Practical Polygonal Mesh Modeling with Discrete Gaussian-Bonnet Theorem

Transcription

1 Practical Polygonal Mesh Modeling with Discrete Gaussian-Bonnet Theorem ERGUN AKLEMAN Visualization Sciences Program Texas A&M Uniersity JIANER CHEN Computer Science Department Abstract In this paper, we introduce a practical modeling approach to improe the quality o polygonal mesh structures. Our approach is based on a discrete ersion o Gaussian-Bonnet theorem on piecewise planar maniold meshes and ertex angle delections that determines local geometric behaior. Based on discrete Gaussian- Bonnet theorem, summation o angle delections o all ertices is independent o mesh structure and it depends on only the topology o the mesh surace. Based on this result, it can be possible to improe organization o mesh structure o a shape according to its intended geometric structure. 1 Motiation Most proessional modelers in animation and special eect industry slowly migrated subdiision suraces. One o the crucial questions or practical modeling with subdiision suraces is to identiy the numbers and types o extraordinary ertices that needs to be used or a successul modeling. In the irst obseration, this question seems easy to answer based on genus o the goal surace. Using Euler s ormula, we can compute the number o extraordinary ertices and their alences (also the number o aces and their alences) or the gien genus. Howeer, extraordinary ertices do not result only rom genus. In most practical cases, we need to consider the geometrical structure o the surace. Tree structures are especially the ones that cause problems. Tree structures are more common than real trees. For instance, een head o a human being can be considered genus- 1 tree that has at least 3 branches that include nose and 2 eyes. The one hole come rom digestie track that start rom mouth. It is possible to model the head as a genus-0 tree with at least our branches by include mouth as a branch. By relating topology and geometry, we gie a ramework to answer such problems coming rom practical modeling. Our results justiy some o the intuitiely deeloped practices in current modeling approaches. For instance, the modelers introduce extraordinary ertices to model eyes, nose and mouth in acial modeling. Our results go urther than and een deies common sense about subdiision modeling. We show that despite the common belie the quality o some suraces can be improed by introducing extraordinary ertices. Corresponding Author: Program Coordinator, Visualization Sciences, Department o Architecture, College o Architecture. Address: 418 Langord Center C, College Station, Texas ergun@iz.tamu.edu. phone: +(979) ax: +(979) Introduction Our approach is based on the Gauss-Bonnet Theorem that says that the integral o the Gaussian curature oer a closed smooth surace is equal to 2π the Euler characteristic o the surace which is 2 2g [Weisstein 2005]. The theorem requires a smooth surace. In this section, we proide a simple proo or Gauss-Bonnet Theorem on Piecewise Linear Meshes. 2.1 Piecewise Linear Meshes We say that a mesh M is piecewise linear i eery edge o M is a straight line segment, and eery ace o M is planar. Piecewise linear meshes are useul since the surace o each ace is well-deined [Williams 1972]. Most common piecewise linear meshes are triangular meshes. Howeer, piecewise linear meshes are a signiicant generalization o triangular meshes, which not only allow nontriangular aces, but also allow aces to hae dierent ace alence. In the discussion o the current paper, we will be ocused on piecewise linear meshes. We start by some intuitions. Let M be a piecewise linear mesh such that all ertices o M hae the same ertex alence m and all aces o M hae the same ace alence n (such a mesh is called a regular mesh). Suppose that M has ertices, e edges, and aces. By Euler Equation [Homann 1989; Mantyla 1988], e + = 2 2g (1) where g is the genus o the mesh M and rom the equalities n = m = 2e, we derie e = nm(2 2g) 2n(2 2g) 2m(2 2g), =, = 2n + 2m nm 2n + 2m nm 2n + 2m nm We would like to generalize the aboe relations to piecewise linear meshes that are not regular. For this, we introduce the ollowing deinitions. Deinition Let M be a piecewise linear mesh, with ertex set {µ 1,..., µ } and ace set {φ 1,...,φ }. Moreoer, or each i, 1 i, let m i be the alence o the ertex µ i, and or each j, 1 j, let n j be the alence o the ace φ j. Then the aerage ertex alence m and the aerage ace alence n are deined, respectiely, to be m = ( m i )/ and n = ( n j )/ Note that m and n are not necessarily integers. Based on these new concepts, the number o ertices, the number e o edges, and the number o aces in a piecewise linear mesh can be gien as unctions in terms o the aerage ertex alence, the aerage ace alence, and the genus o the mesh, as shown in the ollowing lemma.

2 Lemma 2.1 Let M be a piecewise linear mesh. Suppose that M has ertices, e edges, aces, and genus g. Let m and n be the aerage ertex alence and aerage ace alence, respectiely, o the mesh M, then e = nm(2 2g) 2n(2 2g) 2m(2 2g), =, = 2n + 2m nm 2n + 2m nm 2n + 2m nm PROOF. Suppose that the ertex set o M is {µ 1,..., µ } and that the ace set o M is {φ 1,...,φ }. Moreoer, suppose that or each i, 1 i, the ertex µ i has alence m i, and that or each j, 1 j, the ace φ j has alence n j. Since each edge in the mesh M is used exactly twice in ertex alences and is used exactly twice in ace boundaries, we hae m i = n j = 2e On the other hand, by our deinitions o the aerage ertex alence m and the aerage ace alence n, we hae m = m i and n = n j Thereore, we still hae m = n = 2e. Since the mesh M has genus g, Equation (1) holds. Combining Equation (1) and the equalities m = n = 2e, we get immediately e = nm(2 2g) 2n(2 2g) 2m(2 2g), =, = 2n + 2m nm 2n + 2m nm 2n + 2m nm This proes the lemma. 2.2 Angle delection at a ertex To understand the local behaior around a ertex o a piecewise linear meshes, angle delection are great tools. Angle delection at a ertex are ormally deined based on the corner angles around a ertex, as ollows. Figure 1: Ilhan Koman s Deelopable Sculptures. θ(µ i ) = 0: then ertex µ i is planar. θ(µ i ) < 0: then ertex µ i is a saddle point. The actual alues or θ(µ i ) 0 are not particularly useul, they simply tell how sharp is the conex or concae area. The upper limit or positie alues is 2π and it corresponds to the sharpest conex or concae regions. For negatie alues there is no lower limit. In addition, the actual alue o the number gies inormation about the type o saddle. Since there is no lower limit, saddle points can be ery wild, as it can be seen rom Ilhan Koman s sculpture in Figure 2. Deinition Let µ i be a ertex on a piecewise linear mesh M. The angle delection at ertex µ i is deined as θ(µ i ) = 2π m i θ j where θ j is the internal angle o the j-th corner o ertex µ i and m i is the alence o the ertex µ i. The importance o angle delections to describe local behaior can be best understood by iewing deelopable sculptures [Koman 2005] o Sculptor Ilhan Koman. He created a set o sculptures by orcing the total angle in a gien point larger than 2π [Koman 2005]. As shown in Figure 1, when the angle goes beyond 2π, it is possible to create a wide ariety o saddle shapes. We obsere that Koman s deelopable structures can gie ery useul inormation about the local behaior at a ertex or piecewise linear meshes. It is interesting to point out that the alue o θ(µ i ) is somewhat related to curature and based on the alues o corner angles around the ertex µ i. More precisely, θ(µ i ) > 0: then ertex µ i is either conex or concae. Figure 2: A close up iew o Ilhan Koman s highly complex Saddle Sculpture. This is not unexpected since the most popular discrete ersion o Gaussian curature introduced by Calladine or triangular meshes uses angular delection [Calladine 1983]. Calladine s discrete

3 Gaussian curature gien as Angular Delection Area Associated with Vertex 2.3 Sum o Angle Delections Here, we will gie a simple proo o discrete ersion o Gauss- Bonnet theorem as Sum o angle delections (SAD), gien as ollows. Deinition Let M be a piecewise linear mesh with a ertex set {µ 1,..., µ }. The sum o angle delections (SAD) o the mesh M is deined as θ(m) = θ(µ i ) Note that the SAD o a piecewise linear mesh is closely related to the geometric shape o the mesh M. On the other hand, as we will proe in the ollowing theorem, the SAD o a mesh is a topological inariant. Theorem 2.2 Let M be a piecewise linear mesh o genus g. Then the SAD θ(m) o the mesh M is equal to 2π(2 2g). Theorem 2.2 is independent o the number o ertices, the number o edges, the number o aces, and the aerage ertex and ace alences. It depends only on the genus o the piecewise linear mesh M. As a result, any homeomorphic operation (operations that do not change topology, in this case genus) that conerts a piecewise linear mesh to a new piecewise linear mesh does not change the SAD o the mesh. In other words, the eect o a homeomorphic operation on the SAD o a mesh will be zero, which means i we gain angle delections in some ertices, we will lose some angle delections in some other ertices. This is the important result or practical polygonal mesh modeling. 2.4 Visual Intuition To gie a isual intuition about sum o angle delections, we can relate it to critical points in Morse theory [Milnor 1963]. Consider the 3-D objects in Figure 3. The irst two objects hae genus g = 0, the next two objects hae genus g = 1, while the last object has genus g = 2. I we assign +1 to each minima/maxima (conex/concae) type critical point and 1 to each saddle type o critical point (as marked in the igure), the total adds up to (2 2g), as shown in Figure 3. PROOF. Let {µ 1,..., µ }, {ε 1,...,ε e }, and {φ 1,...,φ } be the ertex set, edge set, and ace set o the mesh M, respectiely. Moreoer, or each i, 1 i, let m i be the alence o the ertex µ i, and or each j, 1 j, let n j be the alence o the ace φ j. Let m and n be the aerage ertex alence and the aerage ace alence, respectiely, o the mesh M. Consider each ace φ j o alence n j. Since φ j is a planar polygon, the sum σ(φ j ) o inner angles o φ j is equal to (n j 2)π. Adding this up oer all aces in the mesh M, we obtain ) σ(φ j ) = ( (n j 2)π = n j 2 π = n π 2 π (2) where we hae used the deinition o the aerage ace alence n = ( n j)/. On the other hand, by the deinition o angle delections, the sum o corner angles around each ertex µ i o the mesh M is equal to 2π θ(µ i ). Adding this up oer all ertices in the mesh M, we get (2π θ(µ i )) = 2π θ(m) This alue should be equal to the alue in (2) since both o them are the sum o the total corner angles in the mesh M. Thereore, we get θ(m) = 2π n π + 2 π = 2π( n /2 + ) By the deinition o the aerage ace alence n = ( n j)/, and noticing that 2e = n j, we get immediately n /2 = e. This gies θ(m) = 2π( e + ) By Euler Equation (1), we get This proes the theorem. θ(m) = 2π(2 2g) Figure 3: The total adds up to (2 2g) i we assign minima/maxima points +1 and saddles 1. Figure 3 also gie a isual explanation o the sum o angle delections. In igure, we assume that all angle delections occur only at critical points. I that were really the case, the angle delection in each critical point would hae been +2π or minima/maxima and 2π or each saddle point. So, the total will be equal to 2π(2 2g), which is the sum o angle delections. It is also clear rom Figure 3 why branches do not change the total angle delections. Each branch induces the same numbers o saddle points and maxima/minima points; so total eect becomes zero. On the other hand, each handle (i.e., each hole) introduces two saddle type critical points, which decreases the total angle delections by 4π. 3 Homogenous Operations For branch creations, proessional modelers usually use extrusion [Landreneau et al. 2005] and wrinkle operations (see Section 4). These operations are homogenous, i.e. they do not change mesh topology. Subdiision schemes, which are commonly used to create smooth suraces, are also homogenous operations.

4 To understand why homogenous operations do not change the total angle delections, it is insightul to look at them rom a mesh topological point o iew. Let us assume that the number o aces, edges and ertices o a mesh increase in each application o homogenous operations. Based on this assumption, let us consider the practice in which when homogenous operations are applied to a mesh while the genus o the mesh stays the same, the numbers o aces, edges and ertices o the mesh increase without a limit. Remark 1. This is not an unrealistic assumption. In act, all subdiision schemes, extrusions and wrinkle operations increase the number o aces, edges and ertices. To analyze repeated applications o homogenous operations we can again use the Euler equation. Let us consider speciically meshes o genus g. Let n and m be the aerage ace alence and the aerage ertex alence o a mesh o genus 1. Using Euler equation (1) and equations in Lemma 2.1, we hae 2 2g e = 2 m + 2 n 1 By our assumption e can be arbitrarily large but the mesh genus stays the same, thus 2 m + 2 n 1 0 which gies m 2n n 2 Under the assumption that m 3 and n 3, and replacing the aboe approximation by an equality, we will get 4 Implications on Practical Modeling with Homogenous Operations Our results are particularly insightul or modeling better mesh structures or suraces that include branches. We do not create branches only or modeling trees. Branches are important or modeling and they are eerywhere. Een a simple genus-0 human ace model must include branches such as eye sockets, nose and mouth. Remark 3 With better mesh structures, we mean aces are conex and as regular as possible. It is not always possible to make aces regular. For instance, we cannot get an angle delection larger than π with regular conex aces. Howeer, it is still possible to make aces as regular as possible by using appropriate alence. For instance, i we orce the regular mesh structures such as (4,4), we cannot aoid irregular aces. By introducing saddle and minima/maxima type o critical points we can obtain better mesh structure. In practice, that is what proessional modelers do ery eiciently. Based on our results, it is easy to introduce saddles and minima/maxima. I we use the same type o aces, assuming that all the aces are almost regular, all we hae to do is to careully control the alences o ertices. For instance, i we work only with triangles, we hae to choose 3, 4 and 5 alent ertices in places we want to obtain minima or maxima. For saddle points, we hae to choose alences o ertices larger than 6. Based on how complicated we want to design a saddle point, we can increase the alence. Changing the alence can een improe the mesh structure o genus-1 suraces. The Figure 4 shows an example o mesh improement by decreasing and increasing alence in appropriate regions. 3 n 6 This result says that the aerage ace alence cannot exceed 6 regardless o what we do (i we do not allow alent-2 ertices and polygons with two sides). In other words, i our homogenous operations create only quadrilateral aces (i.e., n = 4), then the aerage ertex alence m goes to 4. I the homogenous operations create only triangles (i.e., n = 3), then the aerage ertex alence m goes to 6. For pentagons (i.e., n = 5), the aerage ertex alence approaches to 10/3 and or hexagons (i.e., n = 6) it goes to 3. Any mixed use o dierent ace types will create a rational number or the aerage ertex alence. This also explain why most subdiision schemes proide (3, 6) [Kobbelt 2000; Loop 1987], (4, 4) [Doo and Sabin 1978; Catmull and Clark 1978; Peters and Rei 1997; Sabin 2000] and (6,3) [Claes et al. 2002; Oswald and Schröder 2003; Akleman and Sriniasan Noember 2002] regular regions in which integer tuple (n, m) satisy the equation m n 1 = 0. Note that in these subdiisions the number o regular ertices and aces (i.e ertices with alence m and aces with alence n) increases, while the number o extraordinary ertices and aces stays the same in each iteration. As a result, the regular regions dominate the mesh. Remark 2. To guarantee that subdiided mesh is at least G 1 continuous each ace o subdiided mesh must approach a conex planar surace. So, we can say that most subdiision suraces consist o piecewise linear approximations o cured aces. Piecewise linear approximations usually are (3,6), (4,4) or (6,3) regular regions or most subdiision schemes [Akleman et al. 2004]. Top part is a Regular (4,4) Structure A Non-Regular Structure Another Non-Regular Structure Figure 4: This example shows that introducing non-regular structures can improe the mesh structure een or a simple genus-1 surace. See the thin quadrilaterals exists in regular (4,4) structure on the let model. By increasing the alence o some ertices in saddle region and decreasing the alence o some ertices in conex region, release the tension and quadrilaterals look more regular.

5 For proessional modeling quadrilateral modeling is a particularly important. In quadrilateral modeling, or minima and maxima, we hae only one choice, 3 alence ertices. For saddles we can use any alence higher than 4. Extrusions and wrinkle operations as shown in Figures 5 and 6 introduce minima/maxima and saddle points simultaneously and thereore, they are commonly by proessional modelers used to create branches. Initial mesh Wrinkle Operation and Vertex Valences Initial mesh Extrusion Operation and Vertex Valences Figure 5: Branch creation with extrusion operation. Beore extrusion we hae a quadrilateral with 4-alent ertices. Ater extrusion 4 ertices hae 5 alence and another 4 ertices hae 3 alence. Aerage ertex alence added is 4. Subdiided by Catmull-Clark One more iteration o Catmull-Clark 5 Conclusions and Future Work In this paper, we hae introduced an approach to improe mesh structures to use in practical modeling with homogenous operations. We show that ertex angle delection can gie a powerul insight about local geometric behaior to proessional modelers. Based on a discrete ersion o Gauss-Bonnet theorem, we hae shown that it is be possible to improe the organization o mesh structures o shapes according to their geometric structure. Based on the results in this paper, it is possible to automatically create a better mesh rom a gien mesh. We think that ertex angle delections can also be used to smooth the suraces. Despite the similarity, angle delections are dierent than Gaussian curature and its discrete ersions. For instance, unlike sum o angle delections, sum o discrete Gaussian Curature may not be equal to 2π(2 2g). Both Gaussian curature and angle delections are rotation and translation inariant. Howeer, Angle delection is also scale inariant which can make it useul or shape retrieal. An unexpected result coming rom ertex angle delection is paper olding. Since we know that the total angle will always stay constant, we can shape papers simply using staples 1. Using this insight, we were able to shape papers intuitiely. And we were able to create a ariety o paper sculptures. Figure 7 shows two examples o paper sculptures we hae created based on the insight coming rom our result. Reerences AKLEMAN, E., AND SRINIVASAN, V. Noember Honeycomb subdiision. In Proceedings o ISCIS 02, 17th Inter- 1 Since papers are deelopable suraces, we can think that papers are piecewise linear meshes that consist o thin stripes o planes. So our theorem applies to papers. Figure 6: Branch creation with wrinkle operation. Ater wrinkle operation, 4 ertices hae 4 alence, 2 ertices hae alence 3 and another 2 ertices hae 5 alence. So, added aerage ertex alence is 4. Subdiided ersion shows how an eye or wrinkle is automatically created with this operation. national Symposium on Computer and Inormation Sciences, ol. 17, AKLEMAN, E., SRINIVASAN, V., MELEK, Z., AND EDMUND- SON, P Semi-regular pentagonal subdiision. In Proceedings o the International Conerence on Shape Modeling and Applications, CALLADINE, C. R Theory o Shell Structures. Cambridge Uniersity Press, Cambridge. CATMULL, E., AND CLARK, J Recursiely generated b- spline suraces on arbitrary topological meshes. Computer Aided Design, 10, CLAES, J., BEETS, K., AND REETH, F. V A corner-cutting scheme or hexagonal subdiision suraces. In Proceedings o Shape Modeling International 2002, Ban, Canada, DOO, D., AND SABIN, M Behaior o recursie subdiision suraces near extraordinary points. Computer Aided Design, 10, DYN, N., LEVIN, D., AND SIMOENS, J Face-alue subdiision schemes on triangulations by repeated aeraging. In Cure and Surace Fitting: Saint-Malo 2002, FERGUSON, H., ROCKWOOD, A., AND COX, J Topological design o sculptured suraces. In Proceedings o SIGGRAPH 1992, ACM Press / ACM SIGGRAPH, Computer Graphics Proceedings, Annual Conerence Series, ACM, FOMENKO, A. T., AND KUNII, T. L Topological Modeling or Visualization. Springer-Verlag, New York.

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