Mean Value Coordinates for Closed Triangular Meshes


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1 Mean Value Coordnates for Closed Trangular Meshes Tao Ju, Scott Schaefer, Joe Warren Rce Unversty (a) (b) (c) (d) Fgure : Orgnal horse model wth enclosng trangle control mesh shown n black (a). Several deformatons generated usng our 3D mean value coordnates appled to a modfed control mesh (b,c,d). Abstract Constructng a functon that nterpolates a set of values defned at vertces of a mesh s a fundamental operaton n computer graphcs. Such an nterpolant has many uses n applcatons such as shadng, parameterzaton and deformaton. For closed polygons, mean value coordnates have been proven to be an excellent method for constructng such an nterpolant. In ths paper, we generalze mean value coordnates from closed D polygons to closed trangular meshes. Gven such a mesh P, we show that these coordnates are contnuous everywhere and smooth on the nteror of P. The coordnates are lnear on the trangles of P and can reproduce lnear functons on the nteror of P. To llustrate ther usefulness, we conclude by consderng several nterestng applcatons ncludng constructng volumetrc textures and surface deformaton. CR Categores: I.3.5 [Computer Graphcs]: Computatonal Geometry and Object Modelng Boundary representatons; Curve, surface, sold, and object representatons; Geometrc algorthms, languages, and systems Keywords: barycentrc coordnates, mean value coordnates, volumetrc textures, surface deformaton Introducton Gven a closed mesh, a common problem n computer graphcs s to extend a functon defned at the vertces of the mesh to ts nteror. For example, Gouraud shadng computes ntenstes at the vertces of a trangle and extends these ntenstes to the nteror usng lnear nterpolaton. Gven a trangle wth vertces {p, p, p 3 } and assocated ntenstes { f, f, f 3 }, the ntensty at pont v on the nteror of the trangle can be expressed n the form ˆf[v] = f j () where w j s the area of the trangle {v, p j, p j+ }. In ths formula, note that each weght w j s normalzed by the sum of the weghts, w to form an assocated coordnate j. The nterpolant ˆf[v] s then smply the sum of the f j tmes ther correspondng coordnate. Mesh parameterzaton methods [Hormann and Grener 000; Desbrun et al. 00; Khodakovsky et al. 003; Schrener et al. 004; Floater and Hormann 005] and freeform deformaton methods [Sederberg and Parry 986; Coqullart 990; MacCracken and Joy 996; Kobayash and Ootsubo 003] also make heavy use of nterpolants of ths type. Both applcatons requre that a pont v be represented as an affne combnaton of the vertces on an enclosng shape. To generate ths combnaton, we smply set the data values f j to be ther assocated vertex postons p j. If the nterpolant reproduces lnear functons,.e.; v = p j, w j the coordnate functons are the desred affne combnaton. For convex polygons n D, a sequence of papers, [Wachspress 975], [Loop and DeRose 989] and [Meyer et al. 00], have proposed and refned an nterpolant that s lnear on ts boundares and only nvolves convex combnatons of data values at the vertces of the polygons. Ths nterpolant has a smple, local defnton as a ratonal functon and reproduces lnear functons. [Warren 996; Warren et al. 004] also generalzed ths nterpolant to convex shapes n hgher dmensons. Unfortunately, Wachspress s nterpolant does not generalze to nonconvex polygons. Applyng
2 (a) (b) such a generalzaton for arbtrary closed surfaces and show that the resultng nterpolants are wellbehaved and have lnear precson. Appled to closed polygons, our constructon reproduces D mean value coordnates. We then apply our method to closed trangular meshes and construct 3D mean value coordnates. (In ndependent contemporaneous work, [Floater et al. 005] have proposed an extenson of mean value coordnates from D polygons to 3D trangular meshes dentcal to secton 3..) Next, we derve an effcent, stable method for evaluatng the resultng mean value nterpolant n terms of the postons and assocated values of vertces of the mesh. Fnally, we consder several practcal applcatons of such coordnates ncludng a smple method for generatng classes of deformatons useful n character anmaton. Mean value nterpolaton (c) Fgure : Interpolatng hue values at polygon vertces usng Wachspress coordnates (a, b) versus mean value coordnates (c, d) on a convex and a concave polygon. the constructon to such a polygon yelds an nterpolant that has poles (dvsons by zero) on the nteror of the polygon. The top porton of Fgure shows Wachspress s nterpolant appled to two closed polygons. Note the poles on the outsde of the convex polygon on the left as well as along the extensons of the two top edges of the nonconvex polygon on the rght. More recently, several papers, [Floater 997; Floater 998; Floater 003], [Malsch and Dasgupta 003] and [Hormann 004], have focused on buldng nterpolants for nonconvex D polygons. In partcular, Floater proposed a new type of nterpolant based on the mean value theorem [Floater 003] that generates smooth coordnates for starshaped polygons. Gven a polygon wth vertces p j and assocated values f j, Floater s nterpolant defnes a set of weght functons w j of the form tan w j = [ α j ] + tan p j v [ ] α j (d). () where α j s the angle formed by the vector p j v and p j+ v. Normalzng each weght functon w j by the sum of all weght functons yelds the mean value coordnates of v wth respect to p j. In hs orgnal paper, Floater prmarly ntended ths nterpolant to be used for mesh parameterzaton and only explored the behavor of the nterpolant on ponts n the kernel of a starshaped polygon. In ths regon, mean value coordnates are always nonnegatve and reproduce lnear functons. Subsequently, Hormann [Hormann 004] showed that, for any smple polygon (or nested set of smple polygons), the nterpolant ˆf[v] generated by mean value coordnates s welldefned everywhere n the plane. By mantanng a consstent orentaton for the polygon and treatng the α j as sgned angles, Hormann also shows that mean value coordnates reproduce lnear functons everywhere. The bottom porton of Fgure shows mean value coordnates appled to two closed polygons. Note that the nterpolant generated by these coordnates possesses no poles anywhere even on nonconvex polygons. Contrbutons Horman s observaton suggests that Floater s mean value constructon could be used to generate a smlar nterpolant for a wder class of shapes. In ths paper, we provde Gven a closed surface P n R 3, let p[x] be a parameterzaton of P. (Here, the parameter x s twodmensonal.) Gven an auxlary functon f[x] defned over P, our problem s to construct a functon ˆf[v] where v R 3 that nterpolates f[x] on P,.e.; ˆf[p[x]] = f[x] for all x. Our basc constructon extends an dea of Floater developed durng the constructon of D mean value coordnates. To construct ˆf[v], we project a pont p[x] of P onto the unt sphere S v centered at v. Next, we weght the pont s assocated value f[x] by p[x] v and ntegrate ths weghted functon over S v. To ensure affne nvarance of the resultng nterpolant, we dvde the result by the ntegral of the weght functon p[x] v taken over S v. Puttng the peces together, the mean value nterpolant has the form x ˆf[v] = w[x,v] f[x]ds v x w[x,v]ds (3) v where the weght functon w[x, v] s exactly p[x] v. Observe that ths formula s essentally an ntegral verson of the dscrete formula of Equaton. Lkewse, the contnuous weght functon w[x, v] and the dscrete weghts w j of Equaton dffer only n ther numerators. As we shall see, the tan [ ] α terms n the numerators of the w j are the result of takng the ntegrals n Equaton 3 wth respect to ds v. The resultng mean value nterpolant satsfes three mportant propertes. Interpolaton: As v converges to the pont p[x] on P, ˆf[v] converges to f[x]. Smoothness: The functon ˆf[v] s welldefned and smooth for all v not on P. Lnear precson: If f[x] = p[x] for all x, the nterpolant ˆf[v] s dentcally v for all v. Interpolaton follows from the fact that the weght functon w[x, v] approaches nfnty as p[x] v. Smoothness follows because the projecton of f[x] onto S v s contnuous n the poston of v and takng the ntegral of ths contnuous process yelds a smooth functon. The proof of lnear precson reles on the fact that the ntegral of the unt normal over a sphere s exactly zero (due to symmetry). Specfcally, p[x] v x p[x] v ds v = 0 snce p[x] v p[x] v s the unt normal to S v at parameter value x. Rewrtng ths equaton yelds the theorem. v = x p[x] / p[x] v ds v x p[x] v ds v
3 Notce that f the projecton of P onto S v s onetoone (.e.; v s n the kernel of P), then the orentaton of ds v s nonnegatve, whch guarantees that the resultng coordnate functons are postve. Therefore, f P s a convex shape, then the coordnate functons are postve for all v nsde P. However, f v s not n the kernel of P, then the orentaton of ds v s negatve and the coordnates functons may be negatve as well. 3 Coordnates for pecewse lnear shapes In practce, the ntegral form of Equaton 3 can be complcated to evaluate symbolcally. However, n ths secton, we derve a smple, closed form soluton for pecewse lnear shapes n terms of the vertex postons and ther assocated functon values. As a smple example to llustrate our approach, we frst rederve mean value coordnates for closed polygons va mean value nterpolaton. Next, we apply the same dervaton to construct mean value coordnates for closed trangular meshes. 3. Mean value coordnates for closed polygons Consder an edge E of a closed polygon P wth vertces {p, p } and assocated values { f, f }. Our frst task s to convert ths dscrete data nto a contnuous form sutable for use n Equaton 3. We can lnearly parameterze the edge E va p[x] = φ [x]p where φ [x] = ( x) and φ [x] = x. We then use ths same parameterzaton to extend the data values f and f lnearly along E. Specfcally, we let f[x] have the form f[x] = φ [x] f. Now, our task s to evaluate the ntegrals n Equaton 3 for 0 x. Let E be the crcular arc formed by projectng the edge E onto the unt crcle S v, we can rewrte the ntegrals of Equaton 3 restrcted to E as xw[x,v] f[x]de x w[x,v]de = w f (4) w where weghts w = φ [x] x p[x] v de. Our next goal s to compute the correspondng weghts w for edge E n Equaton 4 wthout resortng to symbolc ntegraton (snce ths wll be dffcult to generalze to 3D). Observe that the followng dentty relates w to a vector, w (p v) = m. (5) where m = p[x] v x de s smply the ntegral of the outward unt p[x] v normal over the crcular arc E. We call m the mean vector of E, as scalng m by the length of the arc yelds the centrod of the crcular arc E. Based on D trgonometry, m has a smple expresson n terms of p and p. Specfcally, To evaluate the ntegral of Equaton 3, we can relate the dfferental ds v to dx va ds v = p [x].(p[x] v) p[x] v dx where p [x] s the cross product of the n tangent vectors p[x] to P at x p[x]. Note that the sgn of ths expresson correctly captures whether P has folded back durng ts projecton onto S v. m = tan[α/]( (p v) p v + (p v) p v ) where α denotes the angle between p v and p v. Hence we obtan w = tan[α/]/ p v whch agrees wth the Floater s weghtng functon defned n Equaton for D mean value coordnates when restrcted to a sngle edge of a polygon. Equaton 4 allows us to formulate a closed form expresson for the nterpolant ˆf[v] n Equaton 3 by summng the ntegrals for all edges E k n P (note that we add the ndex k for enumeraton of edges): ˆf[v] = k w k f k k w k (6) where w k and f k are weghts and values assocated wth edge E k. 3. Mean value coordnates for closed meshes We now consder our prmary applcaton of mean value nterpolaton for ths paper; the dervaton of mean value coordnates for trangular meshes. These coordnates are the natural generalzaton of D mean value coordnates. Gven trangle T wth vertces {p, p, p 3 } and assocated values { f, f, f 3 }, our frst task s to defne the functons p[x] and f[x] used n Equaton 3 over T. To ths end, we smply use the lnear nterpolaton formula of Equaton. The resultng functon f[x] s a lnear combnaton of the values f tmes bass functons φ [x]. As n D, the ntegral of Equaton 3 reduces to the sum n Equaton 6. In ths case, the weghts w have the form φ [x] w = x p[x] v dt where T s the projecton of trangle T onto S v. To avod computng ths ntegral drectly, we nstead relate the weghts w to the mean vector m for the sphercal trangle T by nvertng Equaton 5. In matrx form, {w,w,w 3 } = m {p v, p v, p 3 v} (7) All that remans s to derve an explct expresson for the mean vector m for a sphercal trangle T. The followng theorem solves ths problem. Theorem 3. Gven a sphercal trangle T, let θ be the length of ts th edge (a crcular arc) and n be the nward unt normal to ts th edge (see Fgure 3 (b)). Then, m = θ n (8) where m, the mean vector, s the ntegral of the outward unt normals over T. Proof: Consder the sold trangular wedge of the unt sphere wth cap T. The ntegral of outward unt normals over a closed surface s always exactly zero [Flemng 977, p.34]. Thus, we can partton the ntegral nto three trangular faces whose outward normals are n wth assocated areas θ. The theorem follows snce m θ n s then zero. Note that a smlar result holds n D, where the mean vector m defned by Equaton 3. for a crcular arc E on the unt crcle can be nterpreted as the sum of the two nward unt normals of the vectors p v (see Fgure 3 (a)). In 3D, the lengths θ of the edges of the sphercal trangle T are the angles between the vectors p v and p + v whle the unt normals n are formed by takng the cross
4 m E (a) n n v Fgure 3: Mean vector m on a crcular arc E wth edge normals n (a) and on a sphercal trangle T wth arc lengths θ and face normals n. product of p v and p + v. Gven the mean vector m, we now compute the weghts w usng Equaton 7 (but wthout dong the matrx nverson) va w = ψ 3 θ θ m n m n (p v) At ths pont, we should note that projectng a trangle T onto S v may reverse ts orentaton. To guarantee lnear precson, these foldedback trangles should produce negatve weghts w. If we mantan a postve orentaton for the vertces of every trangle T, the mean vector computed usng Equaton 8 ponts towards the projected sphercal trangle T when T has a postve orentaton and away from T when T has a negatve orentaton. Thus, the resultng weghts have the approprate sgn. 3.3 Robust mean value nterpolaton The dscusson n the prevous secton yelds a smple evaluaton method for mean value nterpolaton on trangular meshes. Gven pont v and a closed mesh, for each trangle T n the mesh wth vertces {p, p, p 3 } and assocated values { f, f, f 3 },. Compute the mean vector m va Equaton 8. Compute the weghts w usng Equaton 9 3. Update the denomnator and numerator of ˆf[v] defned n Equaton 6 respectvely by addng w and w f To correctly compute ˆf[v] usng the above procedure, however, we must overcome two obstacles. Frst, the weghts w computed by Equaton 9 may have a zero denomnator when the pont v les on plane contanng the face T. Our method must handle ths degenerate case gracefully. Second, we must be careful to avod numercal nstablty when computng w for trangle T wth a small projected area. Such trangles are the domnant type when evaluatng mean value coordnates on meshes wth large number of trangles. Next we dscuss our solutons to these two problems and present the complete evaluaton algorthm as pseudocode n Fgure 4. Stablty: When the trangle T has small projected area on the unt sphere centered at v, computng weghts usng Equaton 8 and 9 becomes numercally unstable due to cancellng of unt normals n that are almost coplanar. To ths end, we next derve a stable formula for computng weghts w. Frst, we substtute Equaton 8 nto Equaton 9, usng trgonometry we obtan T ψ ψ θ 3 (b) n n n 3 v (9) w = θ cos[ψ + ]θ cos[ψ ]θ + sn[ψ + ]sn[θ ] p k v, (0) // Robust evaluaton on a trangular mesh for each vertex p j wth values f j d j p j x f d j < ε return f j u j (p j x)/d j totalf 0 totalw 0 for each trangle wth vertces p, p, p 3 and values f, f, f 3 l u + u // for =,,3 θ arcsn[l /] h ( θ )/ f π h < ε // x les on t, use D barycentrc coordnates w sn[θ ]d d + return ( w f )/( w ) c (sn[h]sn[h θ ])/(sn[θ + ]sn[θ ]) s sgn[det[u,u,u 3 ]] c f, s ε // x les outsde t on the same plane, gnore t contnue w (θ c + θ c θ + )/(d sn[θ + ]s ) totalf+ = w f totalw+ = w f x totalf/totalw Fgure 4: Mean value coordnates on a trangular mesh where ψ ( =,,3) denotes the angles n the sphercal trangle T. Note that the ψ are the dhedral angles between the faces wth normals n and n +. We llustrate the angles ψ and θ n Fgure 3 (b). To calculate the cos of the ψ wthout computng unt normals, we apply the halfangle formula for sphercal trangles [Beyer 987], cos[ψ ] = sn[h]sn[h θ ], () sn[θ + ]sn[θ ] where h = (θ +θ +θ 3 )/. Substtutng Equaton nto 0, we obtan a formula for computng w that only nvolves lengths p v and angles θ. In the pseudocode from Fgure 4, angles θ are computed usng arcsn, whch s stable for small angles. Coplanar cases: Observe that Equaton 9 nvolves dvson by n (p v), whch becomes zero when the pont v les on plane contanng the face T. Here we need to consder two dfferent cases. If v les on the plane nsde T, the contnuty of mean value nterpolaton mples that ˆf[v] converges to the value f[x] defned by lnear nterpolaton of the f on T. On the other hand, f v les on the plane outsde T, the weghts w become zero as ther ntegral defnton φ [x] p[x] v dt becomes zero. We can easly test for the frst case because the sum Σ θ = π for ponts nsde of T. To test for the second case, we use Equaton to generate a stable computaton for sn[ψ ]. Usng ths defnton, v les on the plane outsde T f any of the dhedral angles ψ (or sn[ψ ]) are zero. 4 Applcatons and results Whle mean value coordnates fnd ther man use n boundary value nterpolaton, these coordnates can be appled to a varety of applcatons. In ths secton, we brefly dscuss several of these applcatons ncludng constructng volumetrc textures and surface deformaton. We conclude wth a secton on our mplementaton of these coordnates and provde evaluaton tmes for varous shapes.
5 Fgure 5: Orgnal model of a cow (topleft) wth hue values specfed at the vertces. The planar cuts llustrate the nteror of the functon generated by 3D mean value coordnates. 4. Boundary value nterpolaton As mentoned n Secton, these coordnate functons may be used to perform boundary value nterpolaton for trangular meshes. In ths case, functon values are assocated wth the vertces of the mesh. The functon constructed by our method s smooth, nterpolates those vertex values and s a lnear functon on the faces of the trangles. Fgure 5 shows an example of nterpolatng hue specfed on the surface of a cow. In the topleft s the orgnal model that serves as nput nto our algorthm. The rest of the fgure shows several slces of the cow model, whch reveal the volumetrc functon produced by our coordnates. Notce that the functon s smooth on the nteror and nterpolates the colors on the surface of the cow. 4. Volumetrc textures These coordnate functons also have applcatons to volumetrc texturng as well. Fgure 6 (topleft) llustrates a model of a bunny wth a D texture appled to the surface. Usng the texture coordnates (u,v ) as the f for each vertex, we apply our coordnates and buld a functon that nterpolates the texture coordnates specfed at the vertces and along the polygons of the mesh. Our functon extrapolates these surface values to the nteror of the shape to construct a volumetrc texture. Fgure 6 shows several slces revealng the volumetrc texture wthn. 4.3 Surface Deformaton Surface deformaton s one applcaton of mean value coordnates that depends on the lnear precson property outlned n Secton. In ths applcaton, we are gven two shapes: a model and a control mesh. For each vertex v n the model, we frst compute ts mean value weght functons w j wth respect to each vertex p j n the undeformed control mesh. To perform the deformaton, we move the vertces of the control mesh to nduce the deformaton on the orgnal surface. Let ˆp j be the postons of the vertces from the deformed control mesh, then the new vertex poston ˆv n the deformed model s computed as ˆv = ˆp j. Notce that, due to lnear precson, f ˆp j = p j, then ˆv = v. Fgures and 7 show several examples of deformatons generated wth ths Fgure 6: Textured bunny (topleft). Cuts of the bunny to expose the volumetrc texture constructed from the surface texture. process. Fgure (a) depcts a horse before deformaton and the surroundng control mesh shown n black. Movng the vertces of the control mesh generates the smooth deformatons of the horse shown n (b,c,d). Prevous deformaton technques such as freeform deformatons [Sederberg and Parry 986; MacCracken and Joy 996] requre volumetrc cells to be specfed on the nteror of the control mesh. The deformatons produced by these methods are dependent on how the control mesh s decomposed nto volumetrc cells. Furthermore, many of these technques restrct the user to creatng control meshes wth quadrlateral faces. In contrast, our deformaton technque allows the artst to specfy an arbtrary closed trangular surface as the control mesh and does not requre volumetrc cells to span the nteror. Our technque also generates smooth, realstc lookng deformatons even wth a small number of control ponts and s qute fast. Generatng the mean value coordnates for fgure took 3.3s and.9s for fgure 7. However, each of the deformatons only took 0.09s and 0.03s respectvely, whch s fast enough to apply these deformatons n realtme. 4.4 Implementaton Our mplementaton follows the pseudocode from Fgure 4 very closely. However, to speed up computatons, t s helpful to precompute as much nformaton as possble. Fgure 8 contans the number of evaluatons per second for varous models sampled on a 3GHz Intel Pentum 4 computer. Prevously, practcal applcatons nvolvng barycentrc coordnates have been restrcted to D polygons contanng a very small number of lne segments. In ths paper, for the frst tme, barycentrc coordnates have been appled to truly large shapes (on the order of 00, 000 polygons). The coordnate computaton s a global computaton and all vertces of the surface must be used to evaluate the functon at a sngle pont. However, much of the tme spent s determnng whether or not a pont les on the plane of one of the trangles n the mesh and, f so, whether or not that pont s nsde that trangle. Though we have not done so, usng varous spatal parttonng data structures to reduce the number of trangles that
6 mportant generalzaton would be to derve mean value coordnates for pecewse lnear mesh wth arbtrary closed polygons as faces. On these faces, the coordnates would degenerate to standard D mean value coordnates. We plan to address ths topc n a future paper. Acknowledgements We d lke to thank John Morrs for hs help wth desgnng the control meshes for the deformatons. Ths work was supported by NSF grant ITR References BEYER, W. H CRC Standard Mathematcal Tables (8th Edton). CRC Press. Fgure 7: Orgnal model and surroundng control mesh shown n black (topleft). Deformng the control mesh generates smooth deformatons of the underlyng model. Model Trs Verts Eval/s Horse control mesh (fg ) Armadllo control mesh (fg 7) Cow (fg 5) Bunny (fg 6) Fgure 8: Number of evaluatons per second for varous models. must be checked for coplanarty could greatly enhance the speed of the evaluaton. 5 Conclusons and Future Work Mean value coordnates are a smple, but powerful method for creatng functons that nterpolate values assgned to the vertces of a closed mesh. Perhaps the most ntrgung feature of mean value coordnates s that fact that they are welldefned on both the nteror and the exteror of the mesh. In partcular, mean value coordnates do a reasonable job of extrapolatng value outsde of the mesh. We ntend to explore applcatons of ths feature n future work. Another nterestng pont s the relatonshp between mean value coordnates and Wachspress coordnates. In D, both coordnate functons are dentcal for convex polygons nscrbed n the unt crcle. As a result, one method for computng mean value coordnates s to project the vertces of the closed polygon onto a crcle and compute Wachspress coordnates for the nscrbed polygon. However, n 3D, ths approach fals. In partcular, nscrbng the vertces of a trangular mesh onto a sphere does not necessarly yeld a convex polyhedron. Even f the nscrbed polyhedron happens to be convex, the resultng Wachspress coordnates are ratonal functons of the vertex poston v whle the mean value coordnates are transcendental functons of v. Fnally, we only consder meshes that have trangular faces. One COQUILLART, S Extended freeform deformaton: a sculpturng tool for 3d geometrc modelng. In SIGGRAPH 90: Proceedngs of the 7th annual conference on Computer graphcs and nteractve technques, ACM Press, DESBRUN, M., MEYER, M., AND ALLIEZ, P. 00. Intrnsc Parameterzatons of Surface Meshes. Computer Graphcs Forum, 3, FLEMING, W., Ed Functons of Several Varables. Second edton. Sprnger Verlag. FLOATER, M. S., AND HORMANN, K Surface parameterzaton: a tutoral and survey. In Advances n Multresoluton for Geometrc Modellng, N. A. Dodgson, M. S. Floater, and M. A. Sabn, Eds., Mathematcs and Vsualzaton. Sprnger, Berln, Hedelberg, FLOATER, M. S., KOS, G., AND REIMERS, M Mean value coordnates n 3d. To appear n CAGD. FLOATER, M Parametrzaton and smooth approxmaton of surface trangulatons. CAGD 4, 3, FLOATER, M Parametrc Tlngs and Scattered Data Approxmaton. Internatonal Journal of Shape Modelng 4, FLOATER, M. S Mean value coordnates. Comput. Aded Geom. Des. 0,, 9 7. HORMANN, K., AND GREINER, G MIPS  An Effcent Global Parametrzaton Method. In Curves and Surfaces Proceedngs (Sant Malo, France), HORMANN, K Barycentrc coordnates for arbtrary polygons n the plane. Tech. rep., Clausthal Unversty of Technology, September. hormann/papers/barycentrc.pdf. KHODAKOVSKY, A., LITKE, N., AND SCHROEDER, P Globally smooth parameterzatons wth low dstorton. ACM Trans. Graph., 3, KOBAYASHI, K. G., AND OOTSUBO, K tffd: freeform deformaton by usng trangular mesh. In SM 03: Proceedngs of the eghth ACM symposum on Sold modelng and applcatons, ACM Press, LOOP, C., AND DEROSE, T A multsded generalzaton of Bézer surfaces. ACM Transactons on Graphcs 8, MACCRACKEN, R., AND JOY, K. I Freeform deformatons wth lattces of arbtrary topology. In SIGGRAPH 96: Proceedngs of the 3rd annual conference on Computer graphcs and nteractve technques, ACM Press, MALSCH, E., AND DASGUPTA, G Algebrac constructon of smooth nterpolants on polygonal domans. In Proceedngs of the 5th Internatonal Mathematca Symposum. MEYER, M., LEE, H., BARR, A., AND DESBRUN, M. 00. Generalzed Barycentrc Coordnates for Irregular Polygons. Journal of Graphcs Tools 7,, 3. SCHREINER, J., ASIRVATHAM, A., PRAUN, E., AND HOPPE, H Intersurface mappng. ACM Trans. Graph. 3, 3, SEDERBERG, T. W., AND PARRY, S. R Freeform deformaton of sold geometrc models. In SIGGRAPH 86: Proceedngs of the 3th annual conference on Computer graphcs and nteractve technques, ACM Press, WACHSPRESS, E A Ratonal Fnte Element Bass. Academc Press, New York. WARREN, J., SCHAEFER, S., HIRANI, A., AND DESBRUN, M Barycentrc coordnates for convex sets. Tech. rep., Rce Unversty. WARREN, J Barycentrc Coordnates for Convex Polytopes. Advances n Computatonal Mathematcs 6,
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