Tiling Triangular Meshes

Transcription

1 Tiling Tringulr Meshes Ming-Yee Iu EPFL I&C 1 Introdution Astrt When modelling lrge grphis senes, rtists re not epeted to model minute nd repetitive fetures suh s grss or snd with individul piees of geometry or individul teture imges euse it is simply too timeonsuming. Even if n rtist were inlined to do suh thing, ll the detil would quikly eeed the mount of memory ville on grphis worksttions, mking the sene diffiult to render. Insted, vrious tehniques for using smll numer of prmeters or smll numer of imges to generte lrge tetures for lrge surfes re usully employed insted. Tiling is one suh tehnique, nd it is ommonly used on squre meshes, in prtiulr on terrin modelled used elevtion mps. For squre grid, n rtist need only design single imge where the top edge lines up with the ottom edge of the imge, nd the left edge lines up with the right edge. This imge n then e psted on top of eh squre in the grid to rete n impression of single lrge semless teture s shown in Figure 1. Tiling ritrry tringulr meshes is muh more diffiult though euse the numer of tringulr tiles needed to tile the surfe semlessly n quikly grow to lrge numers. The sme prolem tullly ours during the tiling of ritrry qudrilterl meshes s well even though the speifi se of tiling grid of squres is quite simple. This pper emines pprohes for reduing the numer of tiles tht n rtist needs to design in order to tile ritrry tringle meshes. 2 Previous Work There is signifint ody of work onerning the field of utomted teturing of lrge models. Lpped tetures [4] re sheme for tking ritrrily shped teture pthes nd repetedly psting them over the surfe of model until it is ompletely overed. Other shemes eist for tking lrge teture imge nd serhing over the imge to find tringle piees tht n e used to over tringles on the surfe mesh in semless fshion [2]. And then, of ourse, there re the entire fields of teture synthesis nd proedurl tetures tht llow for the utomti genertion of tetures from smll numer of prmeters. Note to self: mye you should red some of those ppers so tht you n ite them here The primry grphis pper on tiling tringulr meshes though is y Neyret nd Cni [3]. It notes tht the simplest wy of tiling tringulr mesh is to use single tile where eh side of the tile is symmetril nd n e mthed semlessly with ny other side of the tringle, suh s in Figure 2. Although the tiling my end up eing somewht repetitive, it is possile to mitigte this prolem y reting vritions of the single tile with sides tht re the sme s in the originl tile, ut with different ptterns in the enter of the tile. Figure 2: It is possile to tile n ritrry tringle mesh using single tile. Figure 1: A grid of squres n e tetured with single tile. The prolem with using single tile is tht it severely limits the design of the tile. The tile pttern must e isotropi, the oundries must e symmetri, nd it eomes diffiult to use dvned tehniques suh s ni-

2 mted tiles. The idel tileset used to tile mesh should e one where eh tringulr tile n hve three ompletely different oundries on eh of its sides nd where eh edge oundry my e oriented. Figure 3 shows n oriented edge oundry in whih prtiulr tile edge must e mthed with orresponding omplement edge in order to form semless pttern. Unfortuntely, s the numer of possile edge oundries inrese, the numer of possile tiles tht might pper in the mesh inreses drmtilly s well. For emple, given n different edge oundries (where n oriented edge oundry nd its omplement ount s two), there re n wys of reting tile where ll the edge oundries re the sme, n(n 1) wys of reting tiles onsisting of two different edge oundries, nd n(n 1)(n 2)/3 wys of reting tiles onsisting of three different edge oundries. This mens tht there re totl of n + n(n 1) + n(n 1)(n 2)/3 possile teture tiles tht might pper in mesh. Figure 3: When oriented edges re used, one tile edge must e ligned with orresponding edge to form semless pttern. Suh n edge nnot semlessly lign with itself or refletions of itself. So if tringle tileset with only one oriented edge oundry were to e reted (omposed of n edge oundry nd its omplement ), then four different tringles re possile: (,, ), (,, ), (,, ), nd (,, ). Eh tuple desries wht omintions of edge oundries must pper on eh side of tringle tile. In tileset where eh tile side n e different nd ll edge oundries re oriented, n is 6, mening n rtist might need to rete 76 different tiles to suessfully tile tringle mesh. Although Neyret nd Cni desrie these issues with tileset size, they then restrit themselves to onsidering tilings involving only smll numer of edge oundries. They lso disuss wys of tiling n ojet with tringulr tiles independent of the ojet s rel geometry, nd desrie lgorithms for utomtilly generting tringulr tile sets. 3 Minimizing the Numer of Tiles Although the epression given y Neyret nd Cni desries ll possile tiles tht n e formed y generting permuttions of edge oundry types, the tul numer of tiles needed in tiling n e muh less if the tiling is done refully. For emple, we previously stted tht tringle tileset with only one oriented edge oundry results in four different tiles. In relity, it is possile to tile n ritrry tringle mesh using only two of those tiles (,, ) nd (,, ) not four s previously suggested. Note to self: is it worthwhile inluding the proof of this? So is it possile to tile n ritrry tringle mesh without putting n undue urden on tile rtists? And is it possile to lulte in dvne the minimum numer of tiles tht n rtist needs to rete? To llow for the mimum fleiility in tile design, we wnt to llow for tringle tiles where eh side n hve different oriented edge oundries. In ft, we wnt the stronger ondition tht ll sides of eh tringle tile must hve different oriented edge oundries; otherwise, it eomes trivilly esy to minimize the numer of tiles needed y restriting our tiling to only use single oriented edge oundry nd its omplement, therey llowing for tiling using only two different tiles. In order to gin insight into this prolem, we n rest it s edge oloring [6]. First, onsider the tringle mesh to e tiled s grph. We n then tke the dul of this grph where eh tringle fe eomes verte nd edges re drwn etween verties orresponding to djent fes. Sine eh verte in the dul omes from tringle fe in the originl mesh, the mimum degree of verties in the dul is three. An edge oloring of this dul is n ssignment of olors to the edges of the dul suh tht for eh verte in the dul, ll edges djent to it hve different olors. Figure 4 shows n emple of suh n edge oloring. Aording to Vizing s theorem, grph with mimum verte degree three n e edge olored using four different olors. It might lso e possile to edge olor the grph using only three olors, ut finding suh oloring is generlly NP-omplete, whih, given the lrge size of the grph, mens tht finding suh oloring is essentilly infesile. Unfortuntely, the eistene of four edge oloring doesn t relly help ll tht muh. The most ovious wy of mpping the edge oloring k to prtiulr tiling is to hve eh edge olor orrespond to different oriented edge oundry nd its omplement. This mens tht eh tringle tile will hve different edge oundry on eh of its side, ut sine the edge oloring doesn t give ny prtiulr insight into the orienttions of these edges, we hve to ssume the orienttions re ritrry. So given n edge oloring of the dul using four olors A, B, C, nd D we n go k to the originl mesh, nd ssign one of eight oriented edge oundries,,,,,, d, nd d to eh tile side in wy tht orresponds to the four edge oloring suh s in Figure 5. This mens tht we might still require (8 6 4)/3 or 64 different tile types. Interesting things hppens though, if the dul is ipr-

3 Figure 4: An edge oloring of the dul of tringulr mesh. Note to self: this would e ooler if you ould find mesh requiring four edge oloring. tite. Firstly, it eomes possile to edge olor the dul using only three olors, nd this oloring n e found in liner time [5, 1]. Seondly, euse the tringles in the originl mesh re iprtite, it eomes possile to hve the edge oundries of given tringle ll e oriented in only one diretion or the other. If ll orienttions flip etween djent tringles, then onsistent sheme of orienting the edge oundries is possile. Figure 6 shows how the orienttions of edge oundries n e restrited using iprtite dul. This mens tht tringle mesh with iprtite dul n e tiled in polynomil time using only four tiles: (,, ), (,, ), (,, ), nd (,, ). Requiring n rtist to design four seprte tile imges in order to tile tringulr mesh is quite resonle. In ft, in prtie, only three tile imge need to e designed euse one pir of tiles n e reted y strting with squre tile nd utting it in hlf. As noted efore, oriented edge oundries re llowed, nd eh side of tringulr tile is llowed to e different, mening there is lot of freedom in the design of the tileset. If tiling using only four tiles results in repetitive pttern, it is esy to strt with three edge oloring of the dul nd to dd dditionl olors seletively. This wy, the numer of possile edge oundry types n e inresed while ontrolling the numer of permuttions of these edge ound- ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' Figure 5: A mpping of n edge oloring to different tile types with ritrry edge oundry orienttions. ries in tiles, therey preventing the numer of tringle tile permuttions from eploding. Unfortuntely, ll of these gret properties require tht the originl tringle mesh hve iprtite dul. But wht does it men for the dul of the tringulr mesh to e iprtite? Does tht inhiit the geometry of models in some wy? How is n rtist or pplition supposed to know how to rete suh mesh? 4 Tringle Meshes with Biprtite Duls In ft, tringle mesh with iprtite dul is tully equivlent to tringulr mesh where ll interior veries hve n even degree. Here is the proof: = Given tringle mesh with iprtite dul, ssume there eists t lest one interior verte with n odd degree. There re n odd numer of fes djent to n odd degree verte, nd if we trverse these fes, we end up with n odd yle in the dul. But ll yles in iprtite grph re even, resulting in ontrdition, so our ssumption must e flse. Therefore, ll interior verties in the tringulr mesh must e even. = This is nother proof y ontrdition. Given tringle mesh where ll interior verties hve n even degree, ssume tht the dul is not iprtite. Hene, there must e yle in the dul tht is odd. Tke this yle from the dul nd mp it k to the originl mesh. We now hve yle trversing over tringles s demonstrted in Figure 7. This new yle enloses suset of the tringles in the tringle mesh. Sine the originl yle in the dul is odd, the new yle must ross over n odd numer of edges. Eh one of these edges is inident to etly one verte

4 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' Figure 6: If mesh hs iprtite dul, it n e tiled using only four different tiles. from the enlosed re. Let us onsider this enlosed re now. If we wnted to ount the sum of the degrees of ll the verties from the enlosed re, we would first sum up the originl degrees of ll these verties nd then sutrt the length of the dul yle (euse these orrespond to edges tht do not ontriute to the enlosed re). Sine the originl degrees of eh of the enlosed verties re even, the sum of the originl degrees of ll these verties is lso even. But euse the yle length is odd, the totl sum of the degrees of ll the verties from the enlosed re is odd. Every edge in the enlosed re though must e inident to two verties from the enlosed re, mening tht the totl sum of the degrees of ll verties from the enlosed re must e even. This is ontrdition, mening the the originl ssumption must e flse, hene the dul must e iprtite. 5 Trnsforming Meshes When ll the verties in tringulr mesh hve n even degree, it is etremely esy to tile them with smll numer of tringulr tiles. And lthough the onept of suh mesh is esily understood, it is not ovious how n rtist n e epeted to rete 3d models stisfying suh onstrints. Here, we emine two tehniques for utomtilly trnsforming tringulr meshes to mke eh verte hve n even degree. Figure 7: When n odd yle is mpped k to the originl tringle mesh (the tringles in the yle re mrked in grey), it enloses suset of the mesh (mrked with thik lk line) tht n e nlyzed. For open surfes, it is possile to simply dd edges onneting odd interior verties to oundry verties nd oundry edges (sine we do not re out the degree of verties on the oundry). It is lso possile to tkes pirs of odd verties nd join them with edges, therey mking them oth even. Figure 8 illustrtes these two opertions. Unfortuntely, not ll pirs of odd verties n e joined in this wy; for emple, djent odd verties nnot e joined. And it is lso not ler how to hoose whih verties to join so s to minimize the modifitions mde to the mesh. This is importnt euse pplying these opertions results in the splitting of tringles. Not only do split tringles result in inresed rendering time euse of the greter numer of tringles, ut split tringles my lso e degenerte or of non-uniform size leding to inonsistent tiling or visul rtifts. In ny se, for losed surfes where there is no oundry, the ove opertions my not e suffiient to eliminte ll the odd verties from the mesh, mening nother tehnique is needed. Therese Biedl of the University of Wterloo me up with n interesting onstrution tht elimintes ll odd verties in oth open nd losed surfes. With this onstrution, n etr verte is dded to the enter of every tringle in the mesh. These etr verties re then onneted to the three verties of the originl tringle nd to the midpoints of the three edges of the originl tringle. Afterwrds, eh verte in the resulting mesh will hve n even degree, nd, provided tht the originl tringles were well-formed, the resulting tringles will generlly hve uniform sizes nd shpes s well. Figure 9 demonstrtes how the onstrution works. Unfortuntely, the resulting mesh hs si times s

5 mny tringles s efore, mening tht it tkes si times s long to render the sene. There re only 64 wys in whih eh sudivided tringle n e tiled though. And if eh suh tiling is stored s seprte teture, then it eomes possile to render the originl non-divided mesh using this new set of teture tiles, mening there will e no slowdown. If there is insuffiient memory to store tht mny teture, nother lterntive is to tile eh sudivided tringle in etly sme wy, s shown in Figure 10. This single lrge teture is essentilly tringle tile with single non-oriented edge oundry on ll sides, nd it n then e used to teture the entire originl mesh. Then if memory llows, the tiling of the sudivided mesh n e seletively ltered so tht limited numer of dditionl sudivided tringle tilings pper in the mesh. In this wy, one n ontrol the et numer of generted teture tiles tht will e needed to over the originl mesh. So strting with the four originl tiles, it s possile to permute them to generte etween 1 nd 64 lrger tiles tht n e used to tile the mesh. Of ourse, on more modern hrdwre, one n simply omine the originl four tiles on-the-fly to generte the lrger tiles using frgment progrm. Figure 8: Adding edges etween odd verties nd the oundry or etween pirs of odd verties will mke those verties even. ' ' ' ' ' ' ' ' ' Figure 10: Tiling sudivided tringle in this wy, results in single lrge tringulr tile with identil non-oriented edge oundries Figure 9: By dding edges nd verties to every tringle, it is possile to fore every verte to hve n even degree 6 Conlusions nd Future Work Although the tiling of simple grids of squres is quite esy, the tiling of ritrry tringle meshes is tully quite hrd euse if done improperly, n rtist might hve to design lrge numer of tiles for the tiling. If the tiling proess is done refully, however, it is possile to tile the n ritrry tringle mesh using only four tiles. An rtist lso hs signifint ltitude in the design of these tiles. Unfortuntely, this pper only lys out the theoretil groundwork for these lims. To verify the prtility of these tehniques, it will e neessry to uild few implementtions. It is lso unler how these tiling tehniques n e etended to hndle levels of detil nd to void lising rtifts. There re lso theoretil issues tht ould e emined more deeply suh s finding et lower ounds on the numer of tiles needed to tile tringle meshes with non-iprtite duls or looking t the tilings of ritrry qudrilterl meshes, whih might yield further good insights into the prolem Referenes [1] Olivier Bodini nd Eri Rémil. Tilings with trihromti olored-edges tringles. Theor. Comput. Si.,

6 319(1-3):59 70, [2] Sestin Mgd nd Dvid Kriegmn. Fst teture synthesis on ritrry meshes. In Proeedings of the 14th Eurogrphis workshop on Rendering, pges Eurogrphis Assoition, [3] Frie Neyret nd Mrie-Pule Cni. Pttern-sed teturing revisited. In Proeedings of the 26th nnul onferene on Computer grphis nd intertive tehniques, pges ACM Press/Addison- Wesley Pulishing Co., [4] Emil Prun, Adm Finkelstein, nd Hugues Hoppe. Lpped tetures. In Proeedings of the 27th nnul onferene on Computer grphis nd intertive tehniques, pges ACM Press/Addison- Wesley Pulishing Co., [5] Alender Shrijver. Biprtite edge oloring in o(δm) time. SIAM J. Comput., 28(3): , [6] Eri W. Weisstein. Edge hromti numer. EdgeChromtiNumer.html.