fo othogonal polheda, a gid unfolding is a natual median between edge-unfoldings and unesticted unfoldings. Ou main esult is that an othogonal polhedon, without shape estiction ecept that its suface be homeomophic to a sphee, has a gid vete-unfolding. We pesent an algoithm that gid vete-unfolds an othogonal polhedon with n vetices in O(n 2 ) time. We also pesent, along the wa, a simple algoithm fo 3 1 efinement unfolding, a weakening of gid unfolding that we define below. We believe that the techniques in ou algoithms ma help show that all othogonal polheda can be gid edge-unfolded. 2 Definitions k 1 k 2 efinement of a suface [DO05b] patitions each face into a k 1 k 2 gid of faces. We will conside efinements of gid unfoldings, with the convention that 1 efinement is an unefined gid unfolding. We distinguish between a stict net, in which the net bounda does not self-touch, and a net fo which the bounda ma touch, but no inteio points ovelap. The latte coesponds to the phsical model of cutting out the net fom a sheet of pape, with pehaps some cuts epesenting edge ovelap, and this is the model we use in this pape. We also insist as pat of the definition of a vete-unfolding, again keeping in spiit with the phsical model, that the unfolding path neve self-cosses on the suface in the following sense. If (,,, D) ae fou faces incident in that cclic ode to a common vete v, then the net does not include both the connections v and vd. 1 We use the following notation to descibe the si tpe of faces of an othogonal polhedon, depending on the diection in which the outwad nomal points: font: ; back: +; left: ; ight: +; bottom: ; top: +. We take the -ais to define the vetical diection; vetical faces ae paallel to the -plane. Diections clockwise (cw), and counteclockwise (ccw) ae defined fom the pespective of a viewe positioned at =. We distinguish between an oiginal vete of the polhedon, which we call a cone vete o just a vete, and a gidpoint, a vete of the gid (which might be an oiginal vete). gidedge is an edge segment with both endpoint gidpoints, and a gidface is a face of the gid bounded b gidedges. Let O be a solid othogonal polhedon with the suface homeomophic to a sphee (i.e, genus eo). Let be the plane = i othogonal to the -ais. Let Y 0, Y 1,...,,... be a finite sequence of paallel planes passing though eve vete of O, with 0 < 1 < < i <. We define lae i to be the potion of O between planes and +1. Obseve that a lae ma include a collection of disjoint connected components of O; we call each such component a slab. suface piece that suounds a slab is called a band. Refeing to Fig. 1a, lae 0, 1 and 2 each contain one slab (with oute bands, and D, espectivel). Note that each slab is bounded b an oute (suface) band, but it ma also contain inne bands, bounding holes. Oute bands ae called potusions and inne bands ae called dents ( in Fig. 1a). In othe wods, band is a potusion if a tavesal of the 1 This was not pat of the oiginal definition in [DEE + 03] but was achieved b those unfoldings. 2

3 im of in, ccw fom the viewpoint of =, has the inteio of O to the left of, and a dent if this tavesal has the inteio of O to the ight. Fo fied i, define P = O as the potion of the suface of O ling in plane. P + is the potion of P with nomal in the diection + (composed of back faces), and P the potion with nomal in the diection (composed of font faces). convention, band points in P that ae not incident to eithe font o back faces (e.g., when one band aligns with anothe), belong to both P + and P. Thus P = P + P. Fo a band, Let R i () = be the polgon in detemined b the im of band, and i () the closed egion of whose bounda is R i (). Fo an two bands and, let i () = i () i () and let R i () be the bounda of i (). D Y 3 Y 2 Y 1 D Y 0 Figue 1: Definitions. Shaded connected pieces ae bands;, and D ae potusions; is a dent; 2 () coincides with the back face of ; R 2 (D) is maked in dashed lines. The adjacenc stuctue of bands is a tee. 3 Dents vs. Potusions We obseve that dents ma be teated eactl the same as potusions with espect to unfolding, because an unfolding of a 2-manifold to anothe suface (in ou case, a plane) depends onl on the intinsic geomet of the suface, and not on how it is embedded in R 3. Note that we ae onl concened with the final unfolded flat state [DO05a], and not with possible intesections duing a continuous sequence of patiall unfolded intemediate states. Ou unfolding algoithm elies solel on the amount of suface mateial suounding each point: the cclic odeing of the faces incident to a vete, and the pai of faces shaing an edge. ll these local elationships emain unchanged if we conceptuall pop-out dents to become potusions, i.e., a Flatland ceatue living in the suface could not tell the diffeence; no can ou algoithm. We note that the popping-out is conceptual onl, fo it could poduce self-intesecting objects. lso dents ae gidded independentl of the est of the object, so that it would not matte whethe the ae popped out o not. lthough the dent/potusion distinction is ielevant to the unfolding, the inteelationships between dents and potusions touching a paticula do depend on this distinction. To cite just the simplest eample, thee cannot be two nested potusions to the same side of, but a potusion could have a dent in it to the same side of (e.g., potusion encloses dent to the same side of Y 1 in Fig. 1). These elationships ae cucial to the connectivit of the band gaph G b, discussed in Sec. 8. 3

4 4 Oveview The two algoithms we pesent shae a common cental stuctue, with the second achieving a stonge esult; both ae vete-unfoldings that use othogonal cuts onl. We note that it is the estiction to othogonal cuts that makes the vete-unfolding poblem difficult: if abita cuts ae allowed, then a geneal vete-unfolding can be obtained b simpl tiangulating each face and appling the algoithm fom [DEE + 03]. The (3 1)-algoithm unfolds an genus-0 othogonal polhedon that has been efined in one diection 3-fold. The bands themselves ae neve split (unlike in [DD + 98]). The algoithm is simple. The (1 1)-algoithm also unfolds an genus-0 othogonal polhedon, but this time achieving a gid vete-unfolding, i.e., without efinement. This algoithm is moe delicate, with seveal cases not pesent in the (3 1)-algoithm that need caeful detailing. leal this latte algoithm is stonge, and we va the detail of pesentation to favo it. The oveall stuctue of the two algoithms is the same: 1. band unfolding tee T U is constucted b shooting as veticall fom the top of bands. The oot of T U is a fontmost band (of smallest -coodinate), with ties boken abitail. 2. fowad and etun connecting path of vetical faces is identified, each of which connects a paent band to a child band in T U. 3. Each band is unfolded hoiontall as a unit, but inteupted when a connecting path to a child is encounteed. The paent band unfolding is suspended at that point, and the child band is unfolded ecusivel. 4. The vetical font and back faces of each slab ae patitioned accoding to an illumination model, with vaiations fo the moe comple (1 1)-algoithm. These vetical faces ae attached below and above appopiate hoiontal sections of the band unfolding. The final unfolding las out all bands hoiontall, with the vetical faces hanging below and above the bands. Non-ovelap is guaanteed b this stict two-diection stuctue. lthough ou esult is a boadening of that in [DIL04] fom othostacks to all othogonal polheda, we found it necessa to emplo techniques diffeent fom those used in that wok. The main eason is that, in an othostack, the adjacenc stuctue of bands ields a path, which allows the unfolding to poceed fom one band to the net along this path, neve needing to etun. In an othogonal polhedon, the adjacenc stuctue of bands ields a tee (cf. Fig. 1b). Thus unfolding band-b-band leads to a tee tavesal, which equies tavesing each ac of the tee in both diections. It is this aspect which we conside ou main novelt, and which leads us to hope fo an etension to edge-unfoldings as well. 5 (3 1)-lgoithm 5.1 omputing the Unfolding Tee T U Define a -beam to be a vetical ectangle on the suface of O connecting two band ims whose top and bottom edges ae gidedges. In the degeneate case, a -beam has height eo and connects two ims along a section whee the coincide. We sa that two bands and b 2 ae -visible if thee eists a -beam connecting a gidedge of to a gidedge of b 2. Thee can be man -beams connecting two bands, so fo each pai of bands we select a epesentative -beam of minimal (vetical) height. Let G be the gaph that contains a node fo each band of O and an ac fo each pai of -visible bands. It easil follows fom the connectedness of the suface of O that G is connected. Let the 4

5 unfolding tee T U be an spanning tee of G, with the oot selected abitail fom among all bands adjacent to Y 0. ppling the 31 efinement patitions each font, back, top and bottom face of O into a 3 1 gid of faces. This patitions the top and bottom edges of each -beam into thee efined gidedges and divides the beam itself into thee vetical columns of efined gidfaces. Fo a band in T U with paent, let e be the gidedge on s im whee the -beam fom attaches. We define the pivot point to be the 1 3 -point of e (o, in cicumstances to be eplained below, the 2 3-point), and so it coincides with a point of the 3 1-efined gid. The unfolding of O will follow the connecting vetical a that etends fom on to. Note that if e belongs to both and, then the a connecting and degeneates to a point. To eithe side of a connecting a we have two connecting paths of vetical faces, the fowad and etun path. In Fig. 2a, these connecting paths ae the shaded stips on the font face of. 5.2 Unfolding ands into a Net Stating at a fontmost oot band, each band is unfolded as a conceptual unit, but inteupted b the connecting as incident to it fom its font and back faces. In Fig. 2, band is unfolded as a ectangle, but inteupted at the as connecting to (font childen), and (back child). t each such a the paent band unfolding is suspended, the unfolding follows the fowad connecting path to the child, the child band is ecusivel unfolded, then the unfolding etuns along the etun connecting path back to the paent, esuming the paent band unfolding fom the point it left off. Fig. 2 illustates this unfolding algoithm. The cw unfolding of, laid out hoiontal to the ight, is inteupted to tavese the fowad path down to, and is then unfolded as a ectangle (composed of its contiguous faces). The base of the connecting a is called a pivot point because the ccw unfolding of is otated 180 ccw about so that the unfolding of is also hoiontal to the ight. It is onl hee that we use point-connections that ende the unfolding a vete-unfolding. The unfolding of poceeds ccw back to, cosses ove to unfold, then a cw otation b 180 aound the second image of pivot b oients the etun path to so that the unfolding of continues hoiontal to the ight. Note that the unfolding of is itself inteupted to unfold child D. lso note that thee is edge ovelap in the unfolding at each of the pivot points. The eason fo the 3 1 efinement is that the uppe edge e of the back child band has the same (, )-coodinates as the uppe edge e of on the font face. In this case, the faces of band induced b the connecting paths to would be oveutilied if thee wee onl two. Let, a 2, a 3 be the thee faces of induced b the 3 1 efinement of the connecting path to, as in Fig. 2. Then the unfolding path winds aound to, follows the fowad connecting path to, etuns along the etun connecting path to a 2, cosses ove and unfolds on the back face, with the etun path now joining to a 3, at which point the unfolding of esumes. In this case, the pivot point fo is the 2 3 -point of e. Othe such conflicts ae esolved similal. It is now eas to see that the esulted net has the geneal fom illustated in Fig. 2b: 1. The faces of each band fall within a hoiontal ectangle whose height is the band width. 2. These band ectangles ae joined b vetical connecting paths on eithe side, connecting though pivot points. 3. The stip of the plane above and below each band face that is not incident to a connecting path, is empt. 4. The net is theefoe an othogonal polgon monotone with espect to the hoiontal. 5

6 a 2 a 3 ' ' e' e c d ' D (c) a 2 ' ' D ' a 3 d c D c d D Figue 2: Othogonal polhedon. Unfolding tee T U. (c) Unfolding of bands and font (hachued) face pieces. Vete connection though the pivots points,, c, d is shown eaggeated fo clait. 5.3 ttaching Font and ack Faces to the Net Finall, we hang font and back faces fom the bands as follows. The font face of each band is patitioned b imagining to illuminate downwad lightas fom the im in the font face. The pieces that ae illuminated ae then hung veticall downwad fom the hoiontal unfolding of the band. The potions unilluminated will be attached to the obscuing bands. In the eample in Fig. 2, this illumination model patitions the font face of into thee pieces (the stiped pieces in Fig. 2b). These thee pieces ae attached unde ; the potions of the font face obscued b but illuminated downwad b ae hung beneath the unfolding of (not shown in the figue), and so on. ecause the vetical illumination model poduces vetical stips, and because the stips above and below the band unfoldings ae empt, thee is alwas oom to hang the patitioned font face. Thus, an othogonal polgon ma be vete-unfolded with a 3 1 efinement of the vete gid. lthough we believe this algoithm can be impoved to 2 1 efinement, the complications needed to achieve this ae simila to what is needed to avoid efinement entiel, so we instead tun diectl to 1 1 efinement. 6 (1 1)-lgoithm lthough the (1 1)-algoithm follows the same geneal outline as the (3 1)-lgoithm, thee ae significant complications, which we outline befoe detailing. Fist, without the efinement of -beams into thee stips to allow avoidance of conflicts on opposite sides of a slab (e.g., and in Fig. 2a), we found it necessa to eplace the -beams b a pai of -as that ae in some sense the bounda edges of a -beam. Selecting two as pe band pemits a 2-coloing algoithm (Theoem 3) to identif as that avoid conflicts. Geneating the a-pais (Sec ) equies cae to ensue that the band gaph G b is connected (Sec. 8). This gaph, and the 2-coloing, lead 6

7 to an unfolding tee T U (Sec. 6.2). Fom hee on, thee ae fewe significant diffeences compaed to the (3 1)-lgoithm. Without the luu of efinement, thee is moe need to shae vetical paths on the vetical face of a slab (Fig. 11). Finall, the vetical connecting paths obscue the illumination of some gid faces, which must be attached to the connecting paths. We now pesent the details, in this ode: 1. Select Pivot Points (Sec. 6.1) via a. Ra-Pai Geneation (Sec ) b. Ra Gaph (Sec ) 2. onstuct T U (Sec. 6.2) 3. Select onnecting Paths (Sec ) 4. Detemine Unfolding Diections (Sec ) 5. Recuse: a. Unfold ands into a Net (Sec. 6.3) b. ttach Font and ack Faces to the Net (Sec. 6.4) 6.1 Selecting Pivot Points The pivot a fo a band is the gidpoint of whee the unfolding of stats and ends. The -edge of incident to a is the fist edge of that is cut to unfold. Let be an abita band delimited b planes and +1. Sa that two gidpoints u and w +1 ae in conflict if the upwad as emeging fom u and w hit the endpoints of the same -edge of ; othewise, u and v ae conflict-fee. If u lies eithe on a vetical edge, o on a veticall eteme hoiontal edge, then the a at u degeneates to u itself. Ou goal is to select conflict-fee pivots fo all bands in T U, which will help us avoid late competition ove the use of cetain faces in the unfolding, an issue that will become clea in Sec Selecting these pivots is the most delicate aspect of the (1 1)-algoithm. Ultimatel, we epesent pivoting conflicts in the fom of a gaph G (Sec ), fom which T U will be deived Ra-Pai Geneation In ode to avoid pivoting conflicts, fo each band we will need two choices fo its connecting a. Thus the algoithm geneates the as in pais. ecause thee is no efinement, the two as oiginate at gid points on the same band, but the ma teminate on diffeent bands. simple eample is shown in Figue 3a, whee the a pai oiginating on band D hits two diffeent bands, and. This eample also suggests that one cannot conside a pais connecting pais of bands, as in the (3 1)-algoithm (which would connect D to in this eample), but instead we focus on shooting pais of as upwad fom stategic locations on the bounda of each band, and then selecting a subset of these as so that the conflicts can be esolved and T U is connected. To ensue connectedness of all bands, seveal a-pais must be issued upwad fom each band. Figue 3b shows an eample: no pai of as can emanate upwad fom the top of P o P ; one pai of as shoots upwad fom the top of each component of P : ( 1, 2 ) connects to and ( 3, 4 ) connects to ; finall, one pai of as ( 5, 6 ) issues fom the top of P, which connects to D. So, oveall, thee pais of as ae geneated fo band. We now tun to descibing in detail the method fo geneating a-pais. Let band intesect plane. The algoithm is a fo-loop ove all. Let 1, 2,..., m be the components of, defined as follows. Take all the maimal components of P + that contain an -gidedge, and union with all the maimal components of P that contain an -gidedge. 7

8 D 1 5 D 6 Figue 3: The a pai ( 1, 2 ) connects band D to two diffeent bands and. To ensue connectivit, thee pais of as must be issued fo : ( 1, 2 ), ( 3, 4 ) and ( 5, 6 ). We define S( j ) as the set of all potential as shooting upwad fom j. Moe pecisel, S( j ) consists of the set of all segments s = (a, b), with a j, such that 1. Eithe s is a point, with b = a, o s P is vetical (paallel to ), with a below b. 2. b fo some band. 3. The open segment s \ {a, b} ma contain points of (see 2 in Fig. 4b), but no othe band points. Fo each band, fo each component j, if S( j ) is nonempt, we select one a pai ( 1, 2 ), such that (i) 1 is the leftmost segment in S( j ) that is incident to a highest -gidedge in j, and (ii) 2 is the segment one -gidedge to the ight of 1. Fig. 4 shows a few eamples. s mentioned 2 1 ' ' (c) Figue 4: Geneating a-pais: ( 1, 2 ) fo ; S() =. ( 1, 2 ) fo (note that 2 uns along souce band ); degeneate a-pai ( 1, 2 ) fo. (c) S() = ; ( 1, 2 ) fo. above, seveal a pais could be geneated fo an one band, and indeed seveal pais connecting two bands. Let G b be the band gaph whose nodes ae bands. Two bands ae connected b an ac in G b if the a-pai algoithm geneates a a connecting them. We call a collection of bands in G b a- 8

9 connected if the ae in the same connected component of G b. We establish that G b is a connected gaph, i.e., all bands ae a-connected to one anothe, even if onl one a pe pai is emploed: Lemm G b is connected. Futhemoe, the subgaph of G b induced b eactl one a pe apai (abitail selected) is connected. Wheeas the connectedness of bands b -beams in the (3 1)-algoithm is staightfowad, the comple possible elationships between bands makes connectedness via as moe subtle. We elegate the poof to the ppendi (Sec. 8) in ode to not inteupt the main flow of the algoithm. The ove-geneation of a-pais noted above is designed to ensue connectedness. Eventuall man as will be discaded b the time T U is constucted in Sec Ra Gaph G One pai of as pe pai of bands suffices to ensue that all bands ae a-connected. If multiple pais of as eist fo a pai of bands, pick one pai abitail and discad the est. Then define a a gaph G as follows. The nodes of G ae vetical as in a plane, pehaps degeneating to points, connecting gidpoints between two bands that both intesect. The acs of G each ecods a potential pivoting conflict, and ae of two vaieties: (i) The nodes fo the two as issuing fom the top of one band ae adjacent in G. all such acs -acs; geometicall the can be viewed as paallel to the -ais. (ii) The nodes fo two as incident to opposite sides of the im of a band, connected b a -segment on the band, ae adjacent in G. all such acs -acs; geometicall the can be viewed as paallel to the -ais. Fig. 5 shows two simple eamples of G involving nodes on opposite sides of one band. efoe G D G Figue 5: uilding G. G is a 4-ccle; { 1, 2 } and { 3, 4 } ae -acs; an othe ac is a -ac. G is a path; { 2, 5 } and { 3, 6 } ae -acs; an othe ac is an -ac. poceeding, we list the consequences of the two tpes of acs in G. ssuming that we can 2-colo G {ed, blue}, and we select the base of (sa) the ed as as pivots, then: (i) eactl one pivot is selected fo each band, and (ii) no two pivot as ae in conflict acoss a band. So ou goal now is to show that G is 2-coloable. ecause a gaph is 2-coloable if and onl if it is bipatite, and a gaph is bipatite if and onl if eve ccle is of even length, we aim to pove that eve ccle in G is of even length. We stat b listing a few elevant popeties of G : 9

10 1. Eve node G has eactl one incident -ac. The as ae geneated in pais, and the pais ae connected b an -ac. s no such a is shaed between two bands, at most one -ac is incident to an. 2. Nodes have at most degee 3, with the following stuctue: degee-1 nodes have an incident -ac; degee-2 nodes have both an incident - and -ac; and degee-3 nodes have an incident -ac and two incident -acs. 3. Each -ac spans eactl one pai of adjacent -gidlines, and each -ac spans eactl one band im-to-im. The fome is b the definition of a pais, which issue fom adjacent gidpoints, and the latte follows fom the gid patitioning of the object into bands G (c) 5 Figue 6: (a,b) Two side views of an object; -as and -acs ae maked with thick lines. (c) G coodinatied into -plane Π; ( 5, 6, 10, 9 ) is a 4-ccle; ( 1, 3, 4, 8, 7, 11, 12, 9, 5, 2 ) is 0-ccle. Ou net step equies embedding G in an -plane Π. Towad that end, we coodinatie the nodes and acs of G as follows. node G is a -a, and is assigned the (, ) coodinates of the a. Note that this means collinea as get mapped to the same point; howeve we teat them as distinct. The -acs ae then paallel to the -ais, and the -acs ae paallel to the -ais. In essence, this coodinatiation is a view fom = +. Fig. 6 shows a moe comple eample illustating this viewpoint. The object is composed of 7 bands i, one of which ( 3 ) is a dent. Thee ae 12 a nodes, two pais of which lie on the same -vetical line, namel ( 4, 5 ) and ( 8, 9 ). Note that thee ae -acs cossing both the top of and the bottom 2 of 4. The gaph G has a 4-ccle and 0-ccle, both detailed in the caption (as well as 2-ccle not detailed). Lemma 2 Eve ccle in G is of even length. 2 dent is included in this eample pecisel to intoduce such a bottom -ac into G. 10

11 Poof: Let be a ccle in G. The coodinatiation descibed above maps to a (pehaps selfcossing) closed path in the -plane Π, a path which ma visit the same (, ) point moe than once, and/o tavese the same edge in Π moe than once. n such closed path on a gid must have even length, fo the following eason. Fist, b Popet (3) above, each edge of the path in Π connects adjacent gid lines: an edge neve jumps ove one o moe gid lines. Second, an such closed lattice path changes pait with each step, in the following sense. Numbe the - and -gidlines with integes 0, 1, 2,... left to ight and bottom to top espectivel. Define the pait of a gidpoint of Π to be the sum of its - and -gidline coodinates, mod 2. Then each step of the path, necessail in one of the fou compass diections, changes pait, as it changes onl one of o. Retuning to the stat point to close the path must etun to the stating coodinates, and so to the same pait. Thus, thee must be an even numbe of pait changes along an closed path. Theefoe, has an even numbe of edges. We have now established this: Theoem 3 G is 2-coloable. Note that nowhee in the above poof do we assume genus eo, so this theoem holds fo polheda of abita genus. and pivoting. Theoem 3, we can 2-colo the nodes of G {ed,blue}. We choose all ed a-nodes of G to be pivoting as, in that thei base points become pivot points. s emaked befoe, this selection guaantees that each band is pivoted, and no two pivots ae in conflict. 6.2 Unfolding Tee T U The net task is to define a band spanning tee T U, based on the band gaph G b. Define G b, to etain the just the acs of G b coesponding to the ed a nodes (in the above 2-coloing) in G. This maintains the connectivit of Lemm. Then take T U to be an spanning tee of G b ooted at a fontmost band. With T U finall in hand, the emainde of the (1 1)-algoithm follows the oveall stuctue of the 3 1 algoithm, with vaiations as mentioned befoe, as detailed below Selecting onnecting Paths Having established a pivot point fo each band, we ae now ead to define the fowad and etun connecting paths fo a child band in T U. Let be an abita child of a band. If intesects, both fowad and etun connection paths fo educe to the pivot point (e.g., u in Fig. 7). If does not intesect, then a a connects to (Figs. 8a and 10a). The connecting paths ae the two vetical paths sepaated b compised of the gidfaces shaing an edge with (paths and a 2 in Figs. 8a and 10a). The path fist encounteed in the unfolding of is used as a fowad connecting path; the othe path is used as a etun connecting path Detemining Unfolding Diections top-down tavesal of T U assigns an unfolding diection to each band in T U as follows. The oot band in T U ma unfold eithe cw o ccw, but fo definiteness we set the unfolding diection to cw. Let be the band in T U cuentl visited and let be the paent of. If the upwad a incident to connects to a bottom gidpoint of, and if unfolds cw(ccw), then unfolds 11

12 cw(ccw). Othewise, connects to a top o a side (fo degeneate as) gidpoint of ; in this case, if unfolds cw(ccw), then unfolds ccw(cw). In othe wods, and unfold in a same diection if hangs below, and in opposite diection othewise. 6.3 Unfolding ands into a Net Let be a band to unfold, initiall the oot band. The unfolding of stats at a and poceeds in the unfolding diection (cw o ccw) of. Hencefoth we assume w.l.o.g. that the unfolding of poceeds cw (w..t. a viewpoint at = ); the ccw unfolding of is a vetical eflection of the cw unfolding of. In the following we descibe ou method to unfold eve child of ecusivel, which falls natuall into seveal cases. a1 b b a0 0 1 Figue 7: Unfolding when the a connecting to degeneates to. ase 1: Pivot. Then, wheneve the unfolding of eaches, we unfold as in Fig. 7. The unfolding uses the two band faces of incident to ( and in Fig. 7). The gidface of ccw of gets otated aound so that the ccw unfolding of etends hoiontall to the ight. The unfolding of poceeds ccw back to, then the face incident to gets oiented about so that the unfolding of continues hoiontal to the ight. Note that, because the pivots of an two childen of ae conflict-fee, thee is no competition ove the use of and in the unfolding. Note also that the unfolding path does not self-coss. Fo eample, the cclic ode of the faces incident to u in Fig. 7a is (, font,,, back, ), and the unfolding path follows (,,...,, ). ase 2: Pivot and the (fowad, etun) connecting paths fo do not ovelap othe connecting paths (ecept at thei boundaies); we will late see that this ma happen. Let us settle some notation fist (cf. Fig 8a): is the a connecting to ; and a 2 ae fowad and etun connecting paths fo (one to eithe side of ); u 1 is the endpoint of that lies on ; and u 2 is the othe endpoint of the -edge of incident to u 1. We discuss thee situations: ase 2a: u 1 is neithe a efle cone no a bottom cone of. In this case, wheneve the unfolding of eaches, the unfolding of poceeds as in Fig. 8a o Fig. 8b, depending on whethe touches a left face of o not. In eithe case, if is the face of etending ccw left of, otate so that the unfolding of etends hoiontal to the ight, ecusivel unfold, then otate the etun path a 2 about so that the unfolding of continues hoiontal to the ight. 12

13 u a 2 3 u 1 a 2 a u 3 2 a 2 u 1 a 3 a 2 u 2 a 3 b b 1 1 b s a2 Figue 8: Unfolding : u 1 is not a cone vete of incident to a left face of incident to a top face of b. ase 2b: u 1 is a efle cone of. In this case, the unfolding of poceeds as in Fig. 9(a, b). It is the eistence of the vetical stip incident to u 1 (maked t in Fig. 9) that makes handling this case diffeent fom ase 2a above. Note howeve that the eistence of t implies the eistence of at least two gidfaces on eithe the etun path o the fowad path fo, depending on whethe t is a left (Fig. 9a) o a ight (Fig. 9b) stip of faces. In the fome case the unfolding stats as in ase 2a (Fig. 9a), and once the unfolding of etuns to, it continues along the etun path up to u 1, then unfolds t and oients it about u 1 in such a wa that the unfolding of continues hoiontal to the ight. The potion of the etun path that etends above u 1 (a 20 in Fig. 9a) gets attached below the adjacent top face of (a 3 in Fig. 9a). u 2 t a 3 a 20 u 1 a t a3 u 1 a 2 a 21 u 2 t a 3 u 2 t 1 u 1 a 20 0 u 1 Figue 9: Unfolding : u 1 is a cone vete of. t is a left stip t is a ight stip. If t is a stip of ight faces, then t gets unfolded befoe descending along the fowad path down to, as in Fig. 9b (note the vetical smmet with the unfolding in Fig. 9a); the unfolding of then poceeds as in ase 2a (Fig. 8b). 13

14 ase 2c: u 1 is a bottom cone of. In this case, the unfolding poceeds as in Fig. 10a o Fig. 10b, depending on whethe u 1 is a ight o a left bottom cone of. The unfolding illustated in Fig. 10a follows the familia unfolding patten: oient the face of ccw left of so that the unfolding of etends to the ight; once the unfolding of etuns to, follow the etun path back to and unfold the face of cw to the ight of u 1 (a 3 in Fig. 10a) so that the unfolding of continues hoiontal to the ight. simila patten applies to the case illustated in Fig. 10b, ' ' u 1 a 3 u 1 a 3 u 2 u 1 a 2 b0 t a 2 t b b b 0 1 a 2 u 1 a 2 a 3 a 3 Figue 10: Unfolding : u 1 is a bottom cone of ightmost, and leftmost face of veticall aligned with leftmost face of. with one subtle diffeence meant to aid in unfolding font and back faces (discussed in Sec. 6.4): in unfolding bands, we aim at maintaining the vetical position of the (fowad, etun) connecting paths in the unfolding, so that vetical stips hanging below these connecting paths could also hang veticall in the unfolding. Moe on this in Sec Obseve that and a 2 fom Fig. 10a hang downwad in the unfolding. Howeve, if a 2 wee to maintain its vetical position in the unfolding fom Fig. 10b, it would not be possible to oient a 3 aound u 1 so as to continue unfolding hoiontal to the ight of a 2. This is the eason fo emploing the face maked t in the unfolding, so that vetical sides of t emain vetical in the unfolding, and an face stip hanging below t could be attached to t veticall in the unfolding. We note that Fig. 10 illustates onl the situation in which is incident to a left face of, but it should not be difficult to obseve that an eact same idea applies to an top pivot of ; the pivot position onl affects the stat and end unfolding position of, and evething else emains the same. ase 3: Pivot and a connecting path fo ovelaps a connecting path fo anothe descendant of. This case is slightl moe comple, because it involves conflicts ove the use of the connecting paths fo. The following thee situations ae possible. ase 3a: The fowad path fo ovelaps the etun path fo anothe descendant of. This situation is illustated in Fig. 11a. In this case, the unfolding stats as soon as the unfolding 14

15 along the etun path fom to meets a face of incident to (face in Fig. 11a). t this point gets ecusivel unfolded as befoe (see Fig. 11b), then the unfolding continues along the etun path fo back to. Fig. 11b shows face in two positions: we let hang down onl if the net face to unfold is a ight face of a child of (see the tansition fom k 7 to c 5 in Fig. 12); othewise, use in the upwad position, a feedom pemitted to us b otating about vete u. u 2 u c c 21 c 20 a 2 b 3 c 1 c 2 c a 21 a 20 (c) a 2 c 21 c (d) c c 20 a c 1 21 b 3 u a 2 c (e) Figue 11: Retun path fo includes c 20, c 21, ; fowad path fo is. Unfolding fo (c) Retun path fo includes a 20, a 21, c 1 ; fowad path fo is c 1. (d) Retun path fo is a 2 ; fowad path fo includes a 2, c 21. (e) Fowad (etun) paths ae identical fo and. ase 3b: The etun path a 2 fo ovelaps the fowad path fo anothe descendant of. This situation is illustated in Figs. 11c and 11d. The case depicted in Fig. 11c is simila to the one in Fig. 11a and is handled in the same manne. Fo the case depicted in Fig. 11d, notice that a 2 is on both the fowad path fo and the etun path fo. Howeve, no conflict occus hee: fom a 2 the unfolding continues downwad along the fowad path to and unfolds net. ase 3c: The fowad path fo ovelaps the fowad path fo anothe descendant of. This situation occus when eithe o anothe band incident to is a dent, as illustated in Figs. 11e. gain, no conflict occus hee: the ecusive unfolding of, which etuns to c =, is followed b the ecusive unfolding of, which etuns to, then the unfolding continues along the etun path fo () back to. Fig. 12 shows a moe comple eample that emphasies these subtle unfolding issues. Note that the etun path k 1, k 8, k 9 fo ovelaps the fowad path k 9 fo ; and the etun path k 5, k 6 and k 7 fo G ovelaps the fowad path fo H, which includes k 7. The unfolding poduced b the method descibed in this section is depicted in Fig

16 u 1 c 0 c 1 c 2 c 3 c 4 k 9 c 9 k h 3 u 7 3 c h k H 8 c 0 k 6 h u 0 1 u k k 3 k d 4 k 5 k d 0 d D g 0 3 k1 g g G 0 k 0 g 0 g 3 k 1 b 9 k5 g u 2 h 0 h1 k 2 d 0 d 4 k 3 d d k 4 h2 h3 k6 h k 7 g c5 g0 b 9 c 5 c 9 k 8 k9 c u 3 Figue 12: n eample. The vete-unfolding. 6.4 ttaching Font and ack Faces to the Net Font and back faces of a slab ae hung fom bands following the basic idea of the illumination model discussed in Sec Thee ae thee diffeences, howeve, caused b the emploment of some font and back gidfaces fo the connecting paths, which can block illumination fom the bands. 1. We illuminate both upwad and downwad fom each band: each -edge illuminates the vetical face it attaches to. This alone alead suffices to handle the eample in Fig. 12: all vetical faces ae illuminated downwad fom the top of, upwad fom the bottom of, and upwad fom the top of. 2. Some gidfaces still might not be illuminated b an bands, because the ae obscued both above and below b paths in connecting faces. Theefoe we incopoate the connecting faces into the band fo the puposes of illumination. Fo eample, in Fig. 10a, a 2 illuminates downwad and illuminates upwad. The eason this woks is that, with one eception, each vetical connecting stip emains vetical in the unfolding, and so illuminated stips can be hung safel without ovelap. 3. The one eception is the fowad connecting path in Fig. 10b. This paths unfolds on its side, i.e., what is vetical in 3D becomes hoiontal in 2D. Note, howeve, that the face below each of these paths (a face alwas pesent), is oiented veticall. We thus conside to be pat of the connecting path fo illumination puposes, pemitting the stip below to be hung unde. ecause ou cases ae ehaustive, one can see now that all gidfaces of (sa) the font face of ae eithe illuminated b, o b some descendant of on the font face, augmented b the connecting paths as just descibed. (In fact eve gidface is illuminated twice, fom above and below.) Hanging the stips then completes the unfolding. 16

17 6.5 lgoithm ompleit ecause thee ae so few unfolding algoithms, that thee is some algoithm fo a class of objects is moe impotant than the speed of the algoithm. Nevetheless, we offe an analsis of the compleit of ou algoithm. Let n be the numbe of cone vetices of the polhedon, and N = O(n 2 ) the numbe of gidpoints. The vete gid can be easil constucted in O(N) time, leaving a plana suface map consisting of O(N) gidpoints, gidedges, and gidfaces. The computation of connecting as (Sec. 6.2) equies detemining the components of P + and P, fo each. This can be easil ead of fom the plana map b unning though the n vetices of each of the O(n) bands and detemining, fo each vete, whethe it belongs to P + o P. Each of the O(n) band components shoots a vetical a fom one cone vete, in a 2D envionment (the plane ) of n noncossing othogonal segments. Detemining which band a a hits involves a a shooting que. lthough an implementation would emplo an efficient data stuctue, pehaps SP tees [PY92], fo compleit puposes the naive O(n) que cost suffices to lead to O(n 2 ) time to constuct G. Selecting pivots (Sec. 6.1) involves 2-coloing G in O(n) time, and computing the unfolding tee T U in a beadth-fist tavesal of G, which takes O(n) time. Unfolding bands (Sec. 6.3) involves a depth-fist tavesal of T U in O(n) time, and laing out the O(N) gidfaces in O(N) time. Thus, the algoithm can be implemented to un in O(N) = O(n 2 ) time. 7 Futhe Wok Etending these algoithms to abita genus othogonal polheda emains an inteesting open poblem. Holes that etend onl in the and diections within a slab seem unpoblematic, as the simpl disconnect the slab into seveal components. Holes that penetate seveal slabs (i.e, etend in the diection) pesent new challenges. One idea to handle such holes is to place a vitual -face midwa though the hole, and teat each half-hole as a dent (potusion). cknowledgements We thank the anonmous efeees on [DFO06] fo thei caeful eading and insightful comments. Refeences [DD + 98] T. iedl, E. Demaine, M. Demaine,. Lubiw, J. O Rouke, M. Ovemas, S. Robbins, and S. Whitesides. Unfolding some classes of othogonal polheda. In Poc. 10th anad. onf. omput. Geom., pages 70 71, [DEE + 03] E. D. Demaine, D. Eppstein, J. Eickson, G. W. Hat, and J. O Rouke. Veteunfoldings of simplicial manifolds. In ndas edek, edito, Discete Geomet, pages Macel Dekke, Pelimina vesion appeaed in 18th M Smposium on omputational Geomet, acelona, June 2002, pp [DFO06] [DIL04] M. Damian, R. Flatland, and J. O Rouke. Gid vete-unfolding othogonal polheda. In Poc. 23d Smp. on Theoetical spects of omp. Sci., pages , Febua Lectue Notes in omput. Sci., Vol. 3884, Spinge. E. D. Demaine, J. Iacono, and S. Langeman. Gid vete-unfolding of othostacks. In Poc. Japan onf. Discete omp. Geom., pages 76 82, Lectue Notes in omput. Sci., Vol. 3742, Spinge. 17

18 [DO05a] [DO05b] E. D. Demaine and J. O Rouke. suve of folding and unfolding in computational geomet. In J. E. Goodman, J. Pach, and E. Well, editos, ombinatoial and omputational Geomet, pages ambidge Univesit Pess, Eik D. Demaine and Joseph O Rouke. Open poblems fom G In Poc. 17th anad. onf. omput. Geom., pages , [GKK98] S. K. Gupta, D.. oune, K. H. Kim, and S. S. Kishnan. utomated pocess planning fo sheet metal bending opeations. J. Manufactuing Sstems, 17(5): , [O R00] [PY92] Joseph O Rouke. Folding and unfolding in computational geomet. In Discete omput. Geom., volume 1763 of Lectue Notes omput. Sci., pages Spinge- Velag, Papes fom the Japan onf. Discete omput. Geom., Toko, Dec M. S. Pateson and F. F. Yao. Optimal bina space patitions fo othogonal objects. J. lgoithms, 13:99 113, [SSW89] E. L. Schwat,. Shaw, and E. Wolfson. numeical solution to the genealied map-make s poblem: Flattening nonconve poledeal sufaces. IEEE Tans. Patten nal. Mach. Intell., 11(9): , [THM04] Maco Taini, Kai Homann, Paolo ignoni, and laudio Montani. Polcube-maps. M Tans. Gaph., 23(3): , [Wan97].-H. Wang. Manufactuabilit-diven decomposition of sheet metal poducts. PhD thesis, anegie Mellon Univesit, The Robotics Institute,

19 8 ppendi: Poof of Lemm (onnectedness of G b ) Two subsets of P ae path-connected, o just connected, if thee ae points in each that ae connected b a path that lies in P. We need some notation to descibe the potions of () that ae elevantl connected to each band. Fo a potusion, let c () be the subset of () (cf. Sec. 2) that is path-connected to via paths that do not coss an bands. Fo a dent, let c () be the bounda of plus the subset of () that is both path-connected to via paths that do not coss an bands, and is not pat of c (), fo some potusion. onside fo eample Figue 14b. Fo potusion, c ( ) consists of the bounda im of and the potion of the back face of that ovehangs dent. Fo dent, c () consists onl the bounda of, even though the ovehanging potion of can be eached fom without cossing an bands, because that is pat of c ( ). In Figue 16a howeve, the potion of the font face of enclosed b belongs to c (), not to c ( ). The genus-eo assumption implies that, fo potusion and dent on opposite sides of such that c () c () is nonempt, it must be that is nonempt (cf. Figs. 15). Define {, if, and at least one of and is a dent c (, ) = c () c () othewise. This definition is intended to identif gidpoints on eithe o fom which as ae issued b the a-pai geneation algoithm (Sec ). The eason fo teating intesecting dents and potusions diffeentl is a subtle one, and is captued b Fig. 14b: is a dent behind and is a potusion in font of ; c ( ) is the piece of the back face of enclosed b ; u is a highest gidpoint in, while w is a highest gidpoint in c () c ( ); u is a potential a basepoint, while w is not. The above definition eliminates points such as w fom the set c (, ). Ou connectivit poof fo G b poceeds as follows. In geneal, thee ae a numbe of disconnected maimal components P 1, P 2,... of P, with P = P 1 P 2. The bands incident to each of these ae a-connected to each othe via planes othe than. We fist ague that, to pove that G b is a-connected, it suffices to pove that each P j is a-connected. Remove fom O all the slabs S 1, S 2,... incident to Y 0. Establish that the bands in the esulting object O ae a-connected, via induction. Now put back the slabs. Each S j coesponds to a component P j, and we ae assuming we can establish that all bands incident to P j ae a-connected to one anothe. This along with the fact that O itself is connected implies that all bands ae a-connected. Hencefoth we concentate on one such connected component P j, call it Q fo succinctness. Let be the collection of all bands that intesect Q. Then c () = Q. The idea of the connectedness poof is that the bands get connected in upwad chains, and ultimatel to each othe though common ancesto highe bands. We choose to pove it b contadiction, aguing that a highest disconnected component cannot eist. Lemma 4 ll bands in ae a-connected. Futhemoe, if one abita a in each a-pai is discaded, emains a-connected. Poof: Fo the pupose of contadiction, assume that not all bands in ae a-connected. Let 1, 2,... be the distinct maimal subsets of that ae a-connected. Let Q j = j c (). Then Q = j Q j. Since Q is connected, the subsets Q j ae not disjoint, in that fo eve Q j thee is an Q k such that Q j Q k is nonempt. the obsevation above, this means that Q jk = j, k c (, ) 19

20 is also nonempt. Let j and k be such that Q jk contains a highest -gidedge (gidpoint, if Q jk contains onl isolated points) among all Q jk. Let u be the leftmost highest gidpoint in Q jk. Let j and k be such that u c (, ). We have thus identified two bands and, a-disconnected because in diffeent components of Q, which contibute this highest gidpoint u in the highest intesection Q jk. We now eamine in tun the fou potusion/dent possibilities fo these two bands. ase 1. and ae both potusions on opposite sides of. ssume w.l.o.g that is behind, is in font of, and u is on (as depicted in Fig. 13). We discuss two subcases: D u u' ' D u ' u' ' u 1 u' 1 D u Figue 13: ase 1: and ae both potusions on opposite sides of D is a potusion D is a dent with a vetical side incident to u (c) D is a dent with a bottom edge incident to u. a. u is on a top edge of (Figs. 13(a,b)). Then ou a-pai algoithm geneates a a-pai (, ), with incident to u and incident to the gidpoint u cw fom u. onside (the analsis is simila fo ). If hits, then in fact and ae a-connected, contadicting the fact that and belong to diffeent a-connected components of. So let us assume that hits anothe band D l. Fig. 13a illustates the situation when D is a potusion (dent). If D j, then D and ae a-connected in j, and since and D ae a-connected, it follows that and ae a-connected, a contadiction. So assume that D l, with l j. ut then c (, D) (and implicitl Q jl ) has a gidpoint highe than u, contadicting ou choice of j, k and u. b. u is on a vetical (left, ight) edge of (Fig. 13c). Then u must be at the intesection between a dent D and, meaning that D c () is nonempt. Futhemoe, c (, D) has a gidpoint highe than u, meaning that D G j. Let u 1 be the leftmost among the highest gidpoints of D c (). Then ou a-pai algoithm geneates a a-pai (, ) fom u 1 and its ight neighbo u 1. onside (the analsis is simila fo ). If hits, then is a-connected to D, which is a-connected to, a contadiction. If hits a band E othe than D, then it must be that D k, since c (, E) has a gidpoint highe than u 1, which is no lowe than u. This means that is a-connected to E, which is a-connected to D, which is a-connected to, a contadiction. ase 2. is a potusion and is a dent, both on a same side of. The case when and ae both in font of (illustated in Fig. 14a) is identical to ase 1 above, once one conceptuall pops out into a potusion. We now discuss the case when and ae both behind. 20

21 ssume fist that c (, ) contains no top edges of, as depicted in Figue 14b. Let be a potusion in font of coveing the top of. Then c (, ) and c (, ) each contains a gidpoint highe than u. The following two contadicto obsevations settle this case: a. It must be that k ; othewise Q jk would contain a gidpoint in c (, ) highe than u. b. If l, then it must be that l = k; othewise Q lk would contain a gidpoint in c (, ) highe than u. If c (, ) contains at least one top gidedge of, then aguments simila to the ones used fo the case illustated in Fig. 13a (conceptuall popping to become a potusion) settle this case as well. ' u u' w u u' ' Figue 14: ase 2: is a potusion and is a dent behind in font of. ase 3. is a potusion and is a dent on opposite sides of (see Fig. 15). Let be the potusion in font of enclosing. We discuss two subcases: a. c () contains a top edge of (see Fig. 15a). This means that c () () is nonempt, and the a-pai algoithm shoots a a-pai (, ) upwad fom the endpoints of a highest gidedge {u 1, u 1 } of (). onside a (the analsis is simila fo ). If hits, then and ae in fact a-connected, a contadiction. If hits a band D othe than, then aguments simila to the ones fo the case illustated in Fig. 13a (ase 1) lead to a contadiction. w u u 1 ' u' 1 ' u' u ' Figue 15: ase 3: is a potusion behind ; is a dent in, both in font of. 21