Remeshing Schemes for Semi-Regular Tilings

Ergun Akleman∗ Vinod Srinivasan and Esan MandalVisualization Sciences Program

Department of ArchitectureTexas A&M University

Abstract

Most frequently used subdivision schemes such asCatmull-Clark create regular regions after several applica-tion. This paper shows that all semi-regular regions can becreated by subdivision schemes and each semi-regular re-gion type can be created with one application of aparticular subdivision scheme to a particular regular re-gion. Using this property of subdivision schemes it is easyto cover any given surface with semi-regular tiles by ap-plying one semi-regularity creating subdivision afterseveral applications of a regularity creating subdivi-sion.

1. Introduction and Background

Most widely used subdivision schemes such asCatmull-Clark, Doo-Sabin or Loop are regularity creat-ing; i.e. they create regular regions after several applica-tions of the scheme. A topologically regular region is a re-gion where all vertices have the same valence and allfaces have the same number of sides. The Schlafli sym-bol (n,m) is used to characterize regular region, wherenis the number of the sides in each face andm is the va-lence of vertices. For instance,(6,3) means regular re-gions that consist of hexagons with 3-valent vertices[20].

There exists only 3 types of regular regions. These are(3,6), (4,4) and(6,3) [20]. Only genus-1 surfaces can becovered using only regular regions. The other type of sur-faces cannot be covered using only regular regions. How-ever, using subdivision schemes they can be covered bypatches of regular regions. For instance, Figure 1 shows ex-amples of how a sphere can be covered by patches of reg-ular regions. These regular regions can be created by more

∗ Corresponding Author: Address: C418 Langford Center, College Sta-tion, Texas 77843-3137. email: ergun@viz.tamu.edu. phone: +(979)845-6599. fax: +(979) 845-4491.

than one type of subdivision schemes. (Note that these re-gions are only topologically regular, they are not regular ge-ometrically. Geometric entities such as the angles and edgelengths are not the same and do not have to be the same.)

• Regular region(3,6) can be created by both Kobbelt’s√3 subdivision [10] and Loop subdivision [11]. In

√3

subdivision, after one iteration all polygons becometriangles. Each k-gon produces a 2k valence vertex;each k-valence vertex continues to be k-valent. In otherwords, if

√3 is applied to a(3,6) regular region, all

triangles and all valent-6 vertices create valent-6 ver-tices. In Loop subdivision, each k-gon produces a k-gon and k triangles; each k-valence vertex continuesto be k-valence and each edge creates a 6-valence ver-tex. In other words, if Loop is applied to a(3,6) reg-ular region, all 6-valence vertices and all edges createvalent-6 vertices.

• Regular region(6,3) can be created by the dual ofLoop subdivision [13, 16, 14], and the dual of

√3 sub-

division [9, 3, 14]. Dual of a subdivision scheme con-sists of successive applications of three operators to amesh: (1) Dual; (2) Subdivision; (3) Dual. Since(6,3)is the dual mesh of(3,6); it is clear that duals of(3,6)creating subdivision schemes create(6,3) regular re-gions.

• Since (4,4) is self-dual, dual of any subdivi-sion scheme that creates regular region(4,4) canalso create regular region(4,4). For instance, Ver-tex Insertion, such as the remeshing scheme ofCatmull-Clark [6] subdivision, is a(4,4) regular-ity creating scheme. It makes all faces quadrilateraland creates(4,4) regular regions. Its dual, the cor-ner cutting scheme, also creates(4,4) regular regions.An example of corner cutting is the remeshing al-gorithm of Doo-Sabin subdivision [8]. After oneapplication of corner cutting the valence of all the ver-tices becomes 4. Corner cutting is a very usefulremeshing operator that can create interesting struc-tures such as the one shown in Figure??. Similarly,

Simplest and its dual also create(4,4) regular re-gions.

SC(4,4) GD (3,6)

BF (6,3)

Figure 1. Examples of Regular Regions onthe surface of Sphere. (SC) Spherical cube isobtained by applying Catmull-Clark to a cubeand then mapping vertices of subdivided sur-face to a unit sphere. (GD) is a Geodesicdome or Pseudo-Icosahedron. It is created byapplying Loop subdivision to an icosahedronand then mapping vertices of subdivided sur-face to a unit sphere. (BF) is a Buckminster-fullerene or generalized soccer-ball. It is cre-ated by applying Dual of

√3 to a Dodecahe-

dron and then mapping vertices of the subdi-vided surface to a unit sphere.

If we relax the conditions of regularity, we obtain semi-regularity regions. In semi-regular regions, either all facesor all vertices have exactly the same topological structure,but not both. For instance, a semi-regular regions can con-sist of only pentagons but valences of the vertices of eachpentagon follows the same cyclic pattern such as 3,3,4,3,4.The dual of such a semi-regular region consists of only 5-valent vertices. Although, polygons around the vertex donot have same number of sides, the number of sides of poly-gons around the vertex follows the same cyclic pattern: tri-angle, triangle, quadrilateral, triangle and quadrilateral; i.e.3,3,4,3,4. This sequence uniquely defines the structure of

a semi-regular region. In fact, these sequences give Schlaflisymbols for semi-regular regions (instead of comma, we usedots to separate numbers). For instance, the above pentagonexample will be given as (3.3.4.3.4).

In a plane, there are only 7 distinct semi-regular re-gions and their duals [20]. These are given by the follow-ing Schlafli symbols.

1. (4.8.8) : Primary region consists of triangles. Dual re-gion consists of 3-valent vertices.

2. (3.12.12) : Primary region consists of triangles. Dualregion consists of 3-valent vertices.

3. (12.6.4) : Primary region consists of triangles. Dual re-gion consists of 3-valent vertices.

4. (6.4.3.4) : Primary region consists of quadrilateral.Dual region consists of 4-valent vertices.

5. (6.3.6.3) : Primary region consists of quadrilateral.Dual region consists of 4-valent vertices.

6. (3.3.4.3.4) : Primary region consists of pentagons.Dual region consists of 5-valent vertices.

7. (3.3.3.3.6) : Primary region consists of pentagons.Dual region consists of 5-valent vertices.

To create all of these semi-regular tilings, we need to in-troduce only two new (previously-unpublished) remeshingschemes and one of them is simply the dual of the Simplestscheme.

2. Semi-Regularity Creating Remeshing Op-erations

We have developed a system that provides (global) op-erators that are applied to the whole mesh, or what wecall remeshing operators. Under the remeshing category, weonly consider global operators that do not change the topol-ogy of the mesh. These operators do not increase or de-crease the genus of the surface nor do they connect or dis-connect surface components.

The most important remeshing scheme in our system isthe dual operator [20], which is essential for creating dualmeshes. Dual is also very useful operator for smoothing. Ifa dual operation is applied even number of times, in mostcases it will create a smooth mesh. Zorin and Schröder re-cently showed that the dual operation allows creation ofhigher-order quadrilateral subdivision surfaces that providehigher-degree continuity [21]. The same idea is also used toderive the rules for the schemes that create regular(6,3) re-gions [9, 14, 3].

All remeshing operators we provide (except dual) in-crease the number of vertices. Our remeshing operators arenot probabilistic, i.e., they always give the same result for

the given initial mesh. Although, we consider the subdivi-sion schemes as remeshing schemes, a remeshing schemedoes not have to be a subdivision – remeshing schemes maynot necessarily be applied successively and may not smooththe mesh.

Providing a large variety of remeshing operators is par-ticularly useful to making mesh structures ornamental ordecorative. By applying the provided remeshing operatorsto regular regions all major semi-regular regions can be ob-tained. To illustrate the creation of particular semi-regularregions we apply remeshing operators to three initial spher-ical shaped meshes shown in Figure 1. Among these three,spherical cube (SC) includes(4,4) regular regions, andgeodesic dome (GD) [19] includes(3,6) regular regions,and finally Buckminsterfullerene (BF) [19] includes(6,3)regular regions.

The remeshing operators that can create semi-regular re-gions are conversion operators. These operators can changeany given mesh into a mesh that consist of the same type offaces (e.g. all triangles or all pentagons) or vertices (e.g., allvalent-3). Operators are also classified into two additionalcategories: (1) Primary operators that create the same typeof faces, and (2) dual operators that create the same type ofvertices. Although the dual of an operator can simply be ob-tained as dual + operator + dual, in most cases we have im-plemented dual operators separately to achieve faster com-putational speed.

2.1. Triangulization and Valent-3 Conversion

• Vertex Truncation and its dual.Polyhedral vertex trun-cation is a widely used operator in polyhedral mod-eling to obtain Archimedean solids [20]. We have re-cently extended [2] polyhedral vertex truncation byadopting Chaikin’s algorithm [17] to vertex truncation.Vertex truncation can be used to create semi-regular re-gions 4.8.8 and 3.12.12 as shown in Figures 2 and 3. Vertex truncation is a valent-3 conversion scheme.Dual of Vertex truncation, a triangulization scheme, isalso implemented and duals of 4.8.8 and 3.12.12 cre-ated by these operators are also shown in Figures 2and 3.

• 1264 and its dual.We have developed these tworemeshing operators to create semi-regular re-gion 12.6.4 when applied to the regular region (3,6) asshown in Figure 4.

2.2. Quadrilateralization and Valent-4 Conversion

• Vertex Insertion and its dual, Corner Cutting.VertexInsertion is remeshing scheme of Catmull-Clark [6]subdivision. The system provides both Catmull-Clark

(A) (B)

Figure 2. (A) An example of primary 4.8.8obtained by applying dual truncation opera-tor to the subdivided cube. (B) An exampleof dual 4.8.8 obtained by applying truncationoperator to the subdivided cube.

(A) (B)

Figure 3. (A) An example of primary 3.12.12obtained by applying dual truncation opera-tor to the geodesic dome. (B) An example ofdual 3.12.12 obtained by applying truncationoperator to the Buckminsterfullerene.

and linear vertex insertion. This subdivision creates(4,4) regular regions. But, if it is applied once to amesh with a(6,3) region, it creates semi-regular re-gions 6.4.3.4 as shown in Figure 5. The corner cut-ting scheme is the dual of vertex insertion. An exam-ple of corner cutting is remeshing algorithm of Doo-Sabin subdivision [8]. After one application of cornercutting the valence of all the vertices becomes 4. Cor-ner cutting is a very useful remeshing operator that cancreate interesting structures such as the one shown inFigure 5.

• Simplest and its dual.The remeshing scheme of sim-plest subdivision [15] converts any mesh to a meshwith only 4-valent vertices. Simplest creates 6.3.6.3

(A) (B)

Figure 4. (A) An example of primary 12.4.6obtained by applying 1264 operator to thegeodesic dome. (B) An example of dual 12.4.6obtained by applying 1264 operator to theBuckminsterfullerene.

(A) (B)

Figure 5. (A) An example of primary 6.4.3.4obtained by applying vertex insertion oper-ator to the Buckminsterfullerene. (B) An ex-ample of dual 6.4.3.4 obtained by applyingcorner cutting operator to the Buckminster-fullerene.

semi-regular regions when it is applied to (6,3) reg-ular regions as shown in Figure 6. We have also im-plemented the dual of simplest. In the program, Wecall it stellate with edge removal. Dual of simplest isnot a known subdivision scheme and it is not useful asa smoothing scheme. However, it allows the creationof appealing 6.3.6.3 semi-regular regions as shown inFigure 6.

2.3. Pentagonalization and Valent-5 Conversion

We have recently shown that it is possible to convert any2-manifold to a pentagonal mesh [5]. We also showed thatthere is no conversion method that can convert any given

(A) (B)

Figure 6. (A) An example of primary 6.3.6.3obtained by applying dual of simplest opera-tor to the Buckminsterfullerene. (B) An exam-ple of dual 6.3.6.3 obtained by applying sim-plest operator to the Buckminsterfullerene.

mesh to a mesh consisting of only polygons with more thanfive sides.

• Pentagonalization and its dual.This is the only knownpentagonalization algorithm, which we have intro-duced recently This scheme can create two typesof semi-regular regions: 3.4.3.3.4 and 3.3.3.3.6. Wehave also implemented its dual which converts ev-ery vertex of a mesh to valent-5 vertex. (see Figures 7and 8)

(A) (B)

Figure 7. (A) An example of primary 3.3.3.3.6obtained by applying pentagonalization oper-ator to the subdivided cube. (B) An exampleof dual of 3.3.3.3.6 obtained by applying dualof pentagonalization operator to the subdi-vided cube.

(A) (B)

Figure 8. (A) An example of primary3.3.3.3.6obtained by applying pentago-nalization operator to the Buckminster-fullerene. (B) An example of dual of 3.3.3.3.6obtained by applying dual of pentagonaliza-tion operator to the geodesic dome.

3. Conclusion and Future Work

This is a short paper that shows power of mixed usage ofsubdivision and remeshing schemes. Creating semi-regularregions is important especially in shape modeling applica-tions where tiling is important. Note that we use sphere justas an example. The approach presented here can work forany genus and any smooth surface. An example is shownin Figure 9. Using more than one subdivision can also helpto create interesting tessellations that are not semi-regularsuch as the one shown in Figure 11.

One important application of creating semi-regular tilesis for opening holes as seen in Figure 12. We make use ofthe fact that each semi-regular tiling creates distinct groupsof faces on which we can apply any polygonal operator. Forinstance, meshes shown in Figure 12 are created by wiremodeling [12]. The mesh shown in Figure?? is created byrind modeling [4].

An interesting question is how to create Coxeter’s(4,6),(6,6) and(6,4) regular regions [7]. We have recently shownthat [1], these regular mesh structures exists in any genuslarger than 2. Therefore, it can be possible to develop genus-changing subdivision schemes to create such structures.For an example of genus-changing subdivision schemes see[18].

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Figure 9. A series of Genus-3 surfaces cov-ered with semi-regular regions.

Figure 10. A sample of high genus shapesthat are created from surfaces with semi-regular regions using wire modeling.

Figure 11. A Genus-3 surface covered with in-teresting but not a semi-regular region. Thiswas created applying simplest twice to thebottom mesh in Figure 9.

Figure 12. A high genus shape that are cre-ated from surfaces with semi-regular regionsusing rind modeling.