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Computational topology for isotopic surface reconstruction
 Theoretical Computer Science 365 (3) (2006) 184
, 2006
"... Abstract. New computational topology techniques are presented for surface reconstruction of 2manifolds with boundary, while rigorous proofs have previously been limited to surfaces without boundary. This is done by an intermediate construction of the envelope (as defined herein) of the original sur ..."
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Abstract. New computational topology techniques are presented for surface reconstruction of 2manifolds with boundary, while rigorous proofs have previously been limited to surfaces without boundary. This is done by an intermediate construction of the envelope (as defined herein) of the original surface. For any compact C 2 manifold M embedded in R 3, it is shown that its envelope is C 1,1. Then it is shown that there exists a piecewise linear (PL) subset of the reconstruction of the envelope that is ambient isotopic to M, whenever M is orientable. The emphasis of this paper is upon the formal mathematical proofs needed for these extensions. (Practical application examples have already been published in a companion paper.) Possible extensions to nonorientable manifolds are also discussed. The mathematical exposition relies heavily on known techniques from differential geometry and topology, but the specific new proofs are intended to be sufficiently specialized to prompt further algorithmic discoveries.
Mathematical Sciences Letters An International Journal Essentially Copied Topological Spaces @ 2012 NSP Natural Sciences Publishing Cor.
"... Current progress of topology continues to show its applications in different disciplines such as chemistry, information systems, quantum physics, biology and dynamical systems, see [5], [6], [3] and [7]. The basic concept that lies behind all in topology is the idea of being homeomorphic, which fits ..."
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Current progress of topology continues to show its applications in different disciplines such as chemistry, information systems, quantum physics, biology and dynamical systems, see [5], [6], [3] and [7]. The basic concept that lies behind all in topology is the idea of being homeomorphic, which fits with the ideas of congruence and similarity. This idea plays an important role in different applications. For instance, in [4]
A DECOMPOSITION OF δOPEN FUNCTIONS
"... Abstract. In 2008, M. Caldas and G. Navalagi [1] introduced a new class of generalized open functions called weakly δopen functions. By introducing a new type of open functions called relatively weakly δopen together with weakly δopen functions, we establish a new decomposition of δopen function ..."
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Abstract. In 2008, M. Caldas and G. Navalagi [1] introduced a new class of generalized open functions called weakly δopen functions. By introducing a new type of open functions called relatively weakly δopen together with weakly δopen functions, we establish a new decomposition of δopen functions. 1.
Manuscript Adaptive Curve Approximation by Bending Energy
"... Let c denote a C 2 parametric curve c: [0, 1] → R 3.Itisshownthatc has an adaptive piecewise linear (PL) approximation that can be chosen optimally in terms ofthe bending energy. Furthermore, the approximation technique provides bounds upon the Hausdorff distance and the tangency differences betwee ..."
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Let c denote a C 2 parametric curve c: [0, 1] → R 3.Itisshownthatc has an adaptive piecewise linear (PL) approximation that can be chosen optimally in terms ofthe bending energy. Furthermore, the approximation technique provides bounds upon the Hausdorff distance and the tangency differences between the approximant and c. The method leads to efficient algorithms for a broad class of curves that includes splines, as well as many other representations used in computer graphics and animation. Some examples will be given. A novel contribution is the proofthat this method produces an asymptotically optimal number ofapproximating segments as the error bound goes to zero. This approach was motivated by robustness concerns for dynamic visualization in high performance computing environments. Key words: parametric curves, adaptive approximation, compactness, computational topology ∗ Corrresponding author.
Computing Fundamental Group of General 3manifold
"... Abstract. Fundamental group is one of the most important topological invariants for general manifolds, which can be directly used as manifolds classification. In this work, we provide a series of practical and efficient algorithms to compute fundamental groups for general 3manifolds based on CW cel ..."
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Abstract. Fundamental group is one of the most important topological invariants for general manifolds, which can be directly used as manifolds classification. In this work, we provide a series of practical and efficient algorithms to compute fundamental groups for general 3manifolds based on CW cell decomposition. The input is a tetrahedral mesh, while the output is symbolic representation of its first fundamental group. We further simplify the fundamental group representation using computational algebraic method. We present the theoretical arguments of our algorithms, elaborate the algorithms with a number of examples, and give the analysis of their computational complexity. Key words: computational topology, 3manifold, fundamental group, CWcell decomposition. 1
Block Meshes: Topologically Robust Shape Modeling with Graphs Embedded on 3Manifolds
, 2014
"... 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, ..."
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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 3space immersions of higherdimensionalmanifolds, which facilitate our goal of topological robustness. We demonstrate that a specific type of orientable 2manifold mesh, which we call a CMMpattern coverable mesh, can be used to represent structures in higherdimensional manifolds, which we call block meshes. Moreover, the framework includes a set of operations that can preserve CMMpattern coverability. In this sense, CMMpatterncoverable 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 CMMpatterncoverable meshes to CMMpatterncoverable meshes.