Topological Mesh Modeling
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 Topological Mesh Modeling is an umbrella term that covers all our work based on extensions the theory of graph rotation systems. It includes (1) Orientable 2-manifol mesh modeling using graph rotation systems and its computer graphics applications, (2) Knot modeling with immersions of non-orientable manifold meshes and (3) Topological constructions that is based on geometric and physical constraints with graph rotation systems. We recently started to work on immersions of 3-manifolds as a representation to develop shape modeling systems. This webpage provides links to approximately 50 papers that are roughly organized approximately 10 categories. Click links below to directly go to the the categories that provides related papers and manuscripts. Orientable Mesh Modeling: We have provided a solid foundation for orientable 2-manifold mesh modeling using graph rotation systems. Based on this theory, we have developed TopMod , which is is an orientable 2-manifold mesh modeling system. TopMod provides a wide vriety of High Genus Modeling tools, Remeshings & Subdivisions, and Extrusions & Replacements. Using TopMod, one can find a wide variety of ways to create high genus shapes; almost all subdivision algorithms, wide variety of ways to remeshing shapes and new extrusions. These tools are also useful for Architectural applications, Design and Sculpting. We also hav additional tools for Surface Parameterization and Texturing and Tiling. Knots Modeling: We have developed provided a solid foundation for knot, link and cyclic woven object modeling using extended graph rotation systems. If we twist an arbitrary subset of edges of a mesh on an orientable surface, we can obtain non-orientable surfaces. The resulting extended graph rotation system can be used to induce a cyclic weaving on the original surface, that corresponds a 3-space immedding of a non-orientable surface. Topological Constructions: Discrete Gaussian-Bonnet theorem and Gaussian curvatures related mesh topologic concepts to geometry. Using this relationship, we have developed methods to phsyically construct shapes. Immersions of 3-Manifolds: Using an extension of graph rotation systems it is possible to represent 3-space immersions of 3-manifolds by employing a topological graph theory concept called 3D thickening. This work partially supported by the National Science Foundation under Grant No. NSF-CCF-0917288.

 Knot Modeling and Cyclic Woven Objects: Modeling with Immersions of Non-Orientable Manifold Meshes

 If we twist an arbitrary subset of edges of a mesh on an orientable surface, we can obtain non-orientable surfaces. The resulting extended graph rotation system (EGRS) can be used to induce a cyclic weaving on the original surface, that corresponds a 3-space immedding of a non-orientable surface. In extended graph rotation systems, an edge is viewed as a paper strip that can be twisted. The sides of the paper strips provide “two strands” to construct weaving structures. Either these strands are “parallel” to the mesh edge for an “untwisted edge”, or they both cross over the edge and over each other for a “twisted edge”. If an arbitrary subset of edges of a mesh on an orientable surface is twisted in the same helical sense, then the EGRS induces a cyclic plain-weaving on the surface, which consists of cycles that cross other cycles (or themselves) by alternatingly going over and under. For theoretical treatment see this manuscript.

 E. Akleman, J. Chen, Q. Xing and J. Gross, "Cyclic plain-weaving on polygonal mesh surfaces with graph rotation systems", ACM SIGGRAPH, Transactions on Graphics (TOG), Volume 28, Issue 3, pp 78.1-78.8, August 2009. (Paper) (Video) Description: We showed how to create plain-weaving over an arbitrary surface. To create a plain-weaving on a surface, one need to create cycles that cross other cycles (or themselves) by alternatingly going over and under. We use the fact that it is possible to create such cycles, starting from any given manifold-mesh surface by simply twisting every edge of the manifold mesh. We have developed a new method that converts plain-weaving cycles to 3D thread structures. Using this method, it is possible to cover a surface without large gaps between threads by controlling the sizes of the gaps. We have developed a system that converts any manifold mesh to a plain-woven object, by interactively controlling the shapes of the threads with a set of parameters. We have demonstrated that by using this system, we can create a wide variety of plain-weaving patterns, some of which may not have been seen before.

 Q. Xing, E. Akleman, J. Chen, and J. Gross, "Single-Cycle Plain-Woven Objects", Proceedings of Shape Modeling International, 2010, (Paper) Description: In this paper, we show that it is always possible to create a single-cycle plain-weaving starting from a mesh on an arbitrary surface, by selecting an appropriate subset of edges to be twisted. We also demonstrate how, starting from a mesh, to construct a large number of single-cycle plain-woven objects. Interestingly, the single-cycle solutions with a minimal number of edge twists correspond to plainwoven objects that are visually similar to Celtic knots. For converting plain-weaving cycles to 3D thread structures, we extend the original projection method, which previously worked only when all mesh edges are twisted. With the extension described here, our projection method can also be used to handle untwisted edges. We have developed a system that converts any manifold mesh into single-cycle plain-woven objects, by interactively controlling the proportion of edges that are twisted. The system also allows us to change the shapes of the threads with a set of parameters, interactively in real-time. We demonstrate here that by using this system, we can create a wide variety of single-cycle plain-woven objects.

 E. Akleman, J. Chen, Y-L Chen, Q. Xing and J. Gross, "Cyclic Twill Woven Objects", Computers & Graphics 35 (2011) 623–631. (Paper) (Video) Description: Classical (or biaxial) twill is a textile weave in which the weft threads pass over and under two or more warp threads, with an offset between adjacent weft threads to give an appearance of diagonal lines. We ideveloped a theoretical framework for constructing twill-woven objects, i.e., cyclic twill-weavings on arbitrary surfaces. We also presented methods to convert polygonal meshes into twill-woven objects. We also identified a general technique to obtain exact triaxial-woven objects from an arbitrary polygonal mesh surface.

 Parameterization, Texturing & Tiling:

 Shiyu Hu, Qing Xing, Ergun Akleman, Jianer Chen, Jonathan L. Gross, "Pattern Mapping with Quad-Pattern-Coverable Quad-Meshes", Computers & Graphics 36 (2012) 455-465. (Paper) (Video) Description: We show that for every surface of positive genus, there exist many quadrilateral manifold meshes that can be texturemapped with locally translated copies of a single square-texture pattern. This implies, for instance, that every positivegenus surface can be covered seamlessly with any of the 17 plane symmetric wallpaper patterns. We identify su- cient conditions for meshes to be classified as “quad-pattern-coverable”, and we present several methods to construct such meshes. Moreover, we identify some mesh operations that preserve the quad-pattern-coverability property. For instance, since vertex insertion remeshing, which is the remeshing operation behind Catmull-Clark subdivision, preserves quad-pattern-coverability, it is possible to cover any surface of positive genus with iteratively finer versions of the same texture.

 Ergun Akleman, Qing Xing, Pradeep Garigipati, Gabriel Taubin, Jianer Chen, Shiyu Hu, "Hamiltonian Cycle Art: Surface Covering Wire Sculptures and Duotone Surfaces", Computers & Graphics 36 (2012) 455-465. (Paper) (Video1) (Video2) Description: In this work, we present the concept of ”Hamiltonian Cycle Art” that is based on the fact that any mesh surface can be converted to a single closed 3D curve. These curves are constructed by connecting the centers of every two neighboring triangles in the Hamiltonian triangle strips. We call these curves surface covering since they follow the shape of the mesh surface by meandering over it like a river. We show that these curves can be used to create wire sculptures and duotone (two-color painted) surfaces. To obtain surface covering wire sculptures we have developed two methods to construct corresponding 3D wires from surface covering curves. The first method constructs equal diameter wires. The second method creates wires with varying diameter and can produce wires that densely cover the mesh surface. For duotone surfaces, we have developed a method to obtain surface covering curves that can divide any given mesh surface into two regions that can be painted two di erent colors. These curves serve as a boundary that define two visually interlocked regions in the surface. We have implemented this method by mapping appropriate textures to each face of the initial mesh. The resulting textured surfaces look aesthetically pleasing since they closely resemble planar TSP (traveling salesmen problem) art and Truchet-like curves.

 Topological constructions is based on the relation between topology and geometry through Gauss-Bonnet theorem and Euler characteristics. In this work, we turn data structures that is used to represent 2-manifolds into physical data structures. The fundamental Heffter-Edmunds theorem of GRS asserts that there is a bijective correspondence between the set of pure rotation systems of a graph and the set of equivalence classes of embeddings of the graph in the orientable surfaces. As a direct consequence of the theorem, to assemble a structure all construction workers have to do is to attach the corresponding phsyical components. Once all the components are attached to each other, the whole structure is guaranteed to be correctly assembled. Gauss-Bonnet theorem, moreover, asserts that the total Gaussian curvature of a surface is the Euler characteristics times 2pi. Since we use only developable components, Gaussian curvature is zero everywhere on the solid parts. The Gaussian curvature happens only in empty regions and that are determined uniquely. Since, we cor- rectly form Gaussian curvature of holes, the structures always be raise and form in 3-space.

 Edwin Alexander Peraza Hernandez, Shiyu Hu, Han Wei Kung, Darren Hartl, Ergun Akleman, "Towards building smart self-folding structures". Computers & Graphics 37(6): 730-742 (2013) (paper) Description: We report our initial progress on synthesizing complex structures from programmable self-folding active materials, which we call Smart Multi-Use Reconfigurable Forms. We have developed a method to unfold a given convex polygonal mesh into a one-piece planar surface. We analyze the behavior of this surface as if it were constructed from realistic active materials such as shape memory alloys (SMAs), in which sharp creases and folds are not feasible. These active materials can change their shapes when they are heated and have been applied to medical, aerospace, and automotive applications in the engineering realm. We demonstrate via material constitutive modeling and utilization of finite element analysis (FEA) that by appropriately heating the unfolded planar surface it is possible to recover the 3D shape of the original polygonal mesh. We have simulated the process and our finite element analysis simulations demonstrate that these active materials can be raised against gravity, formed, and reconfigured automatically in three dimensions with appropriate heating in a manner that extends previous work in the area of programmable matter. Based on our results, we believe that it is possible to use active materials to develop reprogrammable self-folding complex structures.

 Qing Xing, Gabriel Esquivel, Ergun Akleman, Jianer Chen, Jonathan L. Gross, "Band Decomposition of 2-Manifold Meshes For Physical Construction of Large Structures", Siggraph '2011, Posters & Talks (2011). (extended abstract) Description: In this work, we introduce an approach to automatically create such easily assembled developable components from any given manifold mesh. Our approach is based on classical Graph Rotation Systems (GRS). Each developable component, which we call vertex component, is a physical equiva- lent of a rotation at the vertex v of a graph G. Each vertex component is a star shaped polygon that physically corresponds to the cyclic permutation of the edge-ends incident on v (See Figure 2(a)). We engrave edge-numbers with laser- cutters directly on edge-ends of vertex components to simplify nding corresponding edge ends. When we print edge numbers, we actually define a collection of rotations, one for each vertex in G. This is formally called a pure rotation system of a graph. Using this approach, Architecture students have constructed a large version of Stanford Bunny (see Figure 1) in a design and fabrication course in College of Architecture.

 E. Akleman, J. Chen and J. Gross, "Paper-Strip Sculptures", Proceedings of Shape Modleing International'2010. (Paper) Description: This paper introduces paper-strip sculptures, a physical mesh data-structure used to represent 2- manifold mesh surfaces for understanding topological and geometrical aspects of shape modeling with visual and tactual examples. With paper strips it is possible to construct simple paper sculptures that can convincingly illustrate a variety of ideas in shape modeling — such as 2- manifold mesh surfaces, discrete Gaussian curvature, and the Gauss-Bonnet theorem — with hands-on experiments. Such sculptures can also represent links, knots and weaving. Paper-strip sculptures are also useful to represent and understand non-orientable surfaces such as the projective plane and the Klein bottle.

 E. Akleman and J. Chen, "Practical Polygonal Mesh Modeling with Discrete Gaussian-Bonnet Theorem", Proceedings of Geometry, Modeling and Processing 2006, Pittsburg. (Paper) Description: In this paper, we introduce a practical modeling approach to improve the quality of polygonal mesh structures. Our approach is based on a discrete version of Gaussian-Bonnet theorem on piecewise planar manifold meshes and vertex angle deflections that determines local geometric behavior. Based on discrete Gaussian- Bonnet theorem, summation of angle defects of all vertices is independent of mesh structure and it depends on only the topology of the mesh surface. Based on this result, it can be possible to improve organization of mesh structure of a shape according to its intended geometric structure.

 Using an extension of graph rotation systems it is possible to represent 3-space immersions of 3-manifolds by employing a topological graph theory concept called 3D thickening. (joint work Jianer Chen and Jonathan Gross)

 E. Akleman, "Extended Graph Rotation Systems and Its Applications to Modeling 2-Manifolds, Woven Surfaces and 3-Manifolds", GD/SPM'2013, Presentation at the Minisymposium: Shaping Surfaces. (Presentation) Description: In this talk, I demonstrated that extended graph rotation systems and 3D thickenings has a potential to describe 3-manifolds that can help our understanding and modeling 3-manifold structures. Such a generalized 3-manifold mesh representations can be used in modeling solids, architectural shapes, high-genus surfaces, knots and links. For 3-manifolds, I started with prisms that represents 3D thickened edges of 3-manifold meshes and discussed what kind of models can be constructed using those prisms. I also introduced the concepts of chambers and blocks. Using boundary walk I demonstrated the faces of 3-manifolds can be both one and two-sided. If we want duality, this suggests that 3D thickened edges should also be one or two-sided and 3D thickened vertex boundaries can be any 2-manifold.

 O. Gonen and E. Akleman, "Sketch Based 3D Modeling with Curvature Classification", Computers & Graphics, 2012, 36(5), 521-525. (extended Abstract) Description: In this paper, we introduce a simple approach for sketching 3D models in arbitrary topology. Using this approach, we have developed a system to convert silhouette sketches to 3D meshes that mostly consists of quadrilaterals and 4-valent vertices. Because of their regular structures, these 3D meshes can e ectively be smoothed using Catmull- Clark subdivision. Our approach is based on the identification of corresponding points on a set of curves. Using the structure of correspondences on the curves, we partition curves into junction, cap and tubular regions and construct mostly quadrilateral meshes using these partitions.

 O. Gonen and E. Akleman, "Sketching Knots", Siggraph 2012, Posters. (paper) Description: We present an unexpectedly easy to use interface to create knots by using sketch based modeling. In our interface the only thing the users need to do for creating knots and links is to draw a set of curves. These curves serves the medial axis of the knots to be constructed. To construct knots we rst estimate of the depth {z{ value for every point on the medial axis curve. The depth estimation turns 2D medial axis curve into a 3D medial axis. We then extrude a polygon along the 3D medial axis curve to obtain the tread that forms the physical knot. If the medial axis consists of closed curves, the result is a mathematical knot.

 Theoretical Framework Theoretical Framework for Orientable 2-Manifold Modeling: Topological Graph Embeddings and Its Computer Graphics Applications

 Topologically Robust Mesh Modeling: Concepts, Data Structures and Operations We extend the theory of graph rotation systems and provide a solid foundation for orientable 2-manifold mesh modeling. Based on this theory, we identify a group of simple validity rules, and show that the validity of 2-manifold structures can be tested very efficiently on all existing data structures for mesh modeling. Moreover, the theory enables us to develop very efficient implementations for manifold preserving operations, which are essential for a robust interactive modeling system. For theoretical treatment see this manuscript.

 E. Akleman, J. Chen, "Guaranteeing 2-Manifold Property for Meshes by Using Doubly Linked Face List", International Journal of Shape Modeling, Volume 5, No. 2, pp. 149-177, 2000. (Paper) Description: Meshes, which generalize polyhedra by using non-planar faces, are the most commonly used objects in computer graphics. Modeling 2-dimensional manifold meshes with a simple user interface is an important problem in computer graphics and computer aided geometric design. In this paper, we propose a conceptual framework to model meshes. Our framework guarantees topologically correct 2-dimensional manifolds and provides a new user interface paradigm for mesh modeling systems.

 E. Akleman, J. Chen, V. Srinivasan and F. Eryoldas, "A New Corner Cutting Scheme with Tension and Handle-Face Reconstruction", International Journal of Shape Modeling, Volume 7, No. 2, pp. 111-121, 2001. (Paper) Description: A recently developed topological mesh modeling approach allows users to change topol- ogy of orientable 2-manifold meshes and to create unusual faces. Handle-faces are one of such faces that are commonly created during topology changes. This paper shows that vertex insertion and corner cutting subdivision schemes can effectively be used to recon- struct handle-faces. These reconstructions effectively show the structure of these unusual faces. The paper has three contributions. First, we develop a new corner cutting scheme, which provides a tension parameter to control the shape of subdivided surface. Second, we develop careful and e±cient remeshing algorithms for our corner cutting scheme that use only the basic operations provided by our topological mesh modeling approach. This implementation ensures that our new corner cutting scheme preserves topological robust- ness. Finally, a comparative study shows that the corner cutting schemes create better handles and holes than the well-known Catmull-Clark scheme.

 E. Akleman, J. Chen and V. Srinivasan, "A minimal and complete set of operators for the development of robust manifold mesh modelers", Graphical Models, Volume 65, Issue 5, pp. 286-304, September 2003. (Paper) Description: In this paper, we identify a minimal and complete set of fundamental operators, which is necessary and su±cient for performing all homeomorphic and topological operations on 2-manifold mesh structures. E±cient algorithms are developed for the implementation of these operators. We also developed a set of powerful, user- friendly, and effective operators at the level of user-interface. Using these operators, we have developed a prototype system for robust, interactive and user friendly mod- eling of orientable 2-manifold meshes. Users of our system can perform a large set of homeomorphic and topological changes with these user-interface level operators. Our system is topologically robust in the sense that users will never create invalid 2-manifold mesh structure with these operators. In our system, the homeomorphic and topological surgery operations can be ap- plied alternatively on 2-manifold meshes. With our system,users can blend surfaces, construct rinds and open holes on these rind shapes. With our system, the shapes that look like solid, non-manifold, or 2-manifold with boundary can be manipulated. The system also provides automatic texture mapping during topology changes.

 E. Akleman, J. Chen, V. Srinivasan, "A New Paradigm for Changing Topology During Subdivision Modeling", Proceedings of Pacific Graphics 2000, Hong Kong, China, pp. 192-201, October 2000. (Paper) Description: In this paper, we present a new paradigm that allows dynamically changing the topology of 2-manifold polygonal meshes. Our new paradigm always guarantees topological consistency of polygonal meshes. Based on our paradigm, by simply adding and deleting edges, handles can be created and deleted, holes can be opened or closed, polygonal meshes can be connected or disconnected. These edge insertion and edge deletion operations are highly consistent with subdivision algorithms. In particular, these operations can be easily included into a subdivision modeling system such that the topological changes and subdivision operations can be performed alternatively during model construction. We demonstrate practical examples of topology changes based on this new paradigm and show that the new paradigm is convenient, effective, efficient, and friendly to subdivision surfaces.

 Regular Meshes A Family of Meshes Includes Regular Maps

 E. Akleman and J. Chen, "Regular Meshes", Proceedings of Solid and Physical Modeling 2005, Boston, June 2005. (Paper) Description: This paper presents our preliminary results on regular meshes in which all faces have the same size and all vertices have the same valence. A regular mesh is denoted by (n,m,g) where n is the number of the sides of faces, m is the valence of vertices and g is the genus of the mesh. For g = 0, regular meshes include regular platonic solids, all two sided polygons. For g = 1 regular meshes include regular tilings of infinite plane. Our work shows that there exist infinitely many regular meshes for g > 1. Moreover, we have constructive proofs that describe how to create high genus regular meshes that consist of triangles and quadrilaterals (3,m,g) and (4,m,g).

 E. Akleman and J. Chen, "Regular Meshes Construction Algorithms Using Regular Handles", Proceedings of Shape Modeling International 2006, Matsushima, japan (Paper) Description: We introduce a new concept called regular handles. Using regular handles it is possible to increase genus without increasing the number of vertices. We develop a general procedure based on regular handles. Our procedure allows us to greatly extend ”regular mesh families”. We provide 14 regular mesh families that includes all genus-2 primary regular meshes: (3,7,2), (3,8,2), (3,9,2), (3,10,2), (3,12,2), (3,18,2), (4,5,2), (4,6,2),(4,8,2), (4,12,2) and (5,5,2), (5,10,2), (6,6,2) and (8,8,2). Our regular mesh families are constructed by adding the regular handles to an initial regular mesh M0. By using the same procedure iteratively we construct a series of regular meshes M0,M1,M2, . . .Mn.

 High Genus Modeling: High Level Tools for Modeling High Genus Surfaces

 E. Akleman, V. Srinivasan and J. Chen, "Interactive Rind Modeling", Proceedings of Shape Modeling International 2003, Seoul, Korea, May 2003. (Paper) Description: In this paper, we describe a technique, with roots in topological graph theory, that we call rind modeling. It provides for the easy creation of surfaces resembling peeled and punctured rinds. We show how the method’s two main steps of 1) creation of a shell or crust like the rind of an orange, and 2) opening holes in the crust by punching or peeling can be encapsulated into a real time semi-automatic interactive algorithm. We include a number of worked examples, some by students in a first modeling course, that demonstrate the ease with which a large variety of intricate rind shapes can be created.

 V. Srinivasan, E. Akleman and J. Chen, "Interactive Construction of Multi-Segment Curved Handles", Proceedings of Pacific Graphics 2002, Beijing, China, October 2002. (Paper) Description: In this paper, we present a method to interactively create multi-segment, curved handles between two star-shaped faces of an orientable 2-manifold mesh or to connect two 2-manifold meshes along such faces. The presented algorithm combines a very simple 2D morping algorithm with a Hermite interpolation to construct the handle. Based on the method, we have developed a user interface tool that allows users to simply and easily create multi-segment curved handles. The method can be used for handle creation (i.e., adding a handle to a surface) and for surface blending (i.e. connecting two distinct surfaces). Both applications of the algorithm are useful to designers for creating manifolds of high genus. A handle is not an extrusion (or lofting) and handle creation is not simply an extrusion method. Handle creation is a topological operation and it requires topological consistency. Therefore, handle creation methods are different than extrusion methods since they are required to guarantee topological consistency. Our method not only guarantees 2- manifold property of final mesh, in every stage of our handle creation costructed meshes continue to be 2-manifold.

 V. Srinivasan and E. Akleman, "Connected and Monifold Sierpinsky Polyhedra", Proceedings of Solid Modeling 2004, Genoa, Italy, June 2004. (Paper) Description: In this paper, we present a subdivision-inspired scheme to construct generalized Sierpinski polyhedron. Unlike usual Sierpinski polyhedra construction schemes, which create either an infinite set of disconnected tetrahedra or a non-manifold polyhedron, our robust construction scheme creates one connected and manifold polyhedron. Moreover, unlike the original schemes, this new scheme can be applied to any manifold polyhedral mesh and based on the shape of this initial polyhedra a large variety of Sierpinski polyhedra can be obtained.Our basic scheme can be viewed as applying simplest subdivision scheme [23] to an input polyhedron, but retaining old vertices. The porous structure is then obtained by removing the refined facets of the simplest subdivision.

 V. Srinivasan, E. Mandal and E. Akleman, Solidifying Frames, Bridges: Mathematical Connections in Art, Music, and Science 2004, Banf, Alberta, Canada, August 2005. (Paper) Description: In this paper we present a method to convert a wireframe mesh into a 2-manifold mesh consisting of cylindrical pipes in place of the edges and joints in place of the vertices in the original mesh. Our method allows users to create unique artistic depictions of common objects and structures. The resulting mesh is also more effective at conveying the overall 3D structure and any internal elements of a model when compared to regular wireframe or boundary representations. The input wireframe mesh can be any collection of linear edges; they do not have to form a manifold surface or even be connected to each other. The result is always an orientable 2-manifold surface. Our algorithm replaces every edge in the wireframe mesh with a cylindrical 3D pipe. The pipes are connected to each other using 3D joints created at the vertices in the wireframe where the edges meet. Our method has been implemented as part of a polygonal mesh modeling system and has been used to create artistic models of popular architectural structures as well as to create conceptual sketches for virtual environments.

 E. Mandal, E. Akleman and V. Srinivasan, "Wire Modeling", Visual Proceedings of ACM SIGGRAPH'2003 (Siggraph Sketch), San Diego, California, July 2003. (Sketch) Description: We present a method that will let the users create extremely high genus manifold meshes with minimal human interaction and time. Our method replaces each edge of a given mesh with a {\em 3D pipe}'' by creating a wired look. Our method guarantees that the pipes are connected and the resulting shapes can be physically constructed. We have implemented this method as an extension to an existing modeling system. Our system creates a complex high genus mesh from an input polygonal mesh by converting the edges of the input polygon to 3D pipes which looks like wires or matchsticks. Since the quality final model completely depends the mesh structure of the initial polygonal mesh, we have developed a set of subdivision based approaches to create a wide variety of mesh structures.

 E. Akleman, V. Srinivasan, and E. Mandal, "Remeshing Schemes for Semi-Regular Tilings", Proceedings of Shape Modeling International 2005, Boston, June 2005. (Paper) Description: Most frequently used subdivision schemes such as Catmull-Clark create regular regions after several application. This paper shows that all semi-regular regions can be created by subdivision schemes and each semi-regular region type can be created with one application of a particular subdivision scheme to a particular regular region. Using this property of subdivision schemes it is easy to cover any given surface with semi-regular tiles by applying one semi-regularity creating subdivision after several applications of a regularity creating subdivision

 E. Akleman, V. Srinivasan, Z. Melek and P. Edmundson, Semi-regular Pentagonal Subdivisions, Shape Modeling International 2004, Genoa, Italy, June 2004. (Paper) Description: Triangular and quadrilateral meshes are commonly used in computer graphics applications. In this paper, we analyze the topological existence of meshes that consist of n- sided faces where n is greater than 4 such as pentagonal and hexagonal meshes. We show that it is possible to represent any 2-manifold with a mesh that is made up of only pentagons. We also show that the meshes that consist of only polygons with more than five sides cannot represent all 2-manifolds. We present a pentagonalization (or pentagonal conversion) scheme that can create a pentagonal mesh from any arbitrary mesh structure. We also introduce a pentagonal preservation scheme that can create a pentagonal mesh from any pentagonal mesh.

 E. Akleman, P. Edmundson and O. Ozener, A Vertex Truncation Subdivision Scheme to Create Intriguing Polyhedra, Bridges: Mathematical Connections in Art, Music, and Science 2004, Winfield, Kansas, August 2004. (Paper) Description: In this paper, we present a new class of semi-regular polyhedra. All the faces of these polyhedra are bounded by smooth (quadratic B-spline) curves and the face boundaries are C1 discontinues everywhere. These semi-regular polyhedral shapes are limit surfaces of a simple vertex truncation subdivision scheme. We obtain an approximation of these smooth fractal polyhedra by iteratively applying a new vertex truncation scheme to an initial manifold mesh. Our vertex truncation scheme is based on Chaikin’s construction. If the initial manifold mesh is a polyhedra only with planar faces and 3-valence vertices, in each iteration we construct polyhedral meshes in which all faces are planar and every vertex is 3-valence.

 E. Akleman and V. Srinivasan, "Honeycomb Subdivision", Proceedings of ISCIS'02, 17th International Symposium on Computer and Information Sciences, pp. 137-141, November 2002, Orlando, Florida. (Paper) Description: In this paper, we introduce a new subdivision scheme which we call honeycomb subdivision. After one iteration of the scheme each vertex becomes exactly 3-valent and with consecutive applications regular regions strongly resembles a honeycomb. This scheme can be considered as a dual for triangle schemes. The major advantage of the new scheme is that it creates a natural looking mesh structure. We call this scheme honeycomb since the resulting meshes strongly resemble honeycombs, which is defined as a structure of hexagonal, thin-walled cells constructed from beeswax by honeybees to hold honey and larvae or something resembling this structure in configuration or pattern.

 E. Landreneau, E. Akleman and V. Srinivasan, "Local Mesh Operators: Extrusions Revisited", Proceedings of Shape Modeling International 2004, 2005, Boston, June 2005. (Paper) Description: In this paper, we present a set of generalized “local” mesh operators. Local operators are those that operate on a single face without affecting the rest of the mesh. Boundary edges of the chosen face also stay the same. We have identified two types of local operators: (1) Extrusions that create generalized pipes in which bottom and top polygons have the same number of sides, and (2) Stellations that create generalized pyramids, where there is a top vertex instead of top polygon. Our operators can create extrusions that are regular polyhedra including dodecahedron, icosahedron, octahedron and tetrahedron. The tetrahedron is created using the stellation operator, which is also useful to create generalized versions of Kepler and Poinsot solids. Using these extrusions unusual shapes can be created without changing the genus. The paper also shows how to create non-triangular planar meshes with extrusions.

 E. Landreneau, E. Akleman and J. Keyser, "Iterative Face Replacements for Modeling Detailed Shapes", Proceedings of Geometry, Modeling and Processing 2006, Pittsburg. (Paper) Description: In this paper, we present a method that allows novice users to interactively create partially self-similar manifold surfaces without relying on shape grammars or fractal methods. Moreover, the surfaces created using our method are connected. The modelers that are based on traditional fractal methods or shape grammars usually create disconnected surfaces and restrict the creative freedom of users. In most cases, the shapes are defined by hard-coded schemes that provide only a few parameters that can be adjusted by the users. We present a new approach for modeling such shapes. With this approach, novice users can interactively create a variety of unusual and interesting partially selfsimilar manifold surfaces.

 E. Akleman, A. Kaur and L. Green, Tiled Textures: What if Miro Painted a Sphere, ISAMA'2008, (Paper) Description: We present a simple and practical technique for seamlessly texturing quadrilateral meshes. Using our technique any image can be converted to an isotropic texture that can be mapped to any quadrilateral mesh without any discontinuity or singularity. Using our technique, we can make any abstract painter like Miro to seamlessly paint any smooth manifold surface. The surface can have any number of holes or handles. Our texturing method is to organize a set of tiles that satisfy specific boundary conditions into one texture image file which is called a tiled texture. We have also developed an algorithm to create tiled textures from any image with a simple user interface that allows the users to specify the boundaries. Based on tiled textures, we have developed an extremely simple texture mapping algorithm that assigns one tile to every quadrilateral in any given quadrilateral mesh. Our mapping algorithm provides aperiodicity on the surface of the mesh and yields singularity free textures regardless of the singularities existing in the quadrilateral mesh

 E. Akleman, J. Chen, B. Meric, "Symmetric Tile Design", pp. 283-292 Proceedings of ACADIA 2000, Washington, DC., October 2000. (Paper) Description: This paper presents a new approach for intuitive and effective design of periodic symmetric tiles. We observe that planar graphs can effectively represent symmetric tiles and graph drawing provides an intuitive paradigm for designing symmetric tiles. Moreover, based on our theoretical work to represent hexagonal symmetry by rectangular symmetry, we are able to present all symmetric tiles as graphs embedded on a torus and based on simple modulo operations. This approach enables us to develop a simple and efficient algorithm, which has been implemented in Java. By using this software, designers, architects and artists can create interesting symmetric tiles directly on the web. We also have designed a few examples of symmetric tiles to show the effectiveness of the approach.

 E. Akleman, "Twirling Sculptures", Journal of Mathematics and Arts, vol. 3, no. 1, pp. 1-10, 2009. (Paper) Description: In this paper, I outline a method for constructing aesthetically pleasing sculptures containing spiral shapes. Since every face of an orientable manifold mesh can be given a consistent edge rotation ordering, if one applies an extrusion operator to each face of an orientable manifold mesh using the same rotation and scaling factors, each edge in the original mesh will be converted to an S-shaped region that consists of two spiral arms. The twirling nature of my sculptures results from these S-shaped regions. The nal sculptures are obtained by smoothing the resulting shapes with a subdivision scheme. I discuss several methods for visually emphasizing the twirling nature of S-shaped regions. All the models and virtual sculptures in this paper are created using the Topological Mesh Modeling system TopMod.

 Q. Xing, G. Esquivel and E. Akleman, "Twisted D-Forms: Design and Construction of D-Forms with Twisted Prismatic Handles with Developable Sides", Proceedings of Bridges 2012. (Paper) Description: In this work, we present shapes that are constructed from a set of twisted papers, which we call Twisted D-forms. These shapes consists of twisted prismatic handles with developable sides. We design these handles using the handle creation tool in TopMod. The handle creation tool allows designing twisted handles that consists of strips of long triangles. Using this approach it is possible to design shapes of high genus. This initial triangulated model let us do minor modifications in the designs using commercial software such as Maya without destroying the developable property. We constructed a large number of small scale prototypes using paper.

 Ozgur Gonen, Ergun Akleman and Vinod Srinivasan, "Modeling D-Forms", Proceedings of Bridges 2008. (Paper) Description: Recently, very interesting developable sculptures, called Dforms, were invented by the London designer Tony Wills. D-forms are created by joining the edges of a pair of sheet metal or paper with the same perimeter. Despite its power to construct unusual shapes easily, there are two problems with physical D-form construction. First, the physical construction is limited to only two pieces. It is hard to figure out the perimeter relationships if we try to use more than two pieces. Second problem with D-form construction is that until we finalize the physical construction of the shape we do not exactly know what kind of the shape to be constructed. In this paper we introduce a computation method that provides an alternative to physical D-form construction. Using our method, D-forms can directly be designed with our software. Our D-forms can consist of more than two pieces. Another advantage of our method is that before physical construction of the shape we exactly know what kind of shape to be constructed.

 Jace Miller and E. Akleman, "Edge-Based Intersected Polyhedral Paper Sculptures Constructed by Interlocking Slitted Planar Pieces", Proceedings of Bridges 2008. (Paper) Description: In this paper, we generalize George Hart’s slide-together sculptures as edge-based intersected polyhedral paper sculptures. Edge-based intersected polyhedra are also a conceptual generalization of Kepler’s Small Stellated Dodecahedron. These sculptures are constructed by interlocking slitted planar pieces without using glue. We present a simple procedure to construct slitted planar pieces for any given polyhedron. These sculptures can easily be constructed by children and can be used to teach properties of Platonic or Archimedean Solids through hands-on experience.

 Yutu Liu, Hernan Molina and E. Akleman, "Inout Sculptures", Proceedings of Bridges 2007. (Paper) Description: The people innately find a mysterious beauty in sculptures with smooth saddle regions that exists in hyperbolic sculptures. Well-known examples hyperbolic sculptures are Robert Longhurst’s Arabesque 29th and Brent Collins’s hyperbolic sculptures with many smooth holes and handles. We present a new method to create a new set of hyperbolic sculptures, which we call inout sculptures. Our idea is simply to simultaneously show both inside and outside of an already complicated shape that contains many holes. These sculptures are obtained by showing both inside and outside of a shape with holes. Inout sculptures looks interesting since they allow to simultaneously view both inside and outside of complicated shapes.

 Vinod Sribnivasan, Hernan Molina and E. Akleman, "Multiple Handle Creation and Multiple Hole Opening", Proceedings of Bridges 2007. (Paper) Description: In this paper, we present the concept of multiple handle operation to create complicated high genus virtual sculptures. We have developed and implemented a simple procedure to create multiple handles that connect a set of faces in 3D. To create multiple handles, we first create a connector, which is a convex shaped mesh surface. We then simply connect each selected face to this connector surface with a simple one segment handle. If the connector is inside of the original mesh and the handles goes through the inside of the objects, the result becomes multiple hole.

 E. Akleman, "Designing Symmetric High-Genus Sculptures", Siggraph'2006 Art Exhibition and Presentation. (Extended Abstract) (paper) Description: In this paper, we present a procedure to create a new sculptural family with interactive topological modeling. Using this procedure a large set of sculptures that have a similar conceptual form can easily be created. We have tested the procedure in a computer aided sculpting course. We observe that, using the procedure, students can rapidly create a wide variety of shape. Although these shapes are completely different; they are indistinguishably belong the same family.

 Architectural Applications: Applications of Topological Modeling to Design and Architecture

 E. Akleman, O. Ozener and C. Yuksel, "Designing Symmetric High-Genus Sculptures, Proceedings of Bridges 2006, London . (Paper) Description: This paper introduces a design guideline to construct a family of symmetric, connected sculptures with high number of holes and handles. Our guideline provides users a creative flexibility. Using this design guideline, sculptors can easily create a wide variety of sculptures with a similar conceptual form.

 E. Akleman, O. Ozener and V. Srinivasan, Rind Architecture, International Journal of Architectural Computing, 2005 (Paper) Selected from ECAADE 2004 paper by O. Ozener, E. Mandal and E. Akleman, "Rind Modeling for Architectural Design", Education and Research in Computer Aided Architectural Design in Europe: EcaadE'04, Copenhagen, Denmark, September 2004. (Paper) Description: This paper presents a new modeling technique for architectural design. Rind modeling provides for the easy creation of surfaces resembling peeled and punctured rinds. We show how the method‘s two main steps of 1) creation of a shell or crust 2) opening holes in the crust by punching or peeling can be encapsulated into a real time semi-automatic interactive algorithm. We include a number of worked examples, some by students in a special modeling workshop that demonstrate the ease with which a large variety of intricate rind shapes can be created. Rind modeling method allows us developing a user-friendly tool for designers and architects. The new tool extends the abilities of polygonal modeling and allows designers to work on structured and consistent models for architectural design purposes. Rind modeling gives architects and designers a processing flexibility.

 V. Srinivasan, O. Ozener, E. Mandal and E. Akleman, Solidfying Frames For Architecture, CAAD FUTURES 2005, Vien, Austria, June 2005. (Paper) Description: In this paper we present a method to convert a wireframe mesh into a 2-manifold mesh consisting of cylindrical pipes in place of the edges and joints in place of the vertices in the original mesh. Our method allows users to create unique artistic depictions of common objects and structures. The resulting mesh is also more effective at conveying the overall 3D structure and any internal elements of a model when compared to regular wireframe or boundary representations. The input wireframe mesh can be any collection of linear edges; they do not have to form a manifold surface or even be connected to each other. The result is always an orientable 2-manifold surface. Our algorithm replaces every edge in the wireframe mesh with a cylindrical 3D pipe. The pipes are connected to each other using 3D joints created at the vertices in the wireframe where the edges meet. Our method has been implemented as part of a polygonal mesh modeling system and has been used to create artistic models of popular architectural structures as well as to create conceptual sketches for virtual environments.

 E. Akleman, J. Chen and V. Srinivasan, "An Interactive Shape Modeling System for Robust Design of Functional 3D Shapes", pp. 248-257 Proceedings of ACADIA 2001, Buffalo, NW., October 2001. (Paper) Description: In Architecture, it is essential to design functional and topologically complicated 3D shapes (i.e. shapes with many holes, columns and handles). In this paper, we present a robust and interactive system for the design of functional and topologically complicated 3D shapes. Users of our system can easily change topology (i.e. they can create and delete holes and handles, connect and disconnect surfaces). Our system also provide smoothing operations (subdivision schemes) to create smooth surfaces. Moreover, the system provides automatic texture mapping during topology and smoothing operations. We also present new design approaches with the new modeling system. The new design approaches include blending surfaces, construction of crusts and opening holes on these crusts.

 Topological Repair and Simplification

 V. Srinivasan, E. Akleman and J. Keyser, Topological Construction of 2-Manifold Meshes from Arbitrary Polygonal Data, Technical Report, January 2004. (Paper) Description: In this paper we present a simple algorithm to construct 2-manifold meshes from arbitrary collections of polygons. We form our final data structure using two very basic manifold-preserving operations, thus guaranteeing that the result is a valid manifold. Our algorithm is purely topological and does not consider the geometric properties of the underlying shape. The algorithm automatically and correctly creates the missing faces of manifolds with boundaries. It also eliminates all twogons (2-sided polygons) and converts nonmanifold meshes into one of the possible manifold interpretations. We have implemented this algorithm and we highlight the performance of our algorithm on a number of sample models.

 E. Akleman and J. Chen, Progressive Refinement with Topological Simplification, Technical Report, January 2003. (Paper) Description: This paper presents a theoretical framework for progressive refinement of manifold meshes with topological simplification. We demonstrate that topology changes are not intuitive and therefore a great deal of care is necessary for handling topological simplification. We illustrate nonintutive nature of topology changes with several examples. We also show how to use the non-intutive nature of the topology changes as an advantage and develop a theretical framework for progressive refinement with topological simplification.

 These two manuscripts presents TopMod3D (a.k.a. TopMod). The concepts and algorithms behind the tools in TopMod have been developed, implemented and published by our research group that can be seen above. Many of these tools are unique to TopMod.

 E. Akleman, V. Srinivasan, J. Chen, David Morris, and Stuart Tett, "TopMod3D: An Interactive Topological Mesh Modeler", Computer Graphics International 2008, pp. 10-18. (Paper) Description: This is a description of TopMod 2.0. In August 2007, we released a new version, TopMod 2.0, with an improved user interface and scripting editor. For the interface of the new version, we switched from FLTK to Qt. The new version also runs on Mac, Linux and Windows platforms.We have also developed a web-site to create a community around the software. This experience is a strong example of the impor- tance of creation of a community for the useability of software. For instance, many people discovered ways to create unusually interesting shapes and shared their ex- periences by developing video tutorials. Other users, fol- lowing video tutorials, created similar shapes. Having a community also helped to solve portability problems. For instance, the script editor was initially developed on the Mac platform and we had trouble compiling the code for Windows. One user from Italy provided a solution to this problem. Another user from France translated the user interface from English to French. The model is in the left is created by one of the users, Jonathan Johanson from Germany.

 E. Akleman, V. Srinivasan, E. Mandal, J. Chen, Z. Melek, and E. Lendreneau, "Topmod: Topological Mesh Modeling System", Technical Report, Aug. 2004. (Report) Description: This is description of TopMod 1.0. The initial version of the software, TopMod 1.0, has been available as free software since 2003. Since then, several talented artists created very interesting sculptures using TopMod 1.0. TopMod 1.0 was implemented in C++ using OpenGL and FLTK. It runs on Mac, Linux and Windows platforms. This initial version, although was not user-fiendly, was discovered by a few designers and they created interesting models. The image in the left was created by Torolf Sauermann from Germany