Block Meshes: Topologically Robust Shape Modeling with Graphs Embedded on 3-Manifolds

Transcription

1 Block Meshes: Topologically Robust Shape Modeling with Graphs Embedded on 3-Manifolds Ergun Akleman a, Jianer Chen b, Jonathan L. Gross c a Departments of Visualization & Computer Science and Engineering, Texas A&M University b Department of Computer Science and Engineering, Texas A&M University c Department of Computer Science, Columbia University Abstract We present a unifying framework to represent all topologically distinct shapes in 3D, from solids to surfaces and curves. This framework can be used to build a universal and modular system for the visualization, design, and construction of shapes, amenable to a broad range of science, engineering, architecture, and design applications. Our unifying framework uses 3-space immersions of higher-dimensional-manifolds, which facilitate our goal of topological robustness. We demonstrate that a specific type of orientable 2-manifold mesh, which we call a CMM-pattern coverable mesh, can be used to represent structures in higher-dimensional manifolds, which we call block meshes. Moreover, the framework includes a set of operations that can preserve CMM-pattern coverability. In this sense, CMM-patterncoverable meshes provide an algebraization of shape processing that (1) supports a generalized mesh representation for blocks that may not necessarily be solids, and (2) requires a minimal set of operations that transform CMMpattern-coverable meshes to CMM-pattern-coverable meshes. Keywords: Shape modeling, 3-manifolds, immersions, 3D-thickenings, shape algebra 1. Introduction and Motivation Higher-dimensional manifolds provide a setting in which topologically robust methods can manipulate lower-dimensional shapes without changing their nature. For instance, small geometric perturbations of the plane could unknot a projection of a knotted curve represented by a projection of a closed polygon [77, 84]. On the other hand, if that same curve is represented as knotted in 3D, the geometric perturbations of 3-space that we allow would not change the knot type, which means the topology of the configuration [4]. The name CMM pattern refers to one of the 17 classical wallpaper designs [91]. It specifies a crystallographic pattern, rather than the initials of an originator. By topologically robust we mean that changes in geometry such as changing the positions of vertices do not change the topological structure of the shape. Our framework is based on non-cellular decompositions that are formed from atomic objects called vertices, edges, faces, chambers, and blocks. This framework requires specifying whether given vertices, edges and faces are used for locally representing 2-space or 3-space. Accordingly, we add a prefix; for instance, we write 3-vertex or 2-vertex, rather than just vertex. Otherwise, the meanings of vertex, edge, and face are as usual. Blocks and chambers are key concepts. A block is topologically a connected set in which every point has a neighborhood homeomorphic to 3-space (i.e., Euclidean 3-space). Significantly, a block need not be globally homeomorphic to 3-space. Indeed, blocks can be generalized versions of solids that are not realizable in 3-space without self-intersections. We call such a block non-realizable. We exclude only those blocks whose boundaries cannot be represented by the orientable 2-manifold meshes that we call chambers. We demonstrate here that these meshes of blocks, i.e. block meshes, can be used for robust modeling of 3Dshapes. We present a theoretical framework with (1) Preprint submitted to Computers & Graphics September 30, 2014

2 a generalized representation of 2-manifold meshes and (2) a minimal set of topological operations to surgically transform 3-manifolds into 3-manifolds. Based on this theoretical framework, it is possible to implement a minimal kernel on which to build more complicated systems modularly. The minimal set of operations can greatly simplify the software development and encourage people to add new tools while taking advantage of the powerful underlying structure. Allowing such general blocks during the modeling process lets us algebraize the model, which can lead to simpler and more powerful shape modeling algorithms. Having such a shape algebra is essential to developing systems that can grow quickly from a simple kernel into an advanced modeling system. Because of the robustness and the simplicity coming from such an algebra, software designers with minimal instruction can make the system grow. They can easily add high-level operations that are created as composites of a minimal set of primitive operators. Our approach here is reminiscent of the historical example of finding roots of cubic equations [99]. By allowing square roots of negative numbers, Tartaglia was able to solve certain kinds of otherwise unsolvable cubic equations [51]. That Cardano called complex numbers fictitious was not a problem, since a cubic equation with real coefficients always has at least one real root. Analogously, our present goal is to create solid shapes in 3-space, yet we can let our algorithms manipulate nonrealizable shapes along the way. For instance, one-sided 3-faces can be naturally created from Möbius strip cuts in sculptural practice, as shown in Figure 1. In the robust representation of knots and links, it is essential to allow such non-realizable structures to have robust representations [4]. Our orientable chambers are capable of representing the one-sided non-orientable boundaries shown in Figure 1 (see Section 2). In fact, we are currently using such a shape algebra for 2-manifold modeling. 2D Graph rotation systems (2D-GRS) that represent all possible 2-manifold meshes have been implicitly used in the guise of various mesh data structures, such as half-edges [74], quad-edges [50], winged-edges [19], doubly-connected edge lists (DCEL) [86] and doubly-linked face lists (DLFL) [1]. 2D-GRS with topology preserving operators such as Splice and Twist [50], or Euler operators [74] or InsertEdge, CreateVertex and Twist [3] provide shape algebras for 2-manifolds. Unfortunately, the power of such shape algebras is lim- 2 (a) Two pieces (b) One piece Figure 1: Two toroidal Ushio sculptures, each created by a physical cut operation. In (a) the cut is 2-sided; in (b) the cut is a Möbius-strip type cut, i.e., a 1-sided surface [43]. Note that this 1-sided cut does not separate the block into two parts. ited by their underlying topological structure, namely the restriction to 2-manifolds. Programmers commonly include some exceptions on 2-manifolds and provide operations that can create non-manifolds or solids in software applications. Such exceptions and operations solve immediate practical concerns, yet they make it harder to extend software without professional help and/or laborious effort. Therefore, we propose a new framework to develop a more powerful shape algebra. 2. Previous Work As is well-known, manifolds can be studied and represented through decompositions provided by CWcomplexes (CW stands for closure-finite, weak topology), which are synthesized from basic building blocks called cells. An n-cell is a topological space that is homeomorphic to an n-dimensional closed ball. We say that a CW-complex is an n-complex if n is the largest dimension of any of its cells. If every point of an n-complex has a neighborhood homeomorphic to an n-dimensional open ball, we call it an n-manifold. If every point of an n-dimensional complex has a neighborhood homeomorphic either to an n-dimensional open ball or to an n-dimensional half ball, we call it an n- manifold-with-boundary. Many low-dimensional cases are widely familiar. For instance, a vertex is a 0-cell, and an edge is a 1-cell. A curve segment is a 1-manifold with boundary. A closed cycle of curve segments is a 1-manifold. A 1-manifold is sometimes represented (discretely approximated) by a cyclically ordered list of vertices. A graph is a 1- complex, but not a 1-manifold if any vertex has degree higher than 2. With higher-dimensional manifolds, representations start to get increasingly complicated.

3 One approach to simplifying the representations of higher-dimensional complexes is to use the simplest kind of n-cells, namely the n-simplexes. A CWcomplexes whose cells are simplexes (and that also satisfy a few other rules) are called simplicial complexes. Simplexes have several attractive properties. For instance, an n-simplex always has n + 1 vertices; e.g., a 2-simplex is a triangle, and a 3-simplex is a tetrahedron. Moreover, an n-simplex in a Euclidean space can always be represented by an n-dimensional barycentric linear equation described by its vertex positions. Moreover, the regularity of simplicial complexes also provides opportunities for creating data structures with efficient space complexity. These properties make them very attractive for geometric processing [23]. Accordingly, triangular meshes are one of the most popular representations for 2-manifolds [26]. Moreover, data structures based on tetrahedral decomposition (or tetrahedral meshes) are specifically used for the simplification and acceleration of common queries and operators that are needed to support applications (see [98, 82, 34, 29, 61, 52] as examples). There also exist data structures to represent n-dimensional simplicial complexes, such as Brisson s cell-tuples [24], Lienhardt s n-generalized maps [69, 70], Rossignac and O Connor s selective geometric complexes [89], Paoluzzi s model [83], and De Floriani and Hui s nonmanifold indexed with adjacencies [35]. A detailed discussion of data structures for simplicial complexes can be found in [36] and [64]. Despite their significant advantages for geometry processing [23], we think that simplicial complexes are unnatural and cumbersome for shape modeling applications. For instance, a cube, which can uniquely and naturally be represented by 6 quadrilaterals, is represented by 12 triangles in 2 6 different ways since each quadrilateral can be triangulated in two different ways. Moreover, triangular meshes allow only a very restrictive set of operations. For instance, it is not possible to use the insert-edge operation in a wholly simplicial model, since inserting an edge can combine two triangles into one octagon, while increasing the genus of the 2-manifold surface. Moreover, the dual operation cannot be used in a wholly simplicial model, since the dual of a simplicial complex may be non-simplicial, e.g., the dual of a random triangle mesh consists mostly of hexagons. Higher-dimensional simplicial cells are even more problematic. For instance, tetrahedra, unlike triangles, are not space-filling shapes and they cannot provide a regular subdivision of 3-space [105, 97]. 3 An alternative to simplicial cells for geometric processing is hypercubes. An n-dimensional hypercube always has 2 n vertices, and, like a simplex, it can always be represented in Euclidean space by its vertex positions. Moreover, hypercubes are always space-filling structures that can be subdivided regularly into finer hypercubes. Even the simplest hypercubes, quadrilaterals, are becoming more popular in geometric processing. Recently, a number of significant papers have appeared for converting surfaces to quad-meshes (see [38, 65, 22, 107] as examples). However, For general applications, even hypercubes can be restrictive. In fact, centroidal Voronoi tessellations [68, 71] require hexahedral and more general polyhedral meshes. For restriction-free modeling of shapes, it is always better to allow arbitrary decompositions, even in modeling with surfaces [59, 58]. In fact, most popular polygonal mesh data structures, such as half-edges [74], quad-edges [50], winged-edges [19, 20, 102], doublyconnected edge lists (DCEL) [86], and doubly-linked face lists (DLFL) [1], support general cells and general modeling operations for orientable 2-manifolds. In these general representations, the edges and faces can be curved. Non-orientable surfaces can be supported by allowing edge-twists [46, 50, 2]. It is also interesting to note that the original half-edge structure proposed by Mantyla is even more general [74] and it supports non-cellular decomposition of surfaces disconnected set of polygons. (a) Figure 2: A Non-cellular face is a by including non-cellular A cellular face a 10-sided polygon. (b) A non-cellular face a faces, which are polygons with holes. These square with a square hole. non-cellular faces consist of a disconnected set of polygons (see Figure 2 for an example). Having non-cellular faces can even be advantageous for some practical surface modeling applications. For instance, Havemann particularly chose to use non-cellular faces for modeling Gothic-windows [55, 56]. Supporting non-cellular decompositions is even more important in solid modeling, since connected solids can be non-cellular, and non-cellular structures are unavoidable. Thus, for really restriction-free modeling of solids, we propose that beyond using 3-manifolds as the underlying mathematical structure, we provide the capacity for non-cellular decompositions. This is a significant departure from approaches that offer only cellular decompositions. It greatly extends the power provided by both 2-manifold representations and

4 on cellular-decomposition-based 3-manifold representations. Moreover, including non-cellular decompositions provides new power to the designers. For instance, a cut always separates a 3-ball into two 3-balls. On the other hand, cutting a toroidal block may or may not separate the toroidal block into two, as shown in Figure 1. This toroidal cut operation has been introduced to the art world by sculptor Keizo Ushio; it has been analyzed and formally explained by Séquin and Friedman [43]. Having non-cellular decompositions is not really a major hurdle for geometry processing applications. They can always be converted into CW- or simplicial complexes. There has been a great deal of research for representing solid shapes using structures that are beyond surfaces. The most frequent usage of such mesh representations is for polyhedral meshes. Among polyhedral meshes, the most common ones are tetrahedral representations [37, 25, 72, 104]. Dobkin and Laszlo s facet-edge structure [37] is an extension of the quadedge data structure [50], and Lopes and Tavares s handle-face structure [72] is an extension of the halfedge data structure [74]. On the other hand, there has not been as much investigation into representing general decompositions as we have proposed. The notable extensions are Weiler s radial-edge [103], which is used for representation of non-manifold shapes with non-regularized 2-complexes, and Masuda s extension of Euler operations for non-manifold shapes [75, 76]. However, radial edge representation and Masuda s operations are limited to 2-complexes and non-manifolds, with which it is hard to develop a shape algebra. Instead, our approach extends existing work to a framework with which any strong 2-manifold mesh data structure can be used to represent and manipulate general decompositions with a shape algebra. Therefore, our framework can be implemented using existing data structures. Since we have a shape algebra with associated operators, the users can never create a topologically invalid structure. An additional advantage of our shape algebra is that it allows nested sets of ever more complicated algebras. Therefore, the same underlying model and operations can be used to represent and manipulate curves, surfaces and solids. For instance, if we start with an orientable 2-manifold mesh and apply only certain orientationpreserving operations, then we can obtain nothing but representations of orientable 2-manifold meshes. Thus, this particular set provides an algebra over orientable 2- manifolds. If we additionally include a twist operator, then we can also obtain non-orientable surfaces. Thus, adding a twist operator yields a more powerful algebra 4 that includes non-orientable objects. This is analogous to the hierarchy structure among real algebras, complex algebras, and quaternion algebras. In other words, the new model provides a strong representational power while using existing infrastructure in 2-manifold mesh modeling without causing a significant increase in computational expense. 3. 3D-GRS with 3D-Thickenings Fields Medalist William P. Thurston [100] has written: People have very powerful facilities for taking in information visually or kinesthetically, and thinking with their spatial sense. On the other hand, they do not have a very built-in facility for inverse vision, that is, turning an internal spatial understanding into a twodimensional image. Consequently, mathematicians usually have fewer and poorer figures in their papers and books than in their heads. One solution mathematicians have used for this inverse vision problem of Thurston is combinatorialization. If one can t draw various manifolds, one can at least give a finite set of elementary pieces and rules for assembling those pieces. This began in the 1920s and led to the blooming of algebraic topology in the middle of the 20th century. The trouble is that the resulting combinatorial structures retain the usual problems of all combinatorial structures: they may be easy to list as data structures, but they behave nonetheless in ways that are far more complicated than one would expect. An interesting example of combinatorialization is that of graph embeddings in surfaces. Graph embeddings on higher-genus surfaces may be difficult to draw, or even to picture, but they are easy to describe by graph rotation systems [46]. Some questions, like embeddability in the plane, have quite complete answers (such as Kuratowski s theorem). On the other hand, the analogous combinatorial structure for describing embeddings of graphs in 3-manifolds yet behaves in ways that are far more complicated than one might ever expect. Given these facts, we remark that we are not intended here in analyzing or characterizing graphs embeddings in 3- manifolds. Rather, we are seeking a powerful and easyto-use model that is based on graphs embedded in certain restricted class of 3-manifolds for designing shapes using meshes of blocks. To represent Block Meshes, we employ graphs embedded into 3-manifolds. Our method is based on the concept of 3D-thickening. We introduce 3D-thickening as

5 Figure 3: A visual description of 2D- and 3D-Thickenings. In the last frame (I) the graph with one vertex and three self-loop is 3D-thickened into an orientable genus-3 mesh. Note that the structure of this mesh is not arbitrary. Every face of the mesh is a quadrilateral. Moreover each quadrilateral has two pairs of opposite edges, green and purple. Using our methods, we identified that this particular structure has 4 faces and 2 chambers. 5

6 an extension of 2D-thickening in topological graph theory that can be regarded as a physical implementation of the graph rotation systems [47]. A graph G can have a number of natural 3D-thickenings. Figure 3 illustrates a principal difference between 2D- and 3D-thickening. In a 2D-thickening, each vertex thickens to a disk or a polygon, and each edge thickens to a quadrilateral band embeddable in a 2-manifold [46]. Similarly, the 3Dthickening operation turns lower-dimensional structures into blocks that can also be represented by 2-manifold meshes. In a 3D-thickening, each 3-vertex thickens to a block whose boundary can be represented by an orientable 2-manifold, and each 3-edge thickens to a block that can be represented by a prism. 3D-thickening is just a conceptual idea. In particular, a 3D-vertex is still regarded as a point, and a 3D-edge is still regarded as a curve or a curve segment. With conceptual 3D-thickening of a graph, it is possible to describe the combinatorial structures around the graph. Further procedures operating on 3D-thickened graphs will enable us to construct special 3-manifolds in which the graphs are embedded. The graphs embedded in such 3-manifolds will be called 3-graphs. To analyze combinatorial structures that can be provided by 3-graphs, we first consider graphs embedded in 2-manifolds, which we call 2-graphs Analogy with 2-Graphs and their 2-Thickenings As discussed in Section 2, combinatorial structures around a 2-graph are used to represent graph embeddings in 2-manifold surfaces. The combinatorial structures can be uniquely described by 2-thickening the graph. This structure is not arbitrary. Every m-valent vertex is 2D-thickened into an m-sided polygon and every edge is 2D-thickened into a quadrilateral, as shown in Figure 3(2D). As shown in the figure, edges that turn into quadrilaterals work like handles that connect the sides of the polygons obtained by 2-thickened vertices. We call the structure of handle connections a configuration of handles. In the 2-manifold case, handle configurations can be provided simply by a cyclicly ordered list of edge-ends, which determines how the sides of polygons (thickened 2-vertices) are connected to each other. the name Graph Rotation System (GRS)) actually implies this cyclic ordering that is added to a graph description to provide handle configurations [46]. It is possible to identify all possible handle connections. For instance, the graph shown in Figure 4(a) (the graph with one vertex and two self-loops) has only two possi- 6 (a) A graph (b) Embedding 1 (c) Embedding 2 Figure 4: 2D thickenings of all 2-manifold embeddings of a simple graph. (a) is embedded on a genus-0 surface and (b) is embedded a genus-1 surface. ble handle configurations. Since the valence of the vertex is 4, the vertex 2D-thickens into a quadrilateral. For creating handle connections, we have only two possible configurations: either we can connect opposing edges or we can connect neigboring edges as shown in Figure 4(b) and (c). In these two examples white regions are insides of the faces. By assuming that their boundary completely describes the faces, we can observe that these two types of handle configurations correspond to two types of surfaces, namely genus-0 and genus-1. Accordingly, the particular graph in Figure 4 is capable of representing only genus-0 and genus-1 surfaces. When the graphs are more complicated, they provide a much larger number of combinatorial possibilities. From the perspective of surface modeling, difficulty of enumeration or classification of all possible handle configurations of a graph has never been an issue, since we do not care about the graph embedding problem. Instead, we are interested in the inverse problem: From a graph embedded in a surface, can we reconstruct all vertices, edges and faces? This inverse problem is really easy, and in fact, all 2-manifold mesh data structures were (implicitly or explicitly) developed to represent the possible 2D-thickened structures of 2-graphs and to efficiently reconstruct vertices, edges, and faces from that representation. For instance, the quad-edge data structure uses the fact that each edge of a 2-graph 2-thickens into a quadrilateral (See Figure 3(2D)). The neighborhood relationships of those quadrilaterals provide complete combinatorial structures around any given 2- graph, which in turn define the underlying manifold surface and are used to construct vertices and faces. Our goal in this paper is exactly a generalization of this line of thinking. Thus, we investigate what kind of structures emerge from graphs embedded in 3- manifolds. We identify the properties of emergent structures and call them CMM-pattern coverable meshes. Then, we show how to reconstruct vertices, edges, faces and chambers from any given CMM-pattern coverable mesh. As it will be shown later, this reconstruction turns

7 out to be simple Structure Emerges From 3-Manifold Embeddings of Graphs In this section, we approach the problem of graph embeddings into 3-manifolds like a topological graph theorist, and we investigate the kinds of structures that can emerge from such embeddings. To analyze these structures we use 3D-thickenings of 3-graphs, since they provide nice visualization. In later sections, we show how 3D-thickened 3-graphs induce embeddings of the graphs into 3-manifolds. Simple examples of emergent structures are shown in Figures 3(I) and 5. The structures of 3D-thickened 3- graphs consist of a set of 2-manifold meshes that represent their 3-vertices and a set of prismatic handles that connect the faces of the 3-vertices. These prismatic handles represent 3-edges. There are only two constraints over the structures. 1. An m-valent vertex of the original graph can be 3D-thickened only into a 3-vertex with m faces, which correspond to the edge-ends of the original graph. For instance, the 6-valent vertex in Figure 3(I) is thickened into a cube. 2. A handle is a prism whose two end-faces have the same number of sides. For instance, as shown in Figure 3(I), all faces of a cube are squares, and we can connect any two faces with a prismatic handle. The number of possible combinatorial structures that emerge from a given graph can be very large. It is possible to develop an intuitive understanding how high this number can be. 1. For 3D-thickened vertices, there is no constraint other than the number of faces. Thus, these meshes can have any number of vertices and edges, and its genus can be any number. For instance, the cube shown in Figure 3(I) is one possible 3D-thickening of a 6-valent vertex and the octahedron shown in Figure 5 is one possible 3D-thickening of an 8- valent vertex. 2. As with 2-graph embeddings, the most important issue in enumeration here is the number of possible configurations of handles. Figure 5 shows 5 possible handle configurations when the 3-vertex is an octahedron. By considering the mirror image of the 3-graph shown in Figure 3(I), we see that the actual number of configurations is 4. 7 (a) A graph (b) Example 1 (c) Example 2 (d) Example 3 (e) Example 4 (f) Example 5 Figure 5: A few examples of 3D-thickening of a graph with one vertex and 4 self-loops by assuming that vertex is thickened to an octahedron. 3. A handle connecting two faces with n sides can be realized in n different ways. For instance, the two handles shown in Figure 3(H) connect the same faces, but they are different. Since their ends are pentagons, there are five inequivalent handles. This significantly increases the number of possible combinatorial structures. Even though the possibilities are relatively low for the examples in Figures 5(b) and 3(H), because of their perfect symmetry, the numbers are still significant, 12 and 24, respectively. Figure A.23(a) shows one additional example from the 24 possible cases that correspond to the configuration in Figure 3(H). Figures 5 and 3(I) illustrate the representational power of 3-thickening a 3-graph, which induces embeddings of the graph into 3-manifolds. Even very small graphs can

8 provide a very rich combinatorial information through their 3-thickenings. Moreover, we observe that a very simple structure emerges from all 3D-thickenings of 3- graphs, which we call CMM-pattern coverable (CMM- PC) meshes. In this paper, we employ the underlying structure called CMM-PC meshes for the decomposition of 3-manifolds. We show that we can easily reconstruct vertices, edges, faces, and chambers from any given CMM-PC mesh. Moreover, we develop a shape algebra for CMM-PC meshes. (a) A basic CMM pattern (b) Odd-valence (c) Non-cycle 4. CMM-PC meshes Our basic conjecture in this paper is that a 3-graph can always be 3D-thickened into a specific type of 2- manifold quad-mesh, in which the opposing edges of each quadrilateral can be consistently labeled by 0 and 1 (or colored green and purple). We call such quadmeshes CMM-Pattern Coverable 1 since they can be seamlessly covered by CMM patterns [91]. Definition A CMM-Pattern Coverable (CMM-PC) mesh is a quad-mesh that can be seamlessly covered by a CMM pattern, which is one of the 17 wallpaper patterns [91]. A quad-pattern coverable mesh is a mesh that can be covered seamlessly by any one of the 17 wallpaper patterns [60]. CMM patterns are mirror-symmetric, as in Figure 6(a). Conjecture 4.1. A 3-graph can always be 3Dthickened into a CMM-PC mesh. Based on this conjecture, we demonstrate that from a CMM-PC mesh, it is possible to reconstruct the 2- manifold boundaries of all elements, i.e. the 3-vertices, 3-edges, 3-faces, and chambers of a decomposed 3- manifold This representation is analogous to the classical method of representing an embedding of a graph in a surface by a 2D graph rotation system (2D-GRS), which is based on a cyclic ordering of the edge-ends incident at each vertex. In the new model, cyclic ordering of the edgeends incident at each vertex is replaced by a CMM- Pattern Coverable Mesh that can be used to reconstruct 3-vertices, 3-edges, 3-faces and chambers. Lemma 4.1. We can consistently label edges of a CMM-PC mesh with 0 and 1, such that opposite edges in every quad have the same label. 1 To refer to CMM-Pattern Coverable Quad-Meshes, we also use a shorter form: CMM-PC meshes ) 8 (d) Grid (e) Even-valence (f) Rings Figure 6: All vertices of a CMM-pattern coverable mesh must be even, and same-colored edges must form consistently colored rings. Note that (b) and (c) includes unresolvable cases, shown as question marks. Figure 7: This figure shows the simplest CMM-PC mesh, which is a single quadrilateral embedded on a toroidal surface. It represents a 3-manifold with a single 3-vertex, a single 3-edge, a single 3-face, and a single chamber. Proof of this lemma is directly from the definition. To visually demonstrate labels in this paper we use colors, such as purple and green, as in Figure 6. As shown in that figure, in a CMM-PC mesh, we can consistently color opposite edges of every quadrilateral. Figure 7 shows the simplest CMM-PC mesh, which is a single quadrilateral embedded on a toroidal surface Cycles, Rings and Strips in CMM-PC meshes To reconstruct 3-edges and 3-faces, we use cycle and ring structures that exist in CMM-PC meshes. Definition In a CMM-PC mesh, a set of same-labeled edges that forms a closed polygon is called a cycle (as in Figure 8). Given a half-edge he, the operation Oppose(he) provides an opposing half-edge in the same face, and the operation Other(he) provides the other half-edge of he. We

9 define the operation Next(he) to be a composition of the Oppose and Other operations, which provides the opposing half-edge in the neighboring face for a given half-edge he. deletion Definition In a CMM-PC mesh, a group of edges whose half-edges are closed under the Next(he) operation is called a ring. (See Figure 8.) Figure 8: This figure illustrates the difference between a cycle and a ring. One of the most important advantages of CMM-PC meshes is that every edge belongs to only one ring, and every edge in a ring has the same color. The next lemma states some useful properties of CMM- PC meshes, which are used to represent non-cellular decompositions, which we call block meshes. Lemma 4.2. In a CMM-PC mesh, all vertex valences are even. Each edge belongs to only one ring. Every edge in the same ring has the same color. This lemma is clear from Figure 6. The same labeled rings of CMM-PC mesh is used to describe 3-edges and 3-faces of a block mesh, which are generalized prisms as shown in Figure 3. Without loss of generality, we assume that purple (Label=0) edges correspond to the sides of 3-edges and that green (Label=1) edges correspond to the sides of 3-faces. Definition In a CMM-PC mesh, the set of quadrilaterals that include all edges in a given ring is called a strip. The boundary cycles of strips are special type of cycles that we call boundary cycles. There are two types of strips, each of which is named by the label of its respective ring. In other words, there are purple strips and green strips. Without loss of generality, let us assume that each Label-0 strip (i.e., purple strip) corresponds to a 3-edge of a block mesh and that each Label-1 strip (i.e., green strip) corresponds to a 3- face of a block mesh. If we delete all the edges of a CMM-PC mesh that do not belong to a given strip, we obtain an orientable 2-manifold mesh surface, which consists mostly of quadrilaterals, which we call sides. In addition, there 9 exist one or two general 2-faces that are constructed by boundary cycles, which we call ends. For instance, in the case of purple strips, the quadrilateral faces are called face-sides and the others described by boundary cycles are called face-ends. The same is applicable for green strips. In the case of green strips, the quadrilateral faces are called edge-sides and the others constructed by boundary cycles are called edge-ends. Without loss of generality, we discuss possible strip structures using purple strip examples. We now provide a few of the many interesting structures that can be described by strips, including some physical Möbius strips 2. Figure 9 demonstrates the most common structures for strips. The boundaries of these strips are two nonconnected cycles. If we remove all other edges of a CMM-PC mesh, the resulting mesh is a genus-0 surface. As demonstrated in the figure, these are really prisms. The two non-connected boundary cycles form top and bottom bases of a prism. Figure 9: An example of strips that correspond to prisms: to see this, assume that we are looking at a transparent one from above with a perspective projection. A physical realization of the corresponding genus-0 surface is shown on the right. Figure 10 demonstrates another common structure that can be obtained by twisting a 3-edge between two 3- faces. The boundary of the strip in the figure consists again of two cycles; however, these two cycles are not disconnected in this case. They share two purple edges shown in as dotted lines in the figure. The result is really a representation of a one-sided structure. In other words, two face-sides of this type of 3-faces belong to the same chamber. Figure 11 demonstrates Keizo s cut that does not separate the block into two. This particular cut is shown in Figure 1 earlier. In this case, the boundary of the strip consists of two non-connected cycles. These two cycles form two 2-faces. This defines a physical Möbius strip, as shown on the right image, and again two faceends of this type of 3-faces belong to the same chamber. 2 Topologically, a physical Möbius strip is an annulus that lies on the boundary of a regular neighborhood in 3-space of a 2-dimensional Möbius strip.

10 Figure 10: An example of a genus-1 strip. The two dotted green edges form a tunnel by turning the shape into a toroid. This type of structure can be obtained by using the twist operation. Figure 11: Another example of a genus-1 strip. The left image shows a strip drawn on an unfolded toroidal surface. This particular case correspond to cutting a toroidal block that does not separate the block into two. This cut corresponds to a physical Möbius strip, as depicted in the right image. Figure 12 demonstrates another genus-1 structure that corresponds to a physical Möbius strip. Unlike Figure 11, here the boundary of the strip consists a single cycle. In this case, the boundary cycle does not define a 2-face. In other words, this particular structure has no face-end (or no edge-end for green strips). These examples show that a 3-face (or a 3-edge) can have two, one, or zero ends. This property of our model significantly extends its representational power. For instance, it is possible to include 3-edges that do not connect any vertices or 3-faces that do not connect any chambers. It is also possible to create a special kind of self-loop 3-edge or 3-face that has a single edge-end or a single face-end, respectively. Figure 12: An example of strips with single cycle boundaries. The strip on the left is drawn on a square that represents a toroid. Note that the two green edges form a single cycle on the toroidal surface. This type of strip also corresponds to a physical Möbius strip; however, their structure is different as shown on the right image. These structures are theoretical possibilities that can be obtained during modeling Reconstruction of 3-Vertices and Chambers We now demonstrate how to obtain 3-vertices and chambers from a given CMM-PC mesh. Conjecture 4.2. A CMM-PC mesh can consistently provide 3-vertices, 3-edges, 3-faces and chambers of a block mesh. If we delete all the purple edges from a 2-manifold mesh, what remains is the set of disconnected meshes that correspond to 3-vertices. Deleting all the purple vertices of a CMM-PC mesh effectively removes all connections between individual 3-vertices. Therefore, what remains is all the 3-vertices of the underlying block mesh. Thus, each connected 2-manifold mesh can be used as a representation of a 3-vertex. If we apply the same operation to a un-thickened graph, similarly we obtain the set of un-thickened 3-vertices. That is, deleting all the purple edges of a CMM-PC mesh effectively deletes all the edges of the original graph. Using the analogy with 2-manifolds, the resulting structure consists only of vertices. In fact, this operation results in a number of 2-manifold meshes that consist only of green edges. Therefore, each connected 2-manifold mesh is the boundary of a 3-vertex of a decomposed structure. Similarly, if we delete all the green edges of a CMM-PC mesh, then we obtain a number of 2-manifold meshes that consist entirely of purple edges. In this case, each connected 2-manifold mesh corresponds to a chamber of a decomposed structure. That is, if we delete all the green edges of a CMM-PC mesh, we remove all connections between individual chambers. What remains, therefore, is all the chambers of the block mesh, and each connected 2-manifold mesh corresponds to a single chamber. This also explains why we cannot allow disconnected blocks. In our framework, we do not have a mechanism to consider them a single block when there is no explicit connection between disconnected pieces. Remark: In a CMM-PC mesh, the green edges can also be called 3-corners of the 3-faces, and the purple edges can also be called 3-corners of the 3-faces. In other words, they correspond to corners of 2-manifolds. Each quadrilateral in a CMM-PC mesh corresponds to a half-edge of a 2-manifold mesh. They are also facesides, as in 2-manifold meshes. In the next section, we demonstrate how to create and manipulate CMM-PC meshes that represent block meshes.

11 5. A Shape Algebra for CMM-Coverable Meshes To create initial CMM-PC meshes, we introduce two simple procedures that can be used to convert any 2- manifold into a related block mesh. It is always possible to convert any orientable or non-orientable 2-manifold mesh into a CMM-PC mesh that represents a decomposition with either two blocks or one block, respectively. The original 2-manifold mesh s vertices turn into 3- vertices, and edges turn into 2-vertices, etc. These block meshes that are produced from 2-manifolds can provide an initial set of structures. More complicated structures can be produced from these initial structures by using operations that that can create CMM-PC meshes when applied to CMM-PC meshes. We provide herein a minimal set of operations, from which one can construct more complicated operations. For a minimal set, all we need do is to identify a set of operations that preserve the CMM-Pattern-Coverable property and that cannot be obtained from other operations. The most important operation in a minimal set of operations is the Dual operation that converts 3-vertices to chambers and 3-edges to 3-faces. We demonstrate that obtaining duals is extremely simple with CMM-PC meshes. (a) Opening a hole with insert edge operator. (b) Combining two surfaces with insert edge operator. Figure 13: The InsertEdge operator changes the topology when the edge-ends are inserted between two corners of two faces, thereby combining the two faces into one. If the initial faces are n 1 -sided and n 2 - sided, then the combined face becomes n 1 + n sided. In these examples, the resulting faces are 10-sided since the initial two faces are quadrilaterals. Note that the inserted edge appears twice as halfedges [74] in the same 10-sided face. This can be seen as a virtual tunnel that makes the yellow areas into cylinders. In these examples, the results are subdivided, in order to visually demonstrate holes and handles created by a single edge insertion. For more information for such Euler operators over 2-manifold meshes see [74, 50, 3] Creation of CMM-Coverable Meshes We present three additional operators that can convert any CMM-PC mesh into another CMM-PC mesh. Along with the Dual they provide a shape algebra over the CMM-Pattern Coverable meshes. These three operators, called InsertPipe, Twist, and SplitCycle, are block-mesh counterparts of the 2-manifold operators InsertEdge, Twist, and SplitVertex [50, 2]. More complicated block-mesh operations can be implemented as a combination of these 2-manifold operations. Figure 13 illustrates how we use InsertEdge operator to manipulate the topology of 2-manifolds. An important property of this framework is that all these operations can be applied using a series of 2-manifold mesh modeling operations. Moreover, CMM-Pattern Coverable meshes can be represented by any mesh data structure that can support all and only orientable 2- manifolds. Acccordingly, our framework can be implemented using any existing 2-manifold mesh data structure and Euler operators [74]. Of course the original mesh data structure and original Euler operators must be kept hidden from users, so that they cannot create invalid structures. 11 We propose to create block meshes by converting existing 2-manifold meshes, since there exist many 2- manifold meshes available now. Moreover, we have a good understanding of 2-manifold surfaces. Nonorientable surfaces are not that common in shape modeling; however, it is also useful to discuss how they can be converted into block meshes. Theorem 5.1. Any orientable 2-manifold mesh can be converted into a CMM coverable mesh that represents a related block mesh. Proof. A connected orientable 2-manifold mesh embedded in 3-space with no self-intersections separates 3-space into two blocks, which can be called inside and outside. Our goal is to obtain a CMM-PC mesh that can produce the two chambers that are boundaries of the inside and outside blocks. Consider a 2-vertex v of a given 2-manifold mesh, and assume that v is n-valent. We first obtain an n-sided polygon by 2D-thickening v, as shown in Figure 14. By 3D-thickening that polygon, we obtain a genus-0 regular mesh, denoted by (n, 2), where n is the valence

12 (a) 2D-thickened (b) 3D-thickened (c) Chambers Figure 14: Converting a 2D thickened valence-4 2-vertex of a 2- manifold into a 3D-thickened 3-vertex. Figure 16: Converting a cube into a CMM-PC mesh using 3Dthickening. (c) shows two chambers that are obtained from a CMM- PC mesh. of each vertex of the regular mesh, and 2 is the number of sides of each face. As shown in Figure 14, each face of this regular mesh is, in fact, a 3-edge end. Note that these regular meshes always have two 2-vertices, and we can label them as 0 and 1-vertices, which are colored blue and green, respectively, in the figure. If we insert these 3-edges between appropriate edgeends, we complete the conversion into a CMM-PC mesh. To insert pipes appropriately, we differentiate two 2-corners of these two-sided faces using labeled vertices. If there is a edge between two original 2-edgeends of a 2-manifold, then we insert two edges to create a corresponding 3-edge. These two 2-edges are inserted so that they join the same-labeled corners of the two 3- edge-ends, as shown in Figure 15. Figure 15: Inserting a pipe between two 2-sided 3-edge ends by inserting two 2-edges. This operation creates a pipe. If we replace all original 2-edges with 3-edges by inserting these pipes, the result becomes a CMM-PC mesh. Each original n-valent 2-vertex splits into two 2n-valent vertices. Moreover, each original 2-face of the 2-manifold turns into a cycle of green edges, where each green edge corresponds to a 3-corner of a 3-face. Now, if we delete all these green edges, the CMM-PC mesh separates into two 2- manifold meshes, which correspond to the chambers of the block mesh. These chambers are boundaries of the inside and outside blocks. This concludes our constructive proof. Figure 16 illustrates how a cube can be converted into a CMM-PC mesh with two chambers. 12 Conversion of an orientable 2-manifold provides a noncellular decomposition of 3-space, and it is always possible to avoid self-intersections when the resulting structure is embedded in 3-space. On the other hand, a nonorientable 2-manifold provides a more interesting case, as indicated by our next theorem. Theorem 5.2. Any non-orientable 2-manifold mesh can be converted into a CMM-PC mesh that represents a related block mesh. Proof. A connected non-orientable 2-manifold mesh has only one side. Thus, we do not have inside and outside blocks. In this case, our goal is to obtain a CMM- PC mesh that can produce a single chamber, which is the boundary of a single block. The first part of the conversion process is the same as for orientable surfaces. We first 3D-thicken every 2-vertex of the non-orientable mesh, thereby obtaining genus-0 (n, 2) regular meshes. Each one of these regular meshes still has two 2-vertices, which we still label as 0 and 1-vertices, which are again visualized as blue and green circles/spheres in the figures. Moreover, the procedure again requires insertion of 3-edges between appropriate edge-ends, to complete the conversion into a CMM- PC mesh. A caveat in this case is that some edges are twisted. An immersion of a twisted edge of a 2-manifold surface can be 2-thickened into a shape that can be visualized as a twisted paper strip. A paper strip can be twisted counter-clockwise or clockwise. This difference comes from immersion as briefly mentioned before. In terms of non-orientable surfaces, there should be no difference between these two different immersions. In 3D-thickening, unlike 2D-thickening, there is no way to differentiate twisted pipes from untwisted pipes unless we label the vertices. To insert twisted pipes appropriately, we use our earlier differentiation of the two 2-corners of the two-sided faces. If there is a twisted

13 Figure 17: Note that the vertex labels are created just for the algorithm. In the final CMM-PC mesh, vertices do not have labels. Without vertex labels, there is no way to differentiate between twisted and untwisted pipes. edge between two original 2-edge-ends of 2-manifold, we again insert two edges to create a corresponding 3- edge. However, in this non-orientable case, these two 2- edges are inserted between oppositely labeled corners of the two 3-edge-ends, as shown in Figure 17. That is, we insert one edge between a label-i corner of one 3-edgeend and a label-i + 1 corner of other 3-edge-end, where + is addition modulo 2. This operation again creates a pipe, and if we replace all the original 2-edges with 3- edges by inserting these pipes, the result again becomes a CMM-PC mesh. Moreover, each original 2-vertex of valence n splits into two vertices, each of valence 2N, and each original 2-face of the 2-manifold turns into a cycle of green edges, where each green edge corresponds to a 3-corner of a 3-face. However, if we delete all the green edges, the CMM-PC mesh does not now separate into two 2-manifold meshes. Instead, we obtain a single chamber as an orientable 2-manifold mesh, which is the boundary of the single block described by the original non-orientable 2-manifold mesh. Remark 1: The resulting chamber from this operation is actually the boundary of a 3D-thickened nonorientable surface. If the initial non-orientable surface is a projective plane, then the corresponding chamber is a genus-0 surface, as shown in Figure 18. (a) 2D-thickened (b) 3D-thickened (c) Chambers Figure 18: Converting a projective plane into a CMM-PC mesh using 3D-thickening. (c) shows that a single chamber is obtained from the CMM-PC mesh. This single chamber can be represented as an orientable 2-manifold mesh. 13 If the initial non-orientable surface is a Klein bottle, then the corresponding chamber is a genus-1 surface, and so on... These chambers, although orientable, selfintersect in 3-space. On the other hand, unlike the original non-orientable surface, there always exists a transformation that can remove the self-intersections of these orientable chambers. However, it is not really possible to make such a transformation, since the positions of yellow and blue vertices of the same 3-vertex must be the same, and our 3D-thickening process is just conceptual. Therefore it is not possible in practice to resolve these self-intersections. It is also interesting that this chamber, since it is an orientable surface, also has an inside and an outside. The real block is in the inside. The other infinitely thin block is an image of that nonorientable surface. We see that the major advantage of this conversion is not removing self-intersections, but rather, the ability to represent non-orientable objects with orientable structures, for which there exist plenty of effective mesh data structures that can be used Dual Operation Dual is the only global operation we need for shape algebra. The operation Dual converts all 3-vertices into chambers and all 3-edges into 3-faces; and vice-versa. Using CMM-PC meshes, obtaining duals is extremely simple. The structure of a CMM-PC mesh does not change under the dual operation. We just add 1 modulo 2 to all the edge labels. Then all the purple edges become green, and the green edges become purple. Having such a simple dual operation is crucial for shapemodeling applications. For instance, the quad-edge data structure provides a similarly simple dual operation for 2-manifold meshes [50]. Such a simple dual operation would have been impossible, if we did not allow 2-manifold representation for 3- vertices and for chambers of block mesh. For instance, a toroidal chamber turns into a 3-vertex after the dual operation. If we disallow toroidal vertices, we could not support the dual operation. As we have discussed earlier, allowing a toroidal 3-vertex does not necessarily mean that the final structure always includes noncellular vertices. For instance, we often need to apply the dual operation twice during subdivision. A twicedual does not change the original structure. It only smooths the geometry (see [108] for theoretical basis). Thus, having flexible representation not only simplifies the process but also can make possible some operations that could not be done before.

14 5.3. Local Operations over CMM-Coverable Meshes In this subsection, we present four local topology change operators that produce a new CMM-PC mesh when they are applied to any CMM-PC mesh. These four operators provide intuitive changes in the structure of underlying block mesh. Moreover, these operators are direct generalizations of four well-known 2-manifold topology change operators: CreateVertex, InsertEdge, TwistEdge and Splice. It is easy, therefore, to use these operators and it is also easy to develop algorithms intuitively combining these basic operators. All of these operators except CreateSimplest 3 have primary and dual versions. The dual operators can be considered as primary operators that are applied to dual meshes, or vice versa. Moreover, all primary and dual operators also have their inverses. Therefore, the total number of operations is 14. On the other hand, if we consider dual as an operator, then the total number of operators needed is eight, i.e., a self-dual operator, three primary operators, their three inverses, and the dual operator. The self-dual operator called CreateSimplest creates a simplest CMM-PC mesh that consists of a single quadrilateral, as shown in Figure 7. This can be considered analogous to the CreateVertex operator for 2- manifolds [74, 3]. Since this operator is really simple, we will focus on only other three operators, which are called the Insert Pipe, Twist, and Split Cycle operators Insert Pipe Operator The InsertPipe operator inserts a digonal prism, i.e. a prism with a digon base, between two same color edges as shown in Figure 19. In a CMM-PC mesh, the same color edges are really the same type 3-corners. In other words, this operator is applied to two corresponding 3-corners, like the InsertEdge operator. If the operator is applied to two green edges, it creates a 3- edge, and therefore, it is called InsertEdgePipe. If it is applied to purple edges, it creates a 3-face and is called InsertFacePipe. We consider InsertEdgePipe as primary and InsertFacePipe as its dual, since InsertEdgePipe is more closely related to practical applications. These operators also have inverses, called DeleteEdgePipe and DeleteFacePipe. An InsertPipe operation consists of two steps. In the first step, it splits the selected edges to create two twosided 2-faces. Then the newly created 2-faces are linked 3 In the case of CreateSimplest, the dual and primary operators are the same. 14 Figure 19: This figure shows an InsertEdgePipe operator that is applied to any two green edges which are actually two 3-corners of 3- faces. The operator insert a 3-edge between these two corners. If the two corners belong to the same 3-face, then the operator split the 3-face into two. If the two 3-corners belong to the two different 3- faces, then the operator combine these two faces into one exactly like InsertEdge operator over 2-manifold. to create a new pipe. InsertEdgePipe works exactly like the InsertEdge operator [3] (see Figure 13 for an example). It can separate a 3-face into two if the two green edges belong to the same 3-face. If the green edges belong to two different 3-faces, then the operator combines the two 3-faces. This operation can change the genus of chambers exactly as we requested. For instance, it can transform a genus-0 chamber into genus-1 chamber, which is non-cellular. It can also combine two chambers into one. Its inverse operator DeleteEdgePipe can reverse the effect of InsertEdgePipe. The dual operator, InsertFacePipe, is included only for the sake of theoretical completion. It is not as intuitive or useful as its counterpart. This operator can separate one 3-edge into two, or combine two 3-edges into one. That is, this operator can change a genus-0 3-vertex into a genus-1 3-vertex. It can also combine two disconnected genus-0 3-vertices into a genus-0 3-vertex Twist Operator The Twist(k) operation for block meshes is also simple and analogous to the twist edge operation for 2- manifolds [46, 50, 2]. For the twist, for every edge in the ring, the link that defines the edge is re-linked to the k th consecutive corner. Note that this is not a geometric twist operation. If there are n edges in the given ring, Twist(n) does not change the configuration. The inverse operation is Twist( k), which is of course equivalent to Twist(n k). The primary operation, i.e., the more intuitive operation, is the one that is applied to purple rings that correspond edge twist. Since the primary operation twists the edges, it is appropriate to call it again TwistEdge(k). This is closely related to the 2-manifold version, where n = 2 and k = 1. Like its 2-manifold

15 version, this operation can combine two neighboring 3-faces by turning them into a single twisted strip, as shown in Figure 20. The operation combines two chambers into one. For instance, it can combine two 3-cells into a non-cellular block. Figure 20: This figure demonstrates the effect of the TwistEdge(1) operator over a purple ring. As shown here, each link that defines a purple edge is re-linked to the first consecutive corner. The dual operator is applied to the green edges, i.e., it twists 3-faces. It is appropriate, therefore, to call the dual operator TwistFace(k). In 2-manifolds there is no conceptual counterpart of this operator. In block meshes, this operation can combine two neighboring 3- edges and can turn a 3-vertex into a non-orientable surface Split Cycle Operator The SplitCycle operator and its inverse are the generalization of the Splice operator that provides VertexSplit and EdgeCollapse as the inverse of VertexSplit (See Figure??. This operation is applied to a cycle. In the first of two steps, each 2-vertex on the cycle is split, by using the VertexSplit operator on a 2-manifold. This creates a ring with one missing boundary cycle. In the second step, the missing boundary cycle is inserted with a series of 2-manifold VertexSplit operations. If the chosen cycle is a green cycle, this operation is a Figure 21: Split operation is applied to a cycle. It creates a ring with two boundary cycles. 15 block mesh analogy to VertexSplit, and so it is called VertexSplit. The reason is that any green cycle is always a cycle on the surface of a single 3-vertex. If the cycle is contractible, then it separates the surface of the 3-vertex, and one side of separation is a disk [47]. In other words, when we split the contractible cycle and insert a ring, we effectively subdivide the original 3-vertex into two and connect them with a 3-edge. Moreover, one of the newly created vertices is always genus-0. Every cycle on a genus-0 surface is contractible. Therefore, if the initial 3-vertex is a genus-0 surface, then VertexSplit creates two new genus-0 3-vertices connected by newly created 3-edge. It is easy to see that the 2- manifold operator VertexSplit works exactly the same way by noticing that selection of two corners is equivalent to selecting a cycle that consists of two 3-face corners on (n, 2) regular 3-vertices. If the chosen cycle is noncontractible, it means that the surface of 3-vertex has a positive genus. The VertexSplit operator in this case does not separate the vertex into two. Instead, it reduces its genus by 1. The newly created edge is a self-loop. This operation can be used to reduce genus of the 3-vertex by introducing self-cycles. For the inverse of the VertexSplit operator, it is appropriate to use the same term as for 2-manifolds: EdgeCollapse. The inverse operation EdgeCollapse combines two vertices by collapsing a proper (i.e., not a self-loop) 3-edge that connects them. If the two ends of the 3-edge are both at the same vertex, i.e., if 3-edge is a self-loop, then the operation increases the genus of surface that defines 3-vertex. This particular observation illustrates that collapsing a self-loop on 2-manifolds can also create non-cellular 2-vertices. If the chosen cycle is purple, this operation, which we call ChamberSplit, can be used to cut blocks. Since this operation is the dual of VertexSplit, it works exactly the same way on the dual elements. For instance, if the cycle is contractible, the operation separates the block, and one of the boundary chambers is guaranteed to be genus-0. On the other hand, if the cycle is noncontractible, then the operation can reduce the genus of the chamber by one. The operation can also make a Möbius cut, as we discussed earlier with the Ushio sculptures. The inverse of the this operator is FaceCollapse, which can be used to reunite the blocks. If the VertexSplit operator is applied to a green cycle, which is actually a boundary of ring, the operator actually subdivides the edge (See Figure 22). We call this operator SubdivideEdge, to signify its practical use.

16 Figure 22: An edge can be subdivided by applying the VertexSplit operator to a green boundary cycle. We call this specific application of a split cycle operation SubdivideEdge, to signify its practical usage. In Appendix C, we give three examples that demonstrate how these operations can be used to manipulate the topology of the block meshes. We also show that it is possible to create unresolvable self-intersections, even with simple operations. 6. Conclusions and Future Work In this paper, we present a shape algebra for modeling with block meshes. Our algebra is based on a specific type of quadrilateral mesh, which we call a CMM-PC mesh. We have developed operators under which the space of CMM-PC meshes is closed. Using this algebra, we can describe all block meshes that can be reconstructed using CMM-PC meshes. This approach comes from the 3D-thickening concept that we have introduced in this paper. Our conjecture is that CMM-PC meshes provide 3D-thickenings of graphs that induce embeddings of the graphs in 3-manifolds, for which we can reconstruct the elements of a block mesh that provide decomposition of a higher dimensional structure. CMM-PC meshes are not developed to classify and/or analyze higher dimensional structures such as 3- manifolds [100]. Instead, our goal is to provide a simple algebra that can guarantee all the topology change operations that always lead to a valid structure. This approach helps us to define an algebra over the CMM-PC meshes. We have shown that our operations create only CMM-PC meshes. On the other hand, we did not touch the questions such as whether the operations can create all possible CMM-PC meshes and whether CMM-PC meshes can necessarily represent decompositions of 3- manifolds. If we start with a 2-manifold mesh and apply only the operations InsertEdgePipe, SplitVertex, and their inverses, then the resulting structures will always consists 16 of 3-vertices in the form of regular meshes (n, 2) where n is the valence of the original vertex, and we will always have two chambers. Thus, this particular set provides an algebra over orientable 2-manifolds. If we also include the TwistEdge operator, the 3-vertex types we can create do not change; however, we can now obtain single chamber as well as double chambers. Thus, we see that adding TwistEdge operator yields a more powerful algebra that includes non-orientable objects. This is analogous to the hierarchy structure among real algebras, complex algebras, and quaternion algebras. Our main motivation for writing this paper is to provide all essential elements for the implementation of this framework. From a practical point of view, our most important contribution is the ability to represent higher-dimensional structures, using only orientable 2- manifold meshes. In the shape-modeling community, the creation and manipulation of orientable 2-manifold meshes is very well-understood. As discussed in this paper, there are many robust mesh data structures that can be used for orientable 2-manifolds to support such an algebra. Of course, the extension still needs some careful book-keeping. For instance, although we can always obtain 3-vertices or chambers, it is better to have dynamically updated topological information. Moreover, when an operator combines two chambers into one, it is better to update this information during the modeling. Such extensions, although small, will be helpful in practical applications, especially during interactive modeling and rendering. Although 3-vertices and 3-edges are not really thickened, it is also possible to geometrically thicken them to provide better visualizations. For instance, creation of k-fold fabrics require such thickenings of face boundary walks. One may think it hard to use such block meshes that can self-intersect in 3-space for 3D-modeling since selfintersections might make it hard to understand the corresponding structures. Fortunately, this is not a problem in practice. Any block mesh, even if it is not embeddable in 3-space, can be represented visually without selfintersections, as a 3D-thickening of a graph specified by a 3D-GRS. Our form of 3D-representation of a higherdimensional shape with self-intersections is called a 3- space immersion [95, 92, 93]. In Appendix, we demonstrate that 3-space immersions of block meshes provide a unifying framework that can robustly represent all topologically distinct shapes in 3D from solids to surfaces and curves. The curves (or surfaces) will not really be part of the main data structure. They will be created on demand or on fly from under-

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20 Appendices In these Appendices, we briefly discuss how block meshes can be visualized and used in practical shape modeling applications by employing concept of immersion. These Appendices are mainly to demonstrate the practical usage of block decompositions in robust shape modeling. In Appendix A, we introduce four types of immersions that can be used for visualizing block meshes. In Appendix B, we provide modeling approaches that stems from immersions of block meshes. In the last section, we give three examples that demonstrate how the minimal operations can be used to manipulate the topology of the block meshes. Edges provide additional information about relationships among vertices. The second simplest representation of an immersion is obtained by adding edges to vertices. In 3-space this is always possible without any edge-intersections. The resulting shape actually represents a graph that is embedded in a 3-space. However, if the original shape has twisted edges, then the graph does not provide a good visualization of the immersed block mesh. Therefore, to understand the true structure of the block mesh it is better to use a geometric 3Dthickening. Figure A.23(a) shows a representation of a 3D-thickened graph that is embedded in a 3-space, represented by one vertex, three edges,one face, and one chamber, which is fewer (of course) than the object itself. Appendix A. Immersion Types Immersions are significantly useful for the conceptualization and visual representation of block meshes. Topologically, an immersion means that at every point x of the given shape, there is a neighborhood N(x) such that the restriction of the immersion to N(x) is a homeomorphism. A lower-dimensional example of an immersion is a drawing of a graph (for instance, the Kuratowski graph K 5 ) in the plane with self-intersections. Our representation of a topological immersion omits the self-intersected parts and lets the designer visualize them in the locations defined by their boundaries. Our implementation of an immersion consists of two stages: (1) remove self-intersected or higherdimensional parts, and (2) view the remaining parts with or without a thickening. These visual (or sculptural) representations can be designed, modeled, and manipulated with a minimal set of operators in 3D, as if we are working with 3-space objects. In particular, novice users can manipulate shapes as if they are simply working with 3D-shapes. The simplest implementation of an immersion is to remove everything except vertices. This process results in a type of shape called a point set in computer graphics [15]. In computer graphics, the vertices turn into splats (oriented 3D-circles) or balls (spheres or ellipsoids) for easy visualization [15]. The the former correspond to 2D-thickening and the latter to 3D-thickening. A point set is an appropriate visualization if the geometric positions of vertices reasonably well provide an interpretation of the actual shape. 20 (a) 3D-Thickened Graph (b) Thickened Face Boundary Figure A.23: This figure shows two 3-space immersions of a block mesh with one vertex, three edges, one face, and once chamber. The original graph is given in Figure 3(I). In this case, prisms that correspond to 3-edges are twisted. Another recently invented 3-space immersion method uses face-boundary walks in 3-space to form links and knots in 3D. This property has been used to construct weaving in 3-space. The resulting knots and links are actually representations of immersions of nonorientable 2-manifold meshes in 3-space. Figure A.24 shows a few immersions from some recent works [5, 106]. These woven shapes are 2-fold fabrics that are mapped on surfaces. Here, the number 2 is the thickness of the weaving, which is defined as the maximum number of cycles intersecting at any point. Using block meshes, it is possible to construct k-fold fabrics, where k can be any positive number. Figure A.23(b) shows a k-fold structure that is obtained from the face-boundary walk of a block mesh with one vertex, three edges, one face and one chamber. By using block meshes it is possible to construct more complicated structures such as braids and many-layered fabrics. This representational power can be particularly useful when modeling composite materials for engineering applications.

21 Appendix B.1. Solid Modeling with 3D-GRS (a) Plain-woven Bunny (b) Twill-woven Bunny A straightforward application of 3D-GRS is solid modeling. Representing and modeling solids is particularly useful for engineering applications, such as, in finite element analysis [54, 44], in discrete exterior calculus [21, 41], or in modeling fractures [81, 80]. If we make some chambers transparent, the resulting visual structure describes solids and can be used for modeling solids. Figure A.24: Two 3-space immersions of the same non-orientable mesh. The resulted woven objects can also be considered as 2-fold fabrics on polygonal meshes. An important challenge is that our framework should similarly provide different 3-space immersions of the same manifolds. Work is needed on the representations of immersions of twists. Our current framework cannot differentiate among 3-space immersions of twists. In general, for modeling fabrics, it is important to differentiate among a wide variety of possible immersions. Although this is outside the scope of this paper, we expect that an integer attached to 3-edges and 3-faces, similar to the 2- manifold case, could be used to distinguish among immersions [106]. Appendix B. Modeling Approaches The main objective of 3D-GRS is to provide a unifying method for a wide variety of shape-modeling applications. In this section, we provide modeling approaches that stems from immersions of block meshes. We have identified four types of immersions that lead to four different modeling approaches: (1) Solid Modeling, (2) Architectural Modeling, (3) High Genus Surface Modeling and (4) Knot and link modeling. A major advantage of 3D-GRS is that all of these applications can be based on the same model with the same minimal operation set. The only differences come from the chosen types of immersion. This approach provides a unified paradigm for designing a wide variety of shapes and topological objects, from 3D-fractals to k-fold fabrics and knots and links. Another advantage of 3D-GRS is that the high-level operators can be implemented using a minimal set of low-level operations. Using these high-level operators, designers can add new tools and methods without having full knowledge of the underlying kernel. This flexible structure can encourage scientists, engineers and artists to create objects with properties that are presently unexpected. 21 Having 3D-GRS can extend existing replacement systems into solid modeling by allowing chambers and blocks as additional entities. 2D-GRS is useful for replacement systems, such as fractal geometry or L- systems [73, 17, 87], by guaranteeing the manifold property (e.g., Akleman created generalized, connected, and manifold versions of the Sierpinsky tetrahedron [17] with topological mesh-modeling [96]). 2D-GRS is also useful for face-replacements [67, 66]. Subdivision schemes such as [27, 39]) are also replacement systems, and development of subdivision algorithms with 2D-GRS is straightforward [13, 6, 7, 14]. These examples of replacements work on surfaces, and their power is limited, based on the elements they can replace. To obtain more complicated replacement schemes, there is a need to extend these methods to handle solid structures. Allowing chambers and blocks provides that additional capability to the replacement systems. For instance, a Menger sponge is a 3D-fractal shape. To represent a 3D Menger sponge [73], we need to interactively remove cube-shaped blocks from the initial block. In 2-manifold modeling, it is impossible to keep track of blocks, since they do not exists as separate entities. By way of contrast, in 3D-GRS modeling, we can develop high-level operators that can simply subdivide an initial block into a set of blocks. Then the remove operation deletes some of these blocks. Since the result of the operation is always a set of blocks, we can iterate the process to create fractal shapes. 3D-GRS can also help to develop high-level operations for general subdivision schemes on block meshes. Existing subdivision schemes for solids are overwhelmingly based on hexahedral or tetrahedral meshes (such as [16, 28, 90]). On the other hand, in 3-space there exist a large number of space-filling polyhedra such as the gyrobifastigium [63], which can be used for subdivision. Moreover, some combinations of different kinds of polyhedron (e.g., of tetrahedra and octahedra) can fill 3-space [97]. Having a set of minimal operators and a mesh representation with generalized blocks can facili-

22 tate the development of high-level operators for subdivision schemes that allow iterative subdivision of 3-space into space-filling polyhedra. A 3D-GRS can also specify 1-sided embeddings and immersions of non-orientable surfaces into 3-space. (For instance, any immersion of a Klein bottle in 3- space is 1-sided.) This flexibility of 3D-GRS is useful in dealing with unusual shapes. Since 3D-GRS can represent 1-sided surfaces, it can efficiently represent cuts similar to Ushio s 1-sided cuts (see Figure 1). Since 1- sided surfaces in 3-space can be thickened, we can represent shapes such as Ushio s toroidal sculptures with block meshes. Another advantage of our approach is that these models can be very simple. For instance, both of Ushio s sculptures can be represented by one vertex, four (curved) edges, and two (curved) faces in terms of 3D-GRS. The number of blocks can be either one (when one of the faces is one-sided) or two (when both of the faces are two-sided) by not counting the outside block. Appendix B.2. Architectural Modeling with 3D-GRS Architecture is another important direction for applications, since powerful visualization and modeling systems are needed for the design and construction of revolutionary types of shapes for buildings and sculptures. Architects already explore such interesting forms, but the design and construction process is still very costly. For the design of shapes, architects use a wide variety of approaches. For instance, Frank Gehry s designs are created by using free-form surfaces [18, 85] (i.e., defined by control parameters, such as a Bezier or B-spline surface). Examples of Gehry-designed buildings are the Disney Concert Hall in Los Angeles and the Guggenheim Museum in Bilbao, Spain. The shape of the Kunsthaus, Graz Art Museum, designed by Peter Cook and Colin Fournier is given by an implicitly defined surface [40]. Classical graph rotation systems, 2D-GRS, already have the power to provide both free-form and implicit surfaces. With graph rotation systems along with subdivision schemes [27, 39], it is possible to obtain shapes that correspond to implicit surfaces, by allowing the topology to change during interactive modeling, as described in [10, 12, 11]. A 3D-GRS model is directly applicable to designing architectural shapes, since a 3D-thickened 3D-GRS can represent the kind of architectural structures that can be built. If we model rooms by chambers, then these chambers serve as a design for a building. As discussed before, a 3D-thickening of the 3-vertices, 3-edges, and 3-22 faces turns them into balls, beams, and plates. In architectural modeling, plates can correspond to walls, windows, or doors of a building. More importantly for architectural modeling, we can label the faces to differentiate between different kinds of architectural elements, such as doors or walls. Appendix B.3. High-genus Modeling with 3D-GRS Restricting the geometric 3D-thickening to vertices and edges corresponds to Schlegel diagrams [33], which are used by mathematical sculptors to visualize regular convex 4-dimensional polytopes such as the 120- cell [33]. In 3D Schlegel diagrams, edges are drawn as cylinders, using construction tools such as Zome-tool [57]. In the new method, by way of contrast, thickened edges turn into prisms that may be curved and twisted. This gives us additional modeling capacity, beyond Schlegel diagrams. The resulting shapes can be of very high genus, and reminiscent of the creations of contemporary sculptors, such as George Hart [53], Eva Hild, Rinus Roelofs, Helaman Ferguson [42], Bathsheba Grossman, Brent Collins, and Carlo Séquin [78], who successfully combine art and mathematics to create unusual sculptures such as infinite polyhedra. These sculptures have close relations with mathematical ideas such as space-filling polyhedra [105], periodic and symmetric polyhedra [105, 32], regular maps [30, 31], and regular meshes [8, 9]. We have also developed a method to construct regular meshes. Recently, J. van Wijk discovered another general method to create regular maps [101, 94]. Since space-filling polyhedra and infinite symmetric polyhedra can be used in solid representation for effective partition of 3-space, these methods can eventually be useful in engineering and scientific analysis. Appendix B.4. Knot and link modeling and k-fold woven fabrics Akleman et al. [4] describe how the face-boundary walks of a 2-manifold mesh in 3-space with edgetwisting are cycles (closed 3D-curves) that correspond to knots and links in 3-space [88, 45]. When we 2Dthicken these cycles within the underlying surface for the mesh, the resulting face-boundary walks correspond to threads of a weaving. This allows a 2D-GRS on a non-orientable 2-manifold to be used to specify a 2-fold fabric as a link (or a knot) immersed on a closed orientable surface in 3-space, so that there are never more than two strands crossing each other at any given point

23 Figure B.25: This figure demonstrates how a cube-shaped block can be subdivided into two with our operations. (A) shows the initial structure, where each circle represents a 3-vertex, and each line represents a 3-edge. For the initial structures, we did not draw 3-faces, in order to show the structure. (B) shows four new 3-vertices that are created by SubdivideEdge operations. (C) shows four new 3-edges that are created by InsertEdgePipe operations. These 3-edges subdivide the original 3-faces. (D) shows a 3-face that is created by ChamberSplit operation. This operation subdivides the whole block into two blocks, whose boundaries are rectangular prisms. Figure B.26: This figure demonstrates how a structure with unresolvable self-intersections can be created by our operations. (A) shows the initial structure, which is the shape we obtained in Figure B.25(d). (B) shows four new 3-vertices that are created by SubdivideEdge operations. (C) shows four new 3-edges that are created by InsertEdgePipe operations. Unlike the case of Figure B.25, here insertion of edges combines the two 3-faces. As a result, the two chambers also combined, resulting in a genus-1 surface. Visually there is a self-intersection. We can resolve that self-intersection by transforming the shape. (D) shows a 3-face that is created again by ChamberSplit operation. However, this operation does not subdivide the block into two since it is applied to a non-contractible cycle. The resulting self-intersection cannot be resolved in 3-space. Figure B.27: This figure demonstrates how similar-looking structures can be quite different topologically. (A) shows the initial structure, which is the shape obtained in Figure B.25(d). (B) shows six new 3-vertices created by SubdivideEdge operations. (C) shows six new 3-edges created by InsertEdgePipe operations. Each inserted edge subdivides the face to which it belongs, since it is inserted between two corners of the same 3-face. Note that the 3-face drawn as a green wall is also subdivided into two. (D) shows two 3-faces created by two ChamberSplit operations. In this case, the operations subdivide the underlying blocks, since they are applied to contractible cycles. The result is four blocks inside the original cube. 23