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31
Computational Topology: Ambient Isotopic Approximation of 2Manifolds
 Theoretical Computer Science
, 2001
"... A fundamental issue in theoretical computer science is that of establishing unambiguous formal criteria for algorithmic output. This paper does so within the domain of computeraided geometric modeling. For practical geometric modeling algorithms, it is often desirable to create piecewise linear appr ..."
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Cited by 28 (14 self)
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A fundamental issue in theoretical computer science is that of establishing unambiguous formal criteria for algorithmic output. This paper does so within the domain of computeraided geometric modeling. For practical geometric modeling algorithms, it is often desirable to create piecewise linear approximations to compact manifolds embedded in and it is usually desirable for these two representations to be "topologically equivalent". Though this has traditionally been taken to mean that the two representations are homeomorphic, such a notion of equivalence suffers from a variety of technical and philosophical difficulties; we adopt the stronger notion of ambient isotopy. It is shown here, that for any , compact, 2manifold without boundary, which is embedded in , there exists a piecewise linear ambient isotopic approximation. Furthermore, this isotopy has compact support, with specific bounds upon the size of this compact neighborhood. These bounds may be useful for practical application in computer graphics and engineering design simulations. The proof given relies upon properties of the medial axis, which is explained in this paper. Email address: amenta@cs.utexas.edu. This author acknowledges, with appreciation, funding from the National Science Foundation under grant number CCR9731977, and a research fellowship from the Alfred P. Sloan Foundation. The views expressed herein are those of the author, not of these sponsors.
Polyhedral Perturbations That Preserve Topological Form
, 1995
"... The idea, that we are willing to accept variation in an object but that we insist it should retain its original topological form, has powerful intuitive appeal, and the concept appears in many applied fields. Some of the most important of these are tolerancing and metrology, solid modeling, engineer ..."
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Cited by 21 (14 self)
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The idea, that we are willing to accept variation in an object but that we insist it should retain its original topological form, has powerful intuitive appeal, and the concept appears in many applied fields. Some of the most important of these are tolerancing and metrology, solid modeling, engineering design, finite element analysis, surface reconstruction, computer graphics, path planning in robotics, fairing procedures, image analysis, and medical imaging. In this paper we focus on the field of tolerancing and metrology. The requirement that two objects or sets should have the same topological form requires a precise definition. We specify "same topological form" to mean that there exists a "space homeomorphism" from R 3 onto R 3 that carries a nominal object S onto another design object. In general, establishing the existence of such space homeomorphisms can be considerably more difficult than demonstrating classical topological equivalence by a homeomorphism. In the special ca...
Tutte’s barycenter method applied to isotopies
 Computational Geometry: Theory and Applications
, 2001
"... This paper is concerned with applications of Tutte’s barycentric embedding theorem (Proc. London Math. Soc. 13 (1963), 743–768). It presents a method for building isotopies of triangulations in the plane, based on Tutte’s theorem and the computation of equilibrium stresses of graphs by Maxwell–Cremo ..."
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Cited by 10 (0 self)
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This paper is concerned with applications of Tutte’s barycentric embedding theorem (Proc. London Math. Soc. 13 (1963), 743–768). It presents a method for building isotopies of triangulations in the plane, based on Tutte’s theorem and the computation of equilibrium stresses of graphs by Maxwell–Cremona’s theorem; it also provides a counterexample showing that the analogue of Tutte’s theorem in dimension 3 is false.
Computational topology for reconstruction of surfaces with boundary: integrating experiments and theory
 Proceedings of the IEEE International Conference on Shape Modeling and Applications, June 15 17, 2005
, 2005
"... Abstract. This paper presents new mathematical foundations for topologically correct surface reconstruction techniques that are applicable to 2manifolds with boundary, where provable techniques previously had been limited to surfaces without boundary. This is done by an intermediate construction of ..."
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Cited by 9 (4 self)
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Abstract. This paper presents new mathematical foundations for topologically correct surface reconstruction techniques that are applicable to 2manifolds with boundary, where provable techniques previously had 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 it is shown that its envelope is C 1,1 and this envelope can be reconstructed with topological guarantees. 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 mathematical proofs needed for these extensions, where more practical applications and examples are presented in a companion paper.
A Digital Index Theorem
 Int. J. Pattern Recog. Art. Intell
, 2000
"... Abstract This paper is devoted to prove a Digital Index Theorem for digital (n − 1)manifolds in a digital space (R n, f), where f belongs to a large family of lighting functions on the standard cubical decomposition R n of the ndimensional Euclidean space. As an immediate consequence we obtain the ..."
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Cited by 5 (5 self)
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Abstract This paper is devoted to prove a Digital Index Theorem for digital (n − 1)manifolds in a digital space (R n, f), where f belongs to a large family of lighting functions on the standard cubical decomposition R n of the ndimensional Euclidean space. As an immediate consequence we obtain the corresponding theorems for all (α, β)surfaces of KongRoscoe, with α, β ∈ {6, 18, 26} and (α, β) �= (6, 6), (18, 26), (26, 26), as well as for the strong 26surfaces of BertrandMalgouyres.
Computational topology for geometric design and molecular design
 in Mathematics for Industry: Challenges and Frontiers
, 2005
"... The nascent field of computational topology holds great promise for resolving several longstanding industrial design modeling challenges. Geometric modeling has become commonplace in industry as manifested by the critical use of Computer Aided Geometric Design (CAGD) systems within the automotive, ..."
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Cited by 5 (2 self)
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The nascent field of computational topology holds great promise for resolving several longstanding industrial design modeling challenges. Geometric modeling has become commonplace in industry as manifested by the critical use of Computer Aided Geometric Design (CAGD) systems within the automotive, aerospace, shipbuilding and consumer product industries. Commercial CAGD packages depend upon complementary geometric and topological algorithms. The emergence of geometric modeling for molecular simulation and pharmaceutical design presents new challenges for supportive topological software within Computer Aided Molecular Design (CAMD) systems. For both CAGD and CAMD systems, splines provide relatively mature geometric technology. However, there remain pernicious issues regarding the ‘topology ’ of these models, particularly for support of robust simulations which rely upon the topological characteristics of adjacency, connectivity and nonselfintersection. This paper presents current challenges and frontiers for reliable simulation and approximation of topology for geometric models. The simultaneous consideration of CAGD and CAMD is important to provide unifying abstractions to benefit both domains. In engineering applications it is a common requirement that topological equivalence be preserved during geometric modifications, but in molecular simulations attention is focused upon where topological changes have occurred as indications of important chemical changes. The methods presented here are supportive of both these disciplinary approaches.
Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the DenjoyRees technique
, 2008
"... In [23], Mary Rees has constructed a minimal homeomorphism of the 2torus with positive topological entropy. This homeomorphism f is obtained by enriching the dynamics of an irrational rotation R. We improve Rees construction, allowing to start with any homeomorphism R instead of an irrational rotat ..."
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Cited by 5 (3 self)
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In [23], Mary Rees has constructed a minimal homeomorphism of the 2torus with positive topological entropy. This homeomorphism f is obtained by enriching the dynamics of an irrational rotation R. We improve Rees construction, allowing to start with any homeomorphism R instead of an irrational rotation and to control precisely the measurable dynamics of f. This yields in particular the following result: Any compact manifold of dimension d ≥ 2 which carries a minimal uniquely ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism with positive topological entropy. More generally, given some homeomorphism R of a (compact) manifold and some homeomorphism hC of a Cantor set, we construct a homeomorphism f which “looks like ” R from the topological viewpoint and “looks like ” R × hC from the measurable viewpoint. This construction can be seen as a partial answer to the following realisability question: which measurable dynamical systems are represented by homeomorphisms on manifolds?
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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Cited by 4 (1 self)
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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.
Propagating Topological Tolerances for Rapid Prototyping
, 1996
"... Rapid prototyping technologies permit timely inclusion of experimental verification within an iterative design process. This work introduces topological tolerances as a tool for retaining desired design invariance during the production of successively refined prototypes. These topological tolerance ..."
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Cited by 3 (3 self)
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Rapid prototyping technologies permit timely inclusion of experimental verification within an iterative design process. This work introduces topological tolerances as a tool for retaining desired design invariance during the production of successively refined prototypes. These topological tolerances are applied to the supporting geometric data representations for rapid prototyping. At each successive local modification of the geometric data, these topological tolerances must be dynamically updated. These updates are modeled by objectoriented propagations, leading to investigation of efficient algorithms to recompute these local topological tolerances. Relevant industrial examples will be discussed. Software to compute topological tolerances has been fully implemented and tested, and one purpose of this paper is to report that development. Another emphasis of this paper is to report upon software prototyping of performance enhancements to adapt this code to support iterative redesig...