Results 1 
4 of
4
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 ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
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.
MODELING TIME AND TOPOLOGY FOR ANIMATION AND VISUALIZATION
"... Abstract. The art of animation relies uopn modeling objects that change over time. A sequence of static images is displayed to produce an illusion of motion, which is frequently trusted to be topologically meaningful. A careful analysis exposes that formal topological guarantees are often lacking. T ..."
Abstract
 Add to MetaCart
Abstract. The art of animation relies uopn modeling objects that change over time. A sequence of static images is displayed to produce an illusion of motion, which is frequently trusted to be topologically meaningful. A careful analysis exposes that formal topological guarantees are often lacking. This lack of formal justification can lead to subtle, but significant, flaws regarding topological integrity. A modified approach is proposed that integrates topological rigor with a continuous model of time. Examples will be given for splines widely used in many applications, with particular emphasis upon scientific visualization for molecular modeling. Moreover, the approach of choosing a family of functions and studying their topological properties over time should be broadly applicable to other domains. Prototype animations are available for viewing over the web.
www.elsevier.com/locate/tcs Computational topology:ambient isotopic approximation of 2manifolds
"... 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 app ..."
Abstract
 Add to MetaCart
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 R 3, 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 su ers from a variety of technical and philosophical di culties; we adopt the stronger notion of ambient isotopy. It is shown here, that for any C 2, compact, 2manifold without boundary, which is embedded in R 3, there exists a piecewise linear ambient isotopic approximation. Furthermore, this isotopy has compact support, with speci c 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. c ○ 2002 Elsevier B.V. All rights reserved.
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 ..."
Abstract
 Add to MetaCart
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