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Polygonization of NonManifold Implicit Surfaces
, 1995
"... A method is presented to broaden implicit surface modeling. The implicit surfaces usually employed in computer graphics are two dimensional manifolds because they are defined by realvalued functions that impose a binary regionalization of space (i.e., an inside and an outside). When tiled, these su ..."
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A method is presented to broaden implicit surface modeling. The implicit surfaces usually employed in computer graphics are two dimensional manifolds because they are defined by realvalued functions that impose a binary regionalization of space (i.e., an inside and an outside). When tiled, these surfaces yield edges of degree two. The new method allows the definition of implicit surfaces with boundaries (i.e., edges of degree one) and intersections (i.e., edges of degree three or more). These nonmanifold implicit surfaces are defined by a multiple regionalization of space. The definition includes a list of those pairs of regions whose separating surface is of interest. Also presented is an implementation that converts a nonmanifold implicit surface definition into a collection of polygons. Although following conventional implicit surface polygonization, there are significant differences that are described in detail. Several example surfaces are defined and polygonized. CR Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling  Curve, Surface, Solid, and Object Representations. Additional Keywords and Phrases: Implicit Surface, NonManifold, Polygonization. 1
Modeling and Designing Functionally Graded Material Components for Fabrication with Local Composition Control
, 1999
"... Solid Freeform Fabrication #SFF# processes have demonstrated the ability to produce parts with locally controlled composition. In the limit, processes such as 3D Printing can create parts with composition control on a length scale of 100 #m. To exploit this potential, new methods to model, exchange, ..."
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Cited by 28 (4 self)
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Solid Freeform Fabrication #SFF# processes have demonstrated the ability to produce parts with locally controlled composition. In the limit, processes such as 3D Printing can create parts with composition control on a length scale of 100 #m. To exploit this potential, new methods to model, exchange, and process parts with local composition control need to be developed. An approach to modeling a part's geometry, topology, and composition is presented. This approach is based on subdividing the solid model into subregions and associating analytic composition blending functions with each region. These blending functions de#ne the composition throughout the model as mixtures of the primary materials available to the SFF machine. Design tools basedupon distance functions are also introduced, such as the speci#cation of composition as a function of the distance from the surface of a part. Finally, the role of design rules restricting maximum and minimum concentrations is discussed. Introduc...
Nonmanifold Modeling: An Approach Based on Spatial Subdivision
, 1997
"... This paper deals with the problem of creating and maintaining a spatial ..."
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Cited by 22 (7 self)
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This paper deals with the problem of creating and maintaining a spatial
Maintenance Of Geometric Representations Through Space Decompositions
, 1997
"... The ability to transform between distinct geometric representations is the key to success of multiplerepresentation modeling systems. But the existing theory of geometric modeling does not directly address or support construction, conversion, and comparison of geometric representations. A study of ..."
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Cited by 11 (5 self)
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The ability to transform between distinct geometric representations is the key to success of multiplerepresentation modeling systems. But the existing theory of geometric modeling does not directly address or support construction, conversion, and comparison of geometric representations. A study of classical problems of CSG $ brep conversions, CSG optimization, and other representation conversions suggests a natural relationship between a representation scheme and an appropriate decomposition of space. We show that a hierarchy of space decompositions corresponding to different representation schemes can be used to enhance the theory and to develop a systematic approach to maintenance of geometric representations. 1. Motivation 1.1. Modern theory of representations The modern field of solid modeling owes much of its success to the theoretical foundations laid by members of the Production Automation Project at the University of Rochester in the 1970's. The history of these development...
Boolean operations with implicit and parametric representation of primitives using Rfunctions
 IEEE Transactions, on Visualization and Computer Graphics
, 2005
"... Abstract—We present a new and efficient algorithm to accurately polygonize an implicit surface generated by multiple Boolean operations with globally deformed primitives. Our algorithm is special in the sense that it can be applied to objects with both an implicit and a parametric representation, su ..."
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Abstract—We present a new and efficient algorithm to accurately polygonize an implicit surface generated by multiple Boolean operations with globally deformed primitives. Our algorithm is special in the sense that it can be applied to objects with both an implicit and a parametric representation, such as superquadrics, supershapes, and Dupin cyclides. The input is a Constructive Solid Geometry tree (CSG tree) that contains the Boolean operations, the parameters of the primitives, and the global deformations. At each node of the CSG tree, the implicit formulations of the subtrees are used to quickly determine the parts to be transmitted to the parent node, while the primitives ’ parametric definition are used to refine an intermediary mesh around the intersection curves. The output is both an implicit equation and a mesh representing its solution. For the resulting object, an implicit equation with guaranteed differential properties is obtained by simple combinations of the primitives ’ implicit equations using Rfunctions. Depending on the chosen Rfunction, this equation is continuous and can be differentiable everywhere. The primitives ’ parametric representations are used to directly polygonize the resulting surface by generating vertices that belong exactly to the zeroset of the resulting implicit equation. The proposed approach has many potential applications, ranging from mechanical engineering to shape recognition and data compression. Examples of complex objects are presented and commented on to show the potential of our approach for shape modeling. Index Terms—Computational geometry and object modeling, constructive solid geometry, object representation, volume visualization, Rfunctions, superquadrics, supershapes, Dupin cyclides. æ
Blist: A Boolean list formulation of CSG trees
, 1998
"... Set membership classification algorithms visit nodes of a CSG tree through a recursive divideandconquer process, which stores intermediate results in a stack, whose depth equals the height, H, of the tree. During this process, the candidate sets is usually subdivided into uniform cells, whose inte ..."
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Cited by 8 (2 self)
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Set membership classification algorithms visit nodes of a CSG tree through a recursive divideandconquer process, which stores intermediate results in a stack, whose depth equals the height, H, of the tree. During this process, the candidate sets is usually subdivided into uniform cells, whose interior is disjoint from primitives' boundaries. Cells inside the CSG object are identified by combining the binary results of classifying them against the primitives. In parallel systems, which allocate a different process to each leaf of the tree, and in algorithms that classify large collections of regularly spaced candidate sets (points, pixels, voxels, rays, or crosssections) against the primitives using forward differences, a separate stack is associated with each candidate or cell. Our new representation for CSG trees, called Blist, distributes the merging operation to the primitives and reduces the storage requirement for each cell to log(H+1) bits. Blist can represent any Boolean expr...
Solid modeling
 Handbook of Computer Aided Geometric Design
, 2002
"... inversion for a global shear velocity ..."
Simplification and Compression of 3D Scenes
, 1997
"... INTRODUCTION....................................................................................4 2. A SIMPLE DATASTRUCTURE FOR TRIANGULATED MESHES.................................6 3. TOPOLOGICAL CHARACTERIZATION OF POLYHEDRA........................................7 3.1 TOPOLOGICAL CONCEPTS AND DE ..."
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INTRODUCTION....................................................................................4 2. A SIMPLE DATASTRUCTURE FOR TRIANGULATED MESHES.................................6 3. TOPOLOGICAL CHARACTERIZATION OF POLYHEDRA........................................7 3.1 TOPOLOGICAL CONCEPTS AND DEFINITIONS .......................................................................................7 3.1.1 Topological closure, interior, and boundary............................................................................7 3.1.2 Dimensional homogeneity.................................................................................................8 3.1.3 Regularization and Boolean operations..................................................................................8 3.1.4 Connectedness, holes, and handles.......................................................................................9 3.1.5 NonManifold conditions..............................................................
Feature Methodologies For Heterogeneous Object Realization
 Software for rapid prototyping. UK: Professional Engineering Publishing; 2002. p. 225–61 (Chapter 7
, 2001
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