Fundamentals of Spherical Parameterization for 3D Meshes

Transcription

1 Fundamentals of Sphercal Parameterzaton for 3D Meshes Crag Gotsman Xanfeng Gu Alla Sheffer Technon Israel Inst. of Tech. Harvard Unversty Technon Israel Inst. of Tech. Abstract Parameterzaton of 3D mesh data s mportant for many graphcs applcatons, n partcular for texture mappng, remeshng and morphng. Closed manfold genus-0 meshes are topologcally equvalent to a sphere, hence ths s the natural parameter doman for them. Parameterzng a trangle mesh onto the sphere means assgnng a 3D poston on the unt sphere to each of the mesh vertces, such that the sphercal trangles nduced by the mesh connectvty are not too dstorted and do not overlap. Satsfyng the non-overlappng requrement s the most dffcult and crtcal component of ths process. We descrbe a generalzaton of the method of barycentrc coordnates for planar parameterzaton whch solves the sphercal parameterzaton problem, prove ts correctness by establshng a connecton to spectral graph theory and show how to compute these parameterzatons. CR Categores: I.3.5 [Computer Graphcs] Computatonal Geometry and Object Modelng Curve, Surface and Sold and Object Representatons. Keywords: Trangle mesh, parameterzaton, embeddng. 1. Introducton Parameterzaton of 3D mesh data s mportant for many graphcs applcatons, n partcular for texture mappng, remeshng and morphng. To date, mostly planar parameterzatons have been consdered. The man challenge s to produce a planar trangulaton that best matches the geometry of the 3D mesh, mnmzng some measure of dstorton, yet s stll vald. In ths context, vald means that the ndvdual planar trangles do not overlap. Most of the recent works on the subject of parameterzaton (e.g. [Desbrun et al. 2002; Floater 1997; Levy et al. 2002; Sander et al. 2001; Sheffer and de Sturler 2001]) have focused on defnng the dstorton, and showng how to mnmze t. Whle parameterzng to the plane s the most natural way to perform texture-mappng, ths s less natural for other mesh processng operatons whch also requre a parameterzaton. For applcatons such as morphng [Alexa 2000; Kana et al. 2000; Shapro and Tal 1998] and remeshng [Gu et al. 2002; Kobbelt 1999] t s best to parameterze the mesh over a doman whch s topologcally equvalent to t. Ths sgnfcantly reduces the dstorton ntroduced by the parameterzaton wthout resortng to methods whch ntroduce other artfacts, such as cuttng seams. If the mesh has the topology of a sphere, t s best to use a sphercal parameter doman. Parameterzng a 3D trangle mesh over the sphere s equvalent to embeddng ts connectvty graph on the sphere, such that the resultng sphercal trangles partton the sphere (ther unon s the sphere, and they are dsjont). A classcal result due to Stentz s that a graph may be embedded on the sphere f and only f t s planar and 3-connected. Thus a closed manfold genus-0 trangulaton can always be mapped to a sphercal trangulaton. The smplest way to map a closed trangle mesh to the sphere s to reduce the problem to the planar case. Frst cut out one trangle to serve as a boundary. Then parameterze the resultng open mesh over the unt trangle usng any planar parameterzaton method, and fnally use the nverse stereo projecton to map the plane to the sphere [Haker et al. 2000]. See Fg. 1(b). The man problem wth ths method s severe dstorton, and although the nverse stereo projecton s conformal, namely, preserves angles n the contnuous case, t does not preserve angles (or any other geometrc propertes) n the dscrete case. The projecton also does not guarantee that the result wll be a sphercal trangulaton. Another straghtforward method to parameterze to the sphere s to cut the mesh nto two peces, each topologcally equvalent to a dsk, parameterze each over a planar dsk wth a common boundary, and then map each dsk to a hemsphere (by addng an approprate z component to each vertex). The common boundary guarantees that the two hemspheres ft together at the equator. See Fg. 1(c). Snce ths boundary wll presumably contan more than just three vertces, each of the two dsk parameterzatons wll be less dstorted than the one obtaned by usng a sngle trangle as the boundary, so the sphercal result wll also be less dstorted. However, the result wll depend strongly on the specfc cut used to obtan the two dsks. It s more natural to parameterze a mesh drectly on the sphere wthout gong back and forth to the plane. Several methods for drect parameterzaton on the sphere exst. The only one to date that seems to guarantee a vald sphercal trangulaton (.e. wth no trangle foldovers) s that of Shapro and Tal [1998], smlar to that of Das and Goodrch [1997]. Ths method works by smplfyng the mesh by vertex removal untl only a tetrahedron remans. Permsson to make dgtal/hard copy of part of all of ths work for personal or classroom use s granted wthout fee provded that the copes are not made or dstrbuted for proft or commercal advantage, the copyrght notce, the ttle of the publcaton, and ts date appear, and notce s gven that copyng s by permsson of ACM, Inc. To copy otherwse, to republsh, to post on servers, or to redstrbute to lsts, requres pror specfc permsson and/or a fee ACM /03/ $5.00 (a) (b) (c) Fgure 1: Parameterzng the (a) rabbt by (b) nverse stereo mappng (c) two hemspheres. Colored dots mark correspondng vertces. 358

2 The tetrahedron s easly embedded on the sphere, and then the vertces are nserted back one by one, so that the valdty of the trangulaton s preserved throughout the process. Whle ths s qute an effcent process, t s dffcult to optmze the parameterzaton, due to ts greedy nature, and mpossble to steer t to have any desrable mathematcal propertes. Other drect parameterzaton methods were proposed by Kobbelt et al [1999] and Alexa [2000]. These are heurstc teratve procedures, attemptng to converge to a vald parameterzaton by applyng local mprovement (relaxaton) rules. These work well n many cases, but there s no guarantee that they wll termnate, and, even f they do, that the resultng embeddng wll be vald, or have any desrable mathematcal propertes. A method whch guarantees a vald embeddng was recently proposed by Sheffer et al. [2003]. Ths s a hghly non-lnear optmzaton procedure, workng wth the angles of the sphercal trangulaton (as opposed to the vertex postons), nspred by the angle-based method of Sheffer and de Sturler [2001] for planar parameterzatons. So far t lacks an effcent numercal computaton procedure, so t s not very practcal. 1.1 Our contrbuton The problem of mesh parameterzaton s that of mappng a pecewse lnear surface wth a dscrete representaton onto a contnuous sphercal surface. The theory of mappngs between varous Remann surfaces s well understood n the contnuous case usng classcal dfferental geometry [Do Carmo 1976]. Probably the most notable example of ths s the so-called conformal mappng theory whch shows how to map any contnuous Remann surface to another such that angles are preserved. However, the dscrete case of meshes s much less understood. In the lmt t obvously converges to the contnuous case, but n practcal applcatons the meshes nvolved may be far from ths lmt. Hence there s a need to treat the dscrete case separately n a combnatoral manner, albet nspred by the classcal theory. Ths paper ntroduces a precse mathematcal characterzaton of all possble sphercal parameterzatons of a closed manfold genus-0 trangle mesh. We show that t s a natural non-lnear extenson of the lnear theory of barycentrc coordnates used n the planar case. The correctness of ths methodology s proved by establshng a lnk to the so-called Coln de Verdere matrces assocated wth planar graphs. We also descrbe a computatonal method for generatng and controllng these parameterzatons. These contrbutons are concentrated n Secton 4, after establshng the theory n Secton The Method of Barycentrc Coordnates 2.1 The planar case Floater [1997] descrbed a generc method to embed a manfold 3D mesh wth a boundary n the plane wthout foldovers. Floater's method s a generalzaton of the basc procedure orgnally proposed by Tutte [1963] for a planar graph, whch can be traced back as far as I. Fary n 1948 and J.C. Maxwell n Ths method makes use of so-called barycentrc coordnates (or convex combnatons) and proceeds as follows: 1. To each nteror (drected) edge e = (,, assgn a postve weght w j, such that = 1 w j j N ( ) where N() s the lst of vertces neghborng the 'th vertex. 2. To all other entres (,, assgn w j = Embed the boundary vertces n the plane such that they form a closed convex polygon. 4. Solve the followng two lnear systems for the x and y coordnates of the n nteror vertces: ( I W) x = bx, ( I W ) y = b y, where W s a nxn matrx contanng w j, and b x and b y vectors wth non-zero entres correspondng to vertces adjacent to the boundary. The cornerstone of ths theory s the followng theorem, frst proven by Tutte [1963], and reproven over the years n dfferent ways (e.g. [Floater 2003b, Rchter-Gebert 1996, Chap 3]): Theorem 1: Gven a planar 3-connected graph wth a boundary fxed to a convex shape n R 2, the postons of the nteror vertces form a planar trangulaton (.e. none of the trangles overlap) f and only f each vertex poston s some convex combnaton of ts neghbor's postons. Theorem 1 mples that the method of barycentrc coordnates generates all possble vald embeddngs of the graph n the plane, gven the (convex) postons of the boundary. Tutte proposed usng w j =1/deg() for all edges (,, n effect placng each nteror vertex at the centrod of ts neghbors (deg() s the degree, or valence of the 'th vertex). Ths choce of W does not take nto account the geometry of the mesh, just ts connectvty. When the mesh s gven wth 3D geometry, a number of recpes for W have been proposed, each amng for some effect related to reflectng the geometry of the mesh n the parameterzaton, namely, mnmzng ts metrc dstorton when flattened to the plane. The most popular methods seem to be the shape-preservng method [Floater 1997], the conformal, or harmonc method [Haker et al. 2000; Levy et al. 2002; Pnkall and Polther 1993] and the mean-value method [Floater 2003a]. These methods all have the desrable 2D reproducton property [Floater 1997], namely that when appled to a 2D trangulaton, the embeddng procedure wll produce an output dentcal to the nput. However, some of the methods, most notably the conformal method, do not always result n postve weghts, hence cannot guarantee a vald embeddng. In general, the method of barycentrc coordnates may be formulated as the soluton to the 2D vector Laplace equaton on the nteror vertces: (1) L W x = b wth boundary condtons derved from the convex boundary vertex postons, whch prevents the trval zero soluton. Ths s a lnear system. L W = I-W s the general normalzed Laplacan operator, general because the weghts of W are arbtrary postve values, and normalzed because the rows of W all sum to unty. The specal case of w j =1/deg() proposed by Tutte wll be called the normalzed Tutte Laplacan. A smple numercal procedure to solve (1) s a relaxaton procedure, where the boundary vertces are placed on a convex boundary, and the nteror vertces are repeatedly updated to be at the weghted average of ther neghborng vertex postons, as dctated by W. Snce L W s dagonally domnant, ths Gauss-Sedel procedure s guaranteed to converge. 359

3 2.2 The sphercal case The Laplacan L W has a unt dagonal, negatve entres for each mesh edge, and vanshes otherwse. Also, all rows sum to zero, hence L W s sngular. L W s, however, not symmetrc. In what follows, we restrct the dscusson to the class of symmetrc Laplacans, whch corresponds to sets of barycentrc coordnates whch are edge-symmetrc up to normalzaton of each row. These are: negatve number (, E LW (, = LW (, k) = j k (, E 0 Note that the symmetrc Tutte Laplacan has -1's at entres correspondng to edges, and the vertex degrees along the dagonal. Symmetrc systems such as these can be gven the physcal nterpretaton of a mass-sprng system at rest, where the vertces are pont masses joned by sprngs along the edges. In ths case, the Laplace equatons are just the normal equatons for mnmzng the quadratc sprng energy. Tutte and conformal barycentrc coordnates have the symmetry property, but mean value coordnates unfortunately do not. Generalzng the barycentrc coordnates theory to sphercal embeddngs s not straghtforward. Beng non-planar, t wll be mpossble n general to express a vertex on the sphere as a convex combnaton of ts neghbors (e.g. f a vertex's neghbors are all co-planar, ths wll mply that the vertex should also be on the same plane). Inspred by classcal dfferental geometry operator theory, Gu and Yau [2002] proposed to embed on a curved 3D surface usng the generalzaton of the Laplacan to the Laplace-Beltram operator. Intutvely, ths s just the tangental component of the Laplacan at that pont of the surface, and mples the followng nonlnear system: (2) LW x = 0 s.t. x = 1, = 1,.., n where L s the tangental component of L. The rght hand sde of (2) s zero, as opposed to the non-zero b n (1), because there s no boundary. Whle Gu and Yau showed that solvng (2) results n a bjectve embeddng of a contnuous Remann surface on the sphere, they dd not show that ths also holds for the dscrete case of a pecewse-lnear mesh, n the sense that the result s a vald sphercal trangulaton. In the next secton we show how some recent deep results n spectral graph theory may be appled to establsh ths. 3. Connecton to Spectral Graph Theory We would lke to prove the followng analog of Theorem 1: Theorem 2: Gven a planar 3-connected graph embedded n R 3, the postons of the vertces form a sphercal trangulaton (.e. none of the sphercal trangles overlap) f and only f each vertex poston s some convex combnaton of the postons of ts neghbors, whch s then projected on the sphere. Theorem 2 means that the barycentrc coordnate theory holds also on the sphere up to a radal resdual, consstent wth (2). In Secton 4 we wll prove ths theorem. We start wth some theory. 3.1 The Coln de Verdere number In 1990, Coln de Verdere [1990] establshed an algebrac nvarant for certan famles of graphs. Gven a n-vertex graph G = <V,E>, consder the class M(G) of symmetrc matrces wth element M j such that: negatve number (, E M j = anythng = j 0 (, E Note that M(G) s a superset of the symmetrc Laplacans for G, allowng the dagonal entres to assume arbtrary values (so that the rows do not necessarly sum to zero). Denote by λ(m) = {λ 0,.., λ n-1 } the spectrum of M wth correspondng egenvectors {ξ 0,.., ξ n-1 }. Let r = r(g) be the maxmal nteger such that λ 1 =λ 2 =..=λ r over all matrces n M(G). Let M be a matrx whch attans ths maxmum. Ths r(g) s called the Coln de Verdere (CdV) number of G, the matrx M a CdV matrx for G, ts r dentcal egenvalues CdV egenvalues, and the correspondng egenvectors CdV egenvectors. Coln de Verdere showed that: G s a 3-connected planar graph f and only f r(g) Nullspace embeddng An mportant extenson of the results of Coln de Verdere was obtaned by Lovasz and Schrjver [1999], who showed that CdV egenvectors of a graph G may be used to embed G n R r. For the specal case r(g) = 3, ths translates to: G descrbes the edges of a convex polyhedron n R 3 contanng the orgn f the three egenvectors ξ 1, ξ 2 and ξ 3 of a CdV matrx of G are used as coordnate vectors for ts vertces. The fact that the polyhedron s convex and contans the orgn s a key fact, snce convexty mples star-shapedness. Ths n turn mples that by normalzng the vertces, the polyhedron may be projected onto the unt sphere to form a (vald) sphercal trangulaton. Snce the spectrum of a matrx may be shfted arbtrarly by addng an approprate constant to the dagonal entres, we may assume, wthout loss of generalty, that the three CdV egenvalues are zero. Thus the correspondng CdV matrx has just one negatve egenvalue and co-rank 3. In ths case the three CdV egenvectors are ndependent non-trval solutons to what looks lke Laplace equatons: Mx = 0, or a bass of the nullspace of M. Hence x s called the nullspace embeddng of G. Usng egenvectors of matrces as coordnate vectors for embeddng graphs s not new. The tradtonal way of dong ths s takng the egenvectors correspondng to the smallest postve egenvalues of the Tutte Laplacan (the smallest egenvalue s zero, due to ths matrx beng sngular). Ths dates back to Fedler [1975] and Hall [1970]. See [Koren 2002] for a dscusson of the dfferent ways of usng the Laplacan for "drawng" graphs. Egenvalues and egenvectors of the Tutte Laplacan of a graph are the cornerstone of spectral graph theory [Chung 1997], and have also been used for codng 3D mesh geometry [Karn and Gotsman 2000]. However, the embeddngs resultng from egenvectors of ths Laplacan do not have very appealng geometrc propertes, and, specfcally, the trangles overlap. The Coln de Verdere theory reveals that more powerful generalzatons of the Laplacan must 360

4 be used, yeldng egenvectors whch are more symmetrc, snce they correspond to dentcal egenvalues. On an ntutve level, ths symmetry s what guarantees the valdty of the embeddng. 4. Generatng Sphercal Nullspace Embeddngs It s dffcult to use the Coln de Verdere theory drectly to embed on a sphere, snce, gven a 3-connected planar graph G, nether Coln de Verdere nor Lovasz and Schrjver provded any recpe to generate a CdV matrx for G. In the planar case, once the boundary of the trangulaton has been fxed and the barycentrc coordnates chosen, the postons of the nteror vertces are unquely determned by solvng a lnear system. Ths s not the case for the sphercal scenaro. However, we propose to use a symmetrc Laplacan as the startng pont for constructng a CdV matrx. The off-dagonal values wll not change, but the dagonal of the matrx s stll lackng, and must be corrected. Only then can the embeddng (the nullspace of the CdV matrx) be obtaned. The key observaton of ths paper s that we may solve for the dagonal of the CdV matrx and ts nullspace smultaneously. We also force the resultng nullspace vectors to le vertex-wse on the 3D sphere. Ths may be posed as the followng set of 4n quadratc equatons on the 3n postons of the vertces (x, y, z ) and the n auxlary varables α : (3) x + y + z = 1 α x LW [ ] x = 0 α y LW [ ] y = 0 α z L [ ] z = 0 W = 1,.., n = 1,.., n = 1,.., n = 1,.., n L W [] denotes the 'th row of the matrx L W, and x, y, and z are column vectors. The number of vertces n the mesh s n. Fg. 2 llustrates the geometrc nterpretaton of (3): the vector dfference between the 'th vertex and the weghted average (as dctated by L W ) of ts neghbors s collnear wth the vector dfference between the vertex and the sphere's center. To prove that ths procedure s correct, assume that L s a symmetrc Laplacan for G and that (3) has been solved for column n- vectors x, y, z and α. Ths means that the 'th row L[] of L satsfes L[](x, y, z) = α (x, y, z ). Defne the matrx M as: Lj j M =, j L α = j obtanng M(x, y, z) = 0. Hence M s a CdV matrx for G wth vanshng CdV egenvalue and nullspace spanned by x, y and z, mplyng that x,y,z form a vald sphercal trangulaton when used as coordnate vectors. A smlar argument shows the converse: If the three vectors (x,y,z) are a sphercal trangulaton of a 3-connected planar graph G, then these three vectors span the nullspace of a sutable CdV matrx M, whch may easly be translated nto L W and α satsfyng (3). Ths completes the proof of Theorem 2. Fgure 2: A sphercal trangulaton based on the Tutte Laplacan: The average of the neghbors of vertex V (A, B, C and D) s the pont M, whch s collnear wth the sphere center O and the vertex V: M-O = c(v-o). A smlar relatonshp between a vertex and ts neghbors holds for all mesh vertces. 5. Implementaton Detals Stentz's theorem guarantees that any planar 3-connected mesh admts a vald sphercal trangulaton. The theory of Coln de Verdere guarantees that f a vald sphercal trangulaton exsts, then t can be found by solvng a system of the form (3) for some symmetrc Laplacan L W. However, there s no guarantee that an arbtrary symmetrc Laplacan, when used n (3), wll result n a non-degenerate trangulaton. Degenerate solutons always exst. The case α 0 s the trval soluton when all vertces collapse to one pont on the sphere. Gu and Yau [2002] tred to prevent ths by requrng that the vertces average to zero. However, ths s damagng and can prevent other solutons. Another degenerate soluton can occur when the vertces are mapped to two antpodal ponts on the sphere. In ths case, the vertces are parttoned nto two sets such that the weghted average of the neghbors of a vertex n each set s stll n the same hemsphere as the vertex. A more nterestng stuaton can occur f the connectvty graph contans a Hamltonan cycle. The cycle of vertces may then be mapped to the equator. Beyond the degenerate solutons, a soluton to (3) s not unque, snce any sphercal trangulaton s certanly nvarant to the rotaton group SO(3), whch mmedately gves three degrees of freedom. However, there are more. Ths can be seen by examnng the smple specal case of a tetrahedral mesh connectvty combned wth the Tutte Laplacan. Observe that any rectangular parallelpped crcumscrbed by the unt sphere defnes an equfaced tetrahedron (.e. a tetrahedron whose four faces are dentcal), by takng ts edges to be the opposte dagonals of the sx faces of the parallelpped. It s easy to see that any vertex of the tetrahedron s collnear wth the orgn and the centrod of the three opposte vertces. Ths means that the vertex locatons on the sphere satsfy (3) wth L W = the Tutte Laplacan and α = 4 for 1 4. The CdV matrx s a 4x4 matrx whose entres are all -1's. There are two degrees of freedom n constructng such a rectangular parallelpped, so all together there are fve degrees of freedom. To solve the nonlnear set of equatons (3) for very large meshes, t s mportant to have a stable and effcent numercal procedure. A relaxaton procedure where each vertex s updated to be the weghted average of ts neghbors, and then projected onto the sphere, wll not converge to the desred result, rather collapse to some degenerate confguraton. 361

5 conformal angle-preservng mappng (although these can be negatve), nverse edge lengths for a trangulaton preservng edge lengths, and the prescrpton of Desbrun et al. [2002] for an areapreservng trangulaton. We use the fsolve procedure of MATLAB, a subspace trust regon procedure [Coleman and L 1996]. To better condton the system, t s useful to anchor an arbtrary vertex to a fxed poston on the sphere. Ths elmnates only two degrees of freedom from SO(3), so s not damagng. It appears that anchorng two arbtrary vertces lmts the soluton. Anchorng three vertces s damagng, snce n the case of the equfaced tetrahedron there does not always exst a fourth vertex on the sphere formng the tetrahedron. Fg. 3 shows some sphercal embeddngs generated by our procedure on three sample meshes, and compares them to those generated by the procedure of Alexa [2000], and those based on reducton to the planar case by nverse stereo projecton. The Tutte Laplacan produces an equ-angular trangulaton, whch s smlar n some cases to that generated by Alexa's algorthm. Alexa's algorthm, though, sometmes tends to spread the vertces out over the sphere (as n the Trceratops model). The conformal embeddng preserves many of the features of the 3D geometry, so the eyes and ears of the Rabbt are stll notceable n the result. The stereo embeddng tends to lose most of the geometrc structure. 6. Expermental Results We have used our methodology to generate a varety of embeddngs of closed manfold genus-0 meshes on the unt sphere. The characterstcs of the embeddng may be controlled by the weghts n the Laplacan matrx, smlarly to the planar case. For example, unform (Tutte) weghts should be used f the sphercal trangulaton s requred to be equ-angular, cotangental weghts for a Pawn (154 vertces) Rabbt (543 vertces) Trceratops (1,727 vertces) Tutte Laplacan Conformal Laplacan Stereo Alexa Fgure 3: Some 3D models and ther sphercal parameterzatons. Colored dots mark correspondng vertces. 362

6 In terms of runtmes, solvng (2) usng the MATLAB procedure requred anywhere between a few seconds for the Pawn model of 154 vertces, and a few mnutes for the Trceratops model (1,727 vertces) on a Xeon 2.8 GHz PC wth 1GB RDRAM. 7. Concluson Parameterzng a closed manfold genus-0 mesh to the sphere s of paramount mportance n dgtal geometry processng. It s a fundamental operaton requred for remeshng, morphng, flterng and texture mappng. Whle mappng between Remann surfacees s well understood n the contnuous case, the dscrete case has not receved satsfactory treatment to date. Ths paper has closed ths gap, provdng precse characterzatons of dscrete sphercal parameterzatons and methods to compute them. We have shown that there s a natural extenson of the barycentrc coordnate theory from planar trangulatons to sphercal trangulatons, correspondng to the extenson of the Laplacan operator to the Laplace-Beltram operator n dfferental geometry. Unfortunately, the extenson nvolves a transton from a lnear theory to a non-lnear theory, so s much more dffcult to analyze and compute. The extenson of the theory of contnuous Remann surfaces to a combnatoral treatment of dscrete trangle meshes has been made based on recent results n algebrac and spectral graph theory. Ths translates to a set of equatons whch may be solved wth not too much dffculty. A few questons reman open, most notably on the exstence of non-degenerate solutons and the analyss of the degrees of freedom n the varous sphercal embeddngs, and how to control (or elmnate) them. Extensons to hgher genus s also nterestng. Acknowledgements Thanks to Zach Karn for helpful dscussons on CdV matrces. Ths research was partally funded by European grant HPRN-CT (MINGLE), German-Israel Fund grant I /1999 and Israel Mnstry of Scence grant References ALEXA, M Mergng Polyhedral Shapes wth Scattered Features. The Vsual Computer 16, 1, CHUNG, F.R.K Spectral Graph Theory. CBMS 92, AMS. COLEMAN, T.F., AND LI, Y An Interor Trust Regon Approach for Nonlnear Mnmzaton Subject to Bounds. SIAM Journal on Optmzaton, 6, COLIN DE VERDIERE, Y Sur un Nouvel Invarant des Graphes et un Crtere de Planarte. Journal of Combnatoral Theory B 50, [Englsh translaton: On a New Graph Invarant and a Crteron for Planarty. In Graph Structure Theory (N. Robertson, P. Seymour, Eds.) Contemporary Mathematcs, AMS, ] DAS, G., AND GOODRICH, M.T On the Complexty of Optmzaton Problems for 3-Dmensonal Convex Polyhedra and Decson Trees. Computatonal Geometry, 8, DESBRUN, M., MEYER, M., AND ALLIEZ, P Intrnsc Parameterzatons of Surface Meshes. Computer Graphcs Forum, 21, 3, DO CARMO, M.P Dfferental Geometry of Curves and Surfaces. Prentce-Hall. FIEDLER, M A Property of Egenvectors of Nonnegatve Symmetrc Matrces and Its Applcaton to Graph Theory. Czechoslovak Math. Journal, 25, FLOATER, M.S Parameterzaton and Smooth Approxmaton of Surface Trangulatons. Computer Aded Geometrc Desgn, 14, FLOATER, M.S Mean-value Coordnates. Computer Aded Geometrc Desgn, 20, FLOATER, M.S One-to-one Pecewse Lnear Mappngs Over Trangulatons. Mathematcs of Computaton 2, GU, X., AND YAU, S.-T Computng Conformal Structures of Surfaces. Communcatons n Informaton and Systems, 2, 2, GU, X., GORTLER, S., AND HOPPE, H Geometry Images. ACM Transactons on Graphcs, 21, 3, GUSKOV, I., VIDIMCE, K., SWELDENS, W., AND SCHROEDER, P Normal Meshes. In Proceedngs of ACM SIGGRAPH 2000, ACM Press/ ACM SIGGRAPH, New York, K. Akeley, Ed., Computer Graphcs Proceedngs, Annual Conferences Seres, ACM, HAKER, S., ANGENENT, S., TANNENBAUM, A., KIKINIS, R., AND SAPIRO, G Conformal Surface Parameterzaton for Texture Mappng. IEEE Transactons on Vsualzaton and Computer Graphcs, 6, 2, 1-9. HALL, K.M An r-dmensonal Quadratc Placement Algorthm. Management Scence, 17, KANAI, T., SUZUKI, H., AND KIMURA, F Metamorphoss of Arbtrary Trangular Meshes. IEEE Computer Graphcs and Applcatons, 20, 2, KARNI, Z., AND GOTSMAN, C. Spectral Compresson of Mesh Geometry. In Proceedngs of ACM SIGGRAPH 2000, ACM Press / ACM SIG- GRAPH, New York, K. Ackley, Ed., Computer Graphcs Proceedngs, Annual Conference Seres, ACM, KOBBELT, L.P., VORSATZ, J., LABISK, U., AND SEIDEL, H.-P A Shrnk-wrappng Approach to Remeshng Polygonal Surfaces. Computer Graphcs Forum, 18, 3, KOREN, Y On Spectral Graph Drawng. Preprnt, Wezmann Insttute of Scence. LEVY, B., PETITJEAN, S., RAY, N., AND MAILLOT, J Least Squares Conformal Maps for Automatc Texture Atlas Generaton. ACM Transactons on Graphcs, 21, 3, LOVASZ, L., AND SCHRIJVER, A On the Nullspace of a Coln de Verdere Matrx. Annales de l'insttute Fourer 49, PINKALL, U., AND POLTHIER, K Computng Dscrete Mnmal Surfaces and Ther Conjugates. Expermental Mathematcs, 2, RICHTER-GEBERT, J Realzaton Spaces of Polytopes. Lecture Notes n Math #1643, Sprnger. SANDER, P.V., SNYDER, J., GORTLER S.J., AND HOPPE, H Texture Mappng Progressve Meshes. In Proceedngs of ACM SIGGRAPH 2001, ACM Press/ ACM SIGGRAPH, New York, E. Fume, Ed., Computer Graphcs Proceedngs, Annual Conferences Seres, ACM, SHAPIRO A., AND TAL, A Polygon Realzaton for Shape Transformaton. The Vsual Computer, 14, 8-9, SHEFFER, A., GOTSMAN C., AND DYN, N Robust Sphercal Parameterzaton of Trangular Meshes. In Proceedngs of 4 th Israel-Korea Bnatonal Workshop on Computer Graphcs and Geometrc Modelng, Tel Avv, SHEFFER, A. AND DE STURLER, E Parameterzaton of Faceted Surfaces for Meshng Usng Angle Based Flattenng. Engneerng wth Computers, 17, 3, TUTTE. W.T How to Draw a Graph. Proc. London Math. Soc. 13, 3,