Computational topology for approximations of knots

• Li, Ji ^[1] ; Peters, T. J. ^[1] ; Jordan, K. E. ^[2]
  1. [1] University of Connecticut

     University of Connecticut

     Town of Mansfield, Estados Unidos

     
  2. [2] Cambridge Research Center
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 15, Nº. 2, 2014, págs. 203-220
• Idioma: inglés
• DOI: 10.4995/agt.2014.2281
• Enlaces
  • Texto completo
• Resumen

  • The preservation of ambient isotopic equivalence under piecewise linear (PL) approximation for smooth knots are prominent in molecular modeling and simulation. Sufficient conditions are given regarding:Hausdorff distance, anda sum of total curvature and derivative.High degree Bézier curves are often used as smooth representations, where computational efficiency is a practical concern. Subdivision can produce PL approximations for a given B\'ezier curve, fulfilling the above two conditions. The primary contributions are:       (i) a priori bounds on the number of subdivision iterations sufficient to achieve a PL approximation that is ambient isotopic to the original B\'ezier curve, and       (ii) improved iteration bounds over those previously established. 

• Referencias bibliográficas
  • N. Amenta, T. J. Peters and A. C. Russell, Computational topology: Ambient isotopic approximation of 2-manifolds, Theoretical Computer Science...
  • (http://dx.doi.org/10.1016/S0304-3975(02)00691-6)
  • L. E. Andersson, S. M. Dorney, T. J. Peters and N. F. Stewart, Polyhedral perturbations that preserve topological form, CAGD 12, no. 8 (1995),...
  • (http://dx.doi.org/10.1016/0167-8396(94)00039-7)
  • M. Burr, S. W. Choi, B. Galehouse and C. K. Yap, Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves, Journal...
  • (http://dx.doi.org/10.1016/j.jsc.2011.08.021)
  • F. Chazal and D. Cohen-Steiner, A condition for isotopic approximation, Graphical Models 67, no. 5 (2005), 390-404.
  • (http://dx.doi.org/10.1016/j.gmod.2005.01.005)
  • W. Cho, T. Maekawa and N. M. Patrikalakis, Topologically reliable approximation in terms of homeomorphism of composite Bézier curves, CAGD...
  • (http://dx.doi.org/10.1016/0167-8396(95)00042-9)
  • E. Denne and J. M. Sullivan, Convergence and isotopy type for graphs of finite total curvature, In: A. I. Bobenko, J. M. Sullivan, P. Schröder,...
  • G. E. Farin, Curves and surfaces for computer-aided geometric design: A practical code, Academic Press, Inc., 1996.
  • M. W. Hirsch, Differential topology, Springer, New York, 1976.
  • (http://dx.doi.org/10.1007/978-1-4684-9449-5)
  • K. E. Jordan, L. E. Miller, T. J. Peters and A. C. Russell, Geometric topology and visualizing 1-manifolds, In: V. Pascucci, X. Tricoche,...
  • (http://dx.doi.org/10.1007/978-3-642-15014-2_1)
  • J. Li, Topological and isotopic equivalence with applications to visualization, PhD thesis, University of Connecticut, U.S., 2013.
  • J. Li and T. J. Peters, Isotopic convergence theorem, Journal of Knot Theory and Its Ramifications 22, no. 3 (2013).
  • (http://dx.doi.org/10.1142/S0218216513500120)
  • J. Li, T. J. Peters, D. Marsh and K. E. Jordan, Computational topology counterexamples with 3D visualization of Bézier curves, Applied General...
  • J. Li, T. J. Peters and J. A. Roulier, Isotopy from Bézier curve subdivision, preprint.
  • L. Lin and C. Yap, Adaptive isotopic approximation of nonsingular curves: the parameterizability and nonlocal isotopy approach, Discrete &...
  • (http://dx.doi.org/10.1007/s00454-011-9345-9)
  • T. Maekawa, N. M. Patrikalakis, T. Sakkalis and G. Yu, Analysis and applications of pipe surfaces, CAGD 15, no. 5 (1998), 437-458.
  • (http://dx.doi.org/10.1016/S0167-8396(97)00042-3)
  • D. D. Marsh and T. J. Peters, Knot and Bézier curve visualizing tool.
  • (http://www.cse.uconn.edu/ tpeters/top-viz.html)
  • L. E. Miller, Discrepancy and Isotopy for Manifold Approximations, PhD thesis, University of Connecticut, U.S., 2009.
  • J. W. Milnor, On the total curvature of knots, Annals of Mathematics 52 (1950), 248-257.
  • G. Monge, Application de l'analyse à la géométrie, Bachelier, Paris, 1850.
  • E. L. F. Moore, T. J. Peters and J. A. Roulier, Preserving computational topology by subdivision of quadratic and cubic Bézier curves, Computing...
  • (http://dx.doi.org/10.1007/s00607-006-0208-9)
  • G. Morin and R. Goldman, On the smooth convergence of subdivision and degree elevation for Bézier curves, CAGD 18 (2001), 657-666.
  • (http://dx.doi.org/10.1016/S0167-8396(01)00059-0)
  • J. Munkres, Topology, Prentice Hall, 2nd edition, 1999.
  • D. Nairn, J. Peters and D. Lutterkort, Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon,...
  • (http://dx.doi.org/10.1016/S0167-8396(99)00026-6)
  • M. Reid and B. Szendroi, Geometry and Topology, Cambridge University Press, 2005.
  • (http://dx.doi.org/10.1017/CBO9780511807510)