A Unified Subdivision Scheme for Polygonal Modeling

Transcription

1 EUROGRAPHICS 2 / A. Chalmers and T.-M. Rhyne (Guest Editors) Volume 2 (2), Number 3 A Unified Subdivision Sheme for Polygonal Modeling Jérôme Maillot Jos Stam Alias Wavefront Alias Wavefront 2 King St. E. 28 Third Ave, Suite 8 Toronto, Ontario, Canada M5A J7 Seattle, WA 98 jmaillot@aw.sgi.om jstam@aw.sgi.om Abstrat Subdivision rules have traditionally been designed to generate smooth surfaes from polygonal meshes. In this paper we propose to employ subdivision rules as a polygonal modeling tool, speifially to add additional level of detail to meshes. However, existing subdivision shemes have several undesirable properties making them ill suited for polygonal modeling. In this paper we propose a general set of subdivision rules whih provides users with more ontrol over the subdivision proess. Most existing subdivision shemes are speial ases. In partiular, we provide subdivision rules whih blend approximating spline based shemes with interpolatory ones. Also, we generalize subdivision to allow any number of refinements to be performed in a single step.. Introdution Subdivision surfaes have reently emerged as the most popular modeling tool in omputer graphis. This is not surprising sine these surfaes ombine the benefits of both polygonal and spline NURBS modeling. Subdivision surfaes, like NURBS, allow users to model smooth surfaes by manipulating a small set of ontrol verties. Unlike NURBS, however, there is no onstraint on the onnetivity of the ontrol verties. The first subdivision shemes were proposed in the late seventies 3, while later researh has foused generally on the properties of the limit surfae, suh as smoothness 4 9 and evaluation 6. Properties of subdivision surfaes are now well understood, making them attrative in design appliations. But, subdivision is rarely used as a polygonal modeling tool. Very little attention has been devoted to the influene of the first ouple of subdivision steps on the final shape of the surfae. However, in pratie it turns out that the initial subdivision steps greatly influene the shape of the surfae. In this paper we address this problem by adding more parameters to the subdivision rules thus providing more user ontrol. This is in ontrast with previous researh where the shape of the surfae was improved by hanging the initial ontrol mesh, not the subdivision rules themselves 6. An important appliation of our new subdivision rules is the generation of level of detail on arbitrary meshes. We will The Eurographis Assoiation and Blakwell Publishers 2. Published by Blakwell Publishers, 8 Cowley Road, Oxford OX4 JF, UK and 35 Main Street, Malden, MA 248, USA. Fig. -a Fig. -b Fig. - Figure : Eah refinement step of the Catmull-Clark algorithm shrinks the onvex areas. The differene between two levels is strongest along the silhouette of the models. demonstrate that our shemes an effiiently ompress suh models for transfers over small bandwidth networks. 2. Motivation Current subdivision rules applied to the problem of generating level of detail suffer from two major limitations. Firstly, the most popular spline-based shemes 3 3 produe surfaes that approximate the base mesh. The limit surfae, espeially in loally onvex areas, is smaller than the base mesh. This is beause the refined ontrol meshes progressively shrink towards the limit surfae. Consequently, notieable popping effets our when swithing between meshes at different levels of resolution.

2 Maillot & Stam / A Unified Subdivision Sheme for Polygonal Modeling Fig. 2-a Fig. 2-b Fig. 2- Figure 2: Eah Catmull-Clark refinement inreases the number of faes by four, thus making few levels of detail available in pratie. Fig. 3-a Fig. 3-b Fig. 3- Figure shows one (-a) and two (-b) Catmull-Clark refinements applied to the same ube. Figure - shows the differene between these two meshes. Dark areas orrespond to larger differenes and are mainly onentrated near the silhouette of the mesh. The differene is even larger if we ompare the initial ube with the first subdivision level (3- d). Although the differenes are less important between finer levels of subdivision, they remain substantial for the first few levels. Interpolatory shemes have also been proposed 5 9, but they suffer from the opposite problem. Limit surfaes tend to bulge out of the polygonal ontrol mesh, and suessive subdivision steps onverge to a surfae that is too big. They also introdue undesirable undulations in the refined meshes. Another problem with urrent subdivision shemes is that the number of refined faes grows exponentially with the number of subdivisions. In many appliations, we want to inrease the number of faes by an arbitrary number instead of some power of four (Catmull-Clark) as shown in Figure 2. This problem was partially addressed with the introdution of 3-subdivision shemes, where the number of triangles inreases only by a fator of three at eah subdivision step. However, in these shemes the number of triangles still inreases exponentially by a power of three and they do not seem to extend to quadrilaterals or shemes of higher degree. Finally, these shemes do not preserve the initial triangle boundaries, whih leads to a sudden hange in olor or texture interpolation between two onseutive subdivision levels. In pratie, this means that even a simple objet an only be refined three or four times before the level of detail database beomes too large. It turns out that this onstraint on the number of refinements is generally too small to effiiently ontrol the geometry s resolution in a real time rendering appliation. 3. From Approximation to Interpolation In order to address the silhouette problem, we propose a single parameterized sheme whih is a blend of an approximating and an interpolating sheme. We will show that these intermediate rules do a better job at preserving the silhouette and the volume of the meshes. Fig. 3-d Fig. 3-e Figure 3: Silhouette improvement using our new sheme. Figure 3-b was built using Catmull-Clark algorithm, while 3-b uses our method. The bottom row show the image differene is both ases. Our work is based on subdivision rules built as a suession of a linear subdivide operator followed by a number of smoothing steps 7. This framework is general enough to work for both triangular and quadrilateral meshes, and produes uniform B-splines of any odd degree p on the regular part of the mesh. We introdue an extra vertex moving step on the subdivided geometry to handle interpolation. 3.. The Curve Case As is usually the ase in subdivision papers, we will first desribe our shemes in a urve setting and then generalize them to surfaes Subdivision Into any Number of Piees Stam 7 showed that B-splines of odd degree p ould be subdivided by first linearly subdividing the ontrol mesh and then performing m p 2 smoothing steps. Eah step involves averaging a vertex with its immediate neighbors using the weights. The novelty over the well known Lane-Riesenfeld algorithm 2 is that this sheme performs two averaging steps at one and therefore leaves the ontrol verties in plae. Stam s result relies on the fat that the subdivision masks are related to the binomial oeffiients 2. The binomial oeffiients are easily omputed using Pasal s triangle: The Eurographis Assoiation and Blakwell Publishers 2.

3 Maillot & Stam / A Unified Subdivision Sheme for Polygonal Modeling ontrolled by the user. Newly introdued verties are also adjusted by linear interpolation of the adjusted original verties. We denote by P j the new verties obtained by subdividing the original verties P i. In the rest of the paper, : denotes assignment, while denotes a true equality of two quantities. In these notations, the first step is: P di : P i () P di k : d k d P i k d P i k d (2) This step is followed by a smoothing step that modifies the verties P i : α α 2 α Figure 4: Refinement steps, 2, and limit urve of our sheme, using degree three interpolation and different α values. A division value d 2 was used for eah step. For example, from the last line we obtain the subdivision masks for B-spline urves of degree three. Every old vertex is updated using the weights, while the new verties 4 4 are inserted between the old verties using the 8 8 masks. The ruial observation is that these two masks are obtained by simply applying the mask to the seond row, whih orresponds to the subdivision rules of linear subdivision. In fat, this onstrution is easily generalized to any number of subdivisions d. In this ase we generalize the Pasal triangle to take the average of the d elements in the row diretly above it: d 2 This gives us a reipe similar to the one found in Stam 7 to ompute the orresponding masks for these subdivision shemes. First linearly subdivide eah segment into d piees, then smooth eah vertex using the mask d 2 2 d 2. In the limit this proess generates B- spline urves of degree 2m if the smoothing is applied m times. See Chui s monograph 2 for a rigorous proof Blending Interpolating and Approximating Shemes In order to limit the amount of shrinking harateristi of approximating subdivision shemes, we propose to add an extra step whih updates the position of the verties after smoothing. We all this a push-bak step: eah original vertex is moved bak towards its original position by an amount The Eurographis Assoiation and Blakwell Publishers 2. P i : d 2 k d dp i d k P i k k P i k (3) Finally, the smoothing step is followed by a push-bak of these new verties: i : α P i P di (4) P di : P di i (5) P di k : P di d k k d k i d i k d (6) All evaluations are done in parallel in a Jaobi manner to avoid any side effets. In pratie this requires the use of an intermediate array to store the verties positions. The volume parameter α ontrols the transition from approximation to interpolation. When α there is no push-bak step and the subdivision sheme produes uniform B-splines in the limit. On the other hand, when α our shemes are interpolatory. Figure 4 shows the influene of the parameter α on the subdivided ontrol verties after several refinements. When the degree p is greater than three, the smoothing and push-bak steps are repeated p 2 times. In partiular, when p 3 and d 2, only one smoothing and one pushbak step are performed. In this ase we an expliitly write down the subdivision matrix applied to five onseutive ontrol verties: M 2 2α 4 3 α 2 2α α 8 α 8 α α 2 2α 4 3 α 2 2α α 8 α 8 α α 2 2α 4 3 α 2 2α P 2i 2 P 2i P 2i P 2i P 2i 2 6 M P i 2 P i P i P i P i 2 (7)

4 Maillot & Stam / A Unified Subdivision Sheme for Polygonal Modeling V V V E V E F E 2 V 4 V 2 V 4 E 3 V 2 V 3 E 4 V 3 E 5 Fig. 6-a Fig. 6-b. α Fig. 6-. α 7 Fig. 5-a Fig. 5-b Figure 5: Subdivision of quadrilaterals and triangles. Eah subdivision produes V, E, and F vertex types. Faes with 5 and more verties use the quadrilateral subdivision rule. In partiular, when α, the P 2i are moved bak exatly to their original position P i, and we obtain the well known four point interpolation sheme, with weights 4. Fig. 6-d. α Fig. 6-e. α Generalization to Surfaes The surfae ase is similar to the urve one: we perform one bilinear subdivision step followed by a smoothing step Binary Subdivisions We first define our rules for binary subdivision shemes when d 2. We introdue the following notations. The number of elements in a set A is denoted by A. The verties of the mesh before a subdivision step are denoted by V i. During a subdivision step these verties are transformed into new verties V i. At the same time new verties E i are introdued by splitting eah edge, and new verties F i are introdued for eah fae as in Figure 5. Let P be a vertex of the mesh, then P is the set ontaining all the verties sharing an edge with P. The set P ontains the orner verties: the verties sharing a fae with P not in P. To illustrate these definitions refer to Figure 5 where V 2 E 2 E 3 E 5, and E 3 E E 2. In the rest of this paper we will fous entirely on quadrilateral shemes. However, triangular shemes an be treated in a similar way, with the exeption that there are no fae verties F i, and V is always empty. In fat, our images were produed from meshes ontaining simultaneously triangles and quadrilaterals. Only the subdivision step must distinguish between those two types of faes. Stam provides different smoothing rules for the verties that result in uniform B-spline surfaes in the limit on the regular part of the mesh 7. The simplest smoothing algorithm whih orresponds to repeated averaging 2 replaes eah vertex by a weighted average of its diret neighbors: N i V i (8) Fig. 6-f Fig. 6-g Figure 6: Influene of parameter α on silhouette hanges. V i : 4 V i 2N i P P V i 4N i P (9) P V i Catmull-Clark surfaes are obtained with a different hoie for the weights: V i : N i 3 N i V i 2 N 2 i P P V i Ni 2 P () P V i We observe that Formulae 9 and are idential when N i 4. This omes as no surprise sine both of these shemes produe uniform B-spline surfaes on regular meshes (N i 4 everywhere). We further observe that, when N i 4, the Catmull-Clark rule an be obtained by following Formula 9 with an adjustment of all the extraordinary verties: 4 δ i N V i 4 i V i () N i N i N i 4 V i V i (2) N i V i : V i γδ i (3) The parameter γ allows us to interpolate between the two The Eurographis Assoiation and Blakwell Publishers 2.

5 Maillot & Stam / A Unified Subdivision Sheme for Polygonal Modeling d 2 d 3 Figure 8: Eah fae is divided in d d smaller one. β β 2 V 3 V 2 V 2 V V 2 V V F F 2 F 3 F F 2 F F 2 β 2 β Figure 7: Influene of β parameter. The original ube is displayed as wireframe. shemes. Not only does this adjustment unify these two shemes, but it simplifies the implementation of the Catmull- Clark subdivision: a simple smoothing followed by a vertex update step. Following the urve ase, the simplest push-bak step is to ompute the differenes i between V i and V i, followed by a (bi-)linear interpolation of these differenes for the new E i and F i verties. V i α V i V i (4) E i 2 V V (5) F i F i E k (6) E k F i F i V k (7) V k F i In Figure 6 we demonstrate how the α parameter improves the silhouette differene between a polygonal objet and its refinements. Figure 6-a is the original model, while Figures 6-b through 6-e illustrate the first and seond refinements for α (Catmull-Clark) and α 7. Figure 6-f (resp. 6-g) is the differene between 6-a and 6-d (resp. 6-e). During an animation, the popping effet an be substantially redued by hoosing an appropriate α value. However, lose to very sharp orners our sheme tends to reate flat areas around the fae enters. The reason is that a bilinear interpolation of vetors of the same length with Figure 9: Vertex [fae] neighborhoods k [ k] are defined by adding one more ring to the previous set. different angles produes smaller vetors at the enter of the faes. This is a well know artifat of ertain renderers, whih do not renormalize vertex normals after interpolation, and onsequently produe darker areas in the fae enters. We an fix this problem by introduing a renormalization step for the interpolation of the vetors. This is ahieved by interpolating the length and diretion of the vetors separately. To smooth the transition between these new rules and the ones without the normalization, we introdue a rounding fator parameter β. l E i β V V β (8) 2 E i l F i β V k F i V k β (9) F i F i E i : l E i E i (2) F i : l F i F i (2) When β there is no renormalization, while when β the lengths of the are exatly interpolated. In the ase β our subdivision rules do not reprodue uniform B- splines on the regular part of the mesh in the limit. This doesn t matter sine we do not use our rules to generate limit surfaes. Figure 7 illustrates how our meshes are deformed when β is inreased. For a ube, β 2 produes the most rounded meshes. Note that in this example we have γ (no Catmull-Clark) to emphasize the flattening problem Subdivision Into any Number of Piees For regular meshes the orresponding limit surfaes Σ s t are equal to a tensor produt of uniform B-spline urves. Therefore, the subdivision sheme for these surfaes is The Eurographis Assoiation and Blakwell Publishers 2.

6 Maillot & Stam / A Unified Subdivision Sheme for Polygonal Modeling simply a linear subdivision step followed by a smoothing step with a mask equal to the tensor produt of the mask 2 d 2 derived in Setion 3... d 2 These rules are naturally extended to irregular regions. In pratie, it turns out that it is easier to deompose the smoothing step into two simple averaging steps. The averaging step is different depending on whether d is odd or even. In the odd ase we replae eah vertex by a simple average of its k-ring neighborhood, where k d 2. When d is even, eah averaging step replaes eah fae with a vertex that is the average of the k-ring of verties surrounding it, where d k 2. The new verties after this step form the dual of the initial mesh. In pratie, however, the dual is never expliitly omputed sine the averaging step is always performed twie (an even amount in general sine we onsider only odd degrees p in this paper). Indeed, after two dualizations the verties are again in plae. More formally, let k V i be the set of all verties whih an be reahed from V i by traversing at most k faes and let k V i denote the orresponding set of the faes traversed. See figure 9 for some examples. We also define a set of fae neighborhoods by k F i V F i k V. Using these definitions we an expliitly state the smoothing steps. When d is odd, we apply the following rule p times, where k d 2 : V i : k V i V j (22) V j k V i When d is even the proedure only works for odd degrees p. We set the neighborhood to k 2, and we apply d the rule (23) p 2 times followed by (24): F i : V i : k F i k V i V j (23) V j k F i F j (24) F j k V i In pratie, we limited our appliation to odd degrees only so that no onstraint is neessary on the number of subdivisions d Catmull-Clark Corretion The Catmull-Clark orretion step defined by Formula was introdued for the ase d 2 and is only applied to the extraordinary verties of the mesh. For arbitrary divisions d we observe that this orretion only influenes a small neighborhood around eah extraordinary vertex. More preisely, this orretion never propagates further than two rings of faes around the extraordinary vertex as shown in Figure. Figure : Catmull-Clark orretion weights of the first two steps, along the Ni N 4 i V i V i vetor. Weights in the right image are meant to be divided by 64. In addition, the orretion is only notieable in the first ouple of subdivision steps. The first subdivision step produes the most visible hange whih from Formula is N i 4 N i V i V i. Subsequent subdivisions pro- equal to C i due hanges, w i C i, whih are proportional to the first one by a weight w i. It is possible to ompute these weights exatly for the first ouple of subdivision steps. These sampled weights then define a pieewise bilinear funtion on the unit square that an be used to ompute the orresponding weight values when d is a power of two. For more general d values the weights an be interpolated from this funtion In pratie, however, we found that a similar behavior an be ahieved using the push-bak step desribed in the next setion. The effet of the Catmull-Clark orretion an be emulated by using a higher α value and by adjusting the β parameter. This is apparent in Figure 7, where a value β 2 produes a rounded spherial shape despite the fat that γ Push-bak Step The push-bak is similar to the d 2 ase desribed above: we first ompute the values for the original verties and then update the newly introdued verties using bilinear interpolation. In a similar fashion we an use the normalized interpolation of the values to keep the lengths equal. For even d, the push-bak step an only be applied after Rules (23) and (24) have been applied. This is beause it doesn t make any sense to apply the push-bak to the intermediate verties F i whih are only used temporarily to ompute the new vertex positions. To make our algorithm onsistent for every number of divisions d, we restrit our algorithm to only perform the push-bak for odd d when Rule (22) is applied twie. We first intended to use a smoother interpolation of the values, but after some experimentation with higher order interpolation shemes we onluded that the differenes were too small to justify a more expensive interpolant The Eurographis Assoiation and Blakwell Publishers 2.

7 4. Appliation to Level of Detail The motivation behind our work was to provide users with a simple smoothing tool for polygonal meshes. The smoothing operation allows users to reate refined versions of their models. Cruial to the suess of suh a model is that the transitions between the different resolutions of the meshes are almost impereptible. In pratie, we found that our new subdivision sheme worked best when we used a push-bak step with α 2, β 2 and γ. Indeed, these are the default values of our smoothing tool. Of ourse, we expose these parameters to the artist who an freely explore the effet of varying the parameters to meet her partiular needs. Although this might be triky, it is a huge improvement over urrent pratie, where artists sometimes have to adjust individual verties at eah level of refinement. With our model, on the other hand, artists only have to worry about a few parameters at eah level. Sending the different levels of detail of the mesh to a remote viewer is also muh more effiient with our representation. Instead of sending the oarse mesh and the updates for eah level of resolution as in the progressive meshes ompression sheme 8, we only require a few numerial parameters to be sent for eah level. Consequently, remote viewers an almost instantly view any level of the mesh refinement. Of ourse our sheme is not as general as the progressive meshes ompression sheme, but we found that in pratie it applies to many interesting shapes. Even when the higher resolution meshes are supplemented with a set of detail offsets for eah vertex we believe that our approah offers a better starting point for the base meshes and results in substantial savings in both memory and speed. Maillot & Stam / A Unified Subdivision Sheme for Polygonal Modeling 5. Results The images in Figure show meshes of a game harater at different levels of detail. The first image shows the base mesh. Subsequent images are reated using our new subdivision sheme with different resolutions, ranging from d 2 to d 7. In eah ase we set our parameters to α 5 β γ p 3. Notie that there is very little differene in the silhouette between the levels. Applying standard approximating subdivision shemes to the base mesh would have resulted in more shrinking and restrited us to only three different levels due to the exponential inrease in the number of verties with eah subdivision. We also performed some animation tests where we interatively swith between different levels of detail depending on the distane of the model to the amera. The dog images in the olor plate were extrated from one suh test. Compare the sequene omputed using the standard Catmull- Clark (left olumn) to the ones omputed using our new sheme (right olumn). Figure : Eight different resolutions of the same model The Eurographis Assoiation and Blakwell Publishers 2.

8 Maillot & Stam / A Unified Subdivision Sheme for Polygonal Modeling The bottom two images in the Color Plate demonstrate that texture oordinates an be interpolated in the same manner. These models were used in an atual game where low resolution textures are ruial to keep the texture memory onsumption as low as possible. 6. Conlusion In this paper we have presented a novel set of subdivision rules speifially designed for the problem of reating different levels of detail for meshes. Our models depend on a few parameters that an artist an adjust to ahieve various effets. Existing approximating and interpolating subdivision shemes are speial ases of our new sheme. The advantages of our new sheme over existing ones is that () we provide user ontrol over the amount of shrinking and (2) we allow arbitrary refinements whih do not grow exponentially. These two features make subdivision shemes more attrative in the realm of polygonal modeling. Our new rules are quite easy to implement. The entire subdivision system was oded as a MAYA plug-in in only roughly 5 lines of C++ ode. We were able to interat in real time with our models on a relatively modest desktop mahine (O 2, R5). Of ourse, this was true only when our division number d was approximately between 2 and 6. Our subdivision shemes allow both quadrilaterals and triangles to o-exist in a mesh and its refined versions. This is in ontrast to existing subdivision shemes whih generate meshes whih are either all triangles or all quadrilaterals. Artists often want to keep both triangles and quadrilaterals in their models. Also it is well known that triangles in a base mesh reate artifats in the refined meshes when a quad-based subdivision sheme is used. The reason that we an inorporate both triangles and quadrilaterals simultaneously in our subdivision sheme is that our sheme is based on a deomposition of the rules into a linear step followed by smoothing. 7. Future work A straightforward extension of our model is to allow our parameters to vary on the mesh. These values ould be painted onto the mesh by the artist for additional ontrol, for example. However, we found that our urrent sheme provided enough flexibility with the α β and γ parameters fixed for eah level. On the other hand, allowing the parameters to vary in very loalized regions ould still be helpful in ertain appliations. In this paper we have ignored the properties of the limit surfaes generated by our shemes. The reason for this omission is that we are interested solely in the shape of a finite number of intermediate meshes, not in the limit surfae. On the other hand our new shemes might be useful in surfae design as well. In this ase it is important to study the smoothness of the limit surfae. At this point it is not lear how the smoothness depends on our parameters. We leave this as an open problem for future researh. Also, we did not pay muh attention to reases and objet boundaries. Boundaries and reases an be dealt with by applying our urve rules to the orresponding polylines. Vertex reases are more diffiult to handle. This is beause reased verties are not updated during our smoothing step. Consequently the push-bak vertex is always zero, resulting in undesirable wrinkles in the meshes. More work is needed in this ase to improve the behavior of the refined shape. Finally, we found that the optimal set of parameters, espeially α, depends on the amera position. This is partiularly visible for a simple objet like a ube, where larger values of α are required when viewed from the side. One solution would be to allow the user to speify different values of α for different views of an objet. Our system would then automatially interpolate α values for intermediate viewing positions as the amera moves. Aknowledgments The help and omments of A W artists have been very useful in order to understand the level of detail problems. We would like to thank partiularly C. Chaix, J. Bell, and P. Pattel who built the objets we used. Referenes. E. Catmull, J. Clark. Reursively Generated B-Spline Surfaes On Arbitrary Topologial Meshes. Computer Aided Design, (6):35 355, C.K. Chui. Multivariate Splines. CBMS-NSF Regional Conferene Series in Applied Mathematis. SIAM, Philadelphia, PA, D. Doo, M. A. Sabin. Behaviour Of Reursive Subdivision Surfaes Near Extraordinary Points.. Computer Aided Design, (6):356 36, N. Dyn, D. Levin, J. Gregory. A 4-point interpolatory subdivision sheme for urve design. CAGD 4, , N. Dyn, D. Levin, J. Gregory. A Butterfly Subdivision Sheme for Surfae Interpolation with Tension Control. ACM Trans. Graph. 9, 6 69, M. Halstead, M. Kass, T. DeRose. Effiient, Fair Interpolation Using Catmull-Clark Surfaes. Computer Graphis Proeedings, Annual Conferene Series, 35 44, August, H. Hoppe, T. DeRose, T. Duhamp, M. Halstead, H. Jin, J. MDonald, J. Shweitzer, W. Stuetzle. Pieewise smooth surfae reonstrution. Computer Graphis Proeedings, Annual Conferene Series, , July, 994. The Eurographis Assoiation and Blakwell Publishers 2.

9 8. H. Hoppe. Progressive Meshes. Computer Graphis Proeedings, Annual Conferene Series, 99-8, August, L. Kobbelt. Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Computer Graphis Forum 5(3), 49 42, September, L. Kobbelt. 3-Subdivisions. Computer Graphis Proeedings, Annual Conferene Series, pages 3 2, July, 2.. U. Labsik, G. Greiner. Interpolatory 3-Subdivision. Proeedings of Eurographis 2, Computer Graphis Forum, 9(3):3 38, September, J.M. Lane, R. F. Riesenfeld. A Theoretial Development For the Computer Generation and Display of Pieewise Polynomial Surfaes. IEEE Transations on Pattern Analysis and Mahine Intelligene, 2():35 46, January, C.T. Loop. Smooth Subdivision Surfaes Based on Triangles. M.S. Thesis, Department of Mathematis, University of Utah, August, U. Reif. A Unified Approah To Subdivision Algorithms Near Extraordinary Verties. Computer Aided Geometri Design, 2:53 74, U. Reif. A degree estimate for subdivision surfaes of higher regularity. Proeedings of the Amerian Mathematial Soiety, 24: , J. Stam. Exat Evaluation of Catmull-Clark Subdivision Surfaes at Arbitrary Parameter Values. In Computer Graphis Proeedings, Annual Conferene Series, pages , July, J. Stam. On Subdivision Shemes Generalizing Uniform B-Spline Surfaes of Arbitrary Degree. Computer Aided Geometri Design 8, Speial issue on Subdivision Surfaes (to appear), H. Weimer, J. Warren. Subdivision Shemes for Thin Plate Splines. Proeedings of Eurographis 998, Computer Graphis Forum, 7(3):33 33, D. Zorin. Subdivision and Multiresolution Surfae Representations. PhD thesis, Calteh, Pasadena, California, D. Zorin, P. Shröder. A Unified Framework for Primal/Dual Quadrilateral Subdivision Shemes. Computer Aided Geometri Design 8, Speial issue on Subdivision Surfaes (to appear), 2. Maillot & Stam / A Unified Subdivision Sheme for Polygonal Modeling The Eurographis Assoiation and Blakwell Publishers 2.

10 Maillot & Stam / A Unified Subdivision Sheme for Polygonal Modeling Catmull-Clark α 5 oarse mesh α 7 d 3 The Eurographis Assoiation and Blakwell Publishers 2.