Topological Integrity for Dynamic Spline Models During Visualization of Big Data

  • Hugh P. Cassidy
  • Thomas J. Peters
  • Horea Ilies
  • Kirk E. Jordan
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


In computer graphics and scientific visualization, B-splines are common geometric representations. A typical display method is to render a piecewise linear (PL) approximation that lies within a prescribed tolerance of the curve. In dynamic applications it is necessary to perturb specified points on the displayed curve. The distance between the perturbed PL structure and the perturbed curve it represents can change significantly, possibly changing the underlying topology and introducing unwanted artifacts to the display. We give a strategy to perturb the curve smoothly and keep track of the error introduced by perturbations. This allows us to refine the PL curve when appropriate and avoid spurious topological changes. This work is motivated by applications to visualization of Big Data from simulations on high performance computing architectures.


Piecewise Linear Molecular Simulation Junction Point Piecewise Linear Approximation Perturbation Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors thank the referees, both for the conference presentation, as well as for this final book, for their helpful and constructive comments, which led to many improvements. The three UConn authors were partially supported by NSF grants CMMI 1053077 and CNS 0923158. T. J. Peters was also partially supported by an IBM Faculty Award and IBM Doctoral Fellowships. All statements here are the responsibility of the author, not of the National Science Foundation nor of IBM.


  1. 1.
    L.-E. Andersson, T.J. Peters, N.F. Stewart, S.M. Doney, Polyhedral perturbations that preserve topological form. Comput. Aided Geom. Des. 12(8), 785–799 (1995)CrossRefMATHGoogle Scholar
  2. 2.
    Anonymous, The protein data bank (2013),
  3. 3.
    H. Cassidy, T. Peters, K. Jordan, Dynamic computational topology for piecewise linear curves, in Proceedings of the Canadian Conference on Computational Geometry 2012, Charlottetown, 8–10 Aug 2012, pp. 279–284Google Scholar
  4. 4.
    T. Etiene, L.G. Nonato, C.E. Scheidegger, J. Tierny, T.J. Peters, V. Pascucci, R.M. Kirby, C.T. Silva, Topology verification for isosurface extraction. IEEE Trans. Vis. Comput. Graph. 18(6), 952–965 (2012)CrossRefGoogle Scholar
  5. 5.
    G. Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practicle Guide, 2nd edn. (Academic, San Diego, 1990)Google Scholar
  6. 6.
    R. Gal, O. Sorkine, N.J. Mitra, D. Cohen-Or, iwires: an analyze-and-edit approach to shape manipulation. ACM Trans. Graph. 28(3), 33:1–33:10 (2009)Google Scholar
  7. 7.
    H. Ilies, MRI: development of a gesture based virtual reality system for research in virtual worlds. NSF award 0923158 (2009),
  8. 8.
    K.E. Jordan, L.E. Miller, E.L.F. Moore, T.J. Peters, A.C. Russell, Modeling time and topology for animation and visualization with examples on parametric geometry. Theor. Comput. Sci. 405, 41–49 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    R.M. Kirby, C.T. Silva, The need for verifiable visualization. IEEE Comput. Graph. Appl. 28(5), 78–83 (2008)CrossRefGoogle Scholar
  10. 10.
    R. Kolodny, L. Guibas, M. Levitt, P. Koehl, Inverse kinematics in biology: the protein loop closure problem. J. Robot. Res. 24, 151–162 (2005)CrossRefGoogle Scholar
  11. 11.
    R. Kolodny, P. Koehl, L. Guibas, M. Levitt, Small libraries of protein fragments model native protein structures accurately. J. Mol. Biol. 323, 297–307 (2002)CrossRefGoogle Scholar
  12. 12.
    E. Martz, T.D. Kramer, World index of molecular visualization resources (2010),
  13. 13.
    E. Martz, T.D. Kramer, Molviz (2013),
  14. 14.
    L. Miller, E. Moore, T. Peters, A. Russell, Topological neighborhoods for spline curves: practice and theory, in Reliable Implementation of Real Number Algorithms: Theory and Practice, ed. by P. Hertling et al. Volume 5045 of LNCS (Springer, New York, 2008), pp. 149–161Google Scholar
  15. 15.
    D. Nairn, J. Peters, D. Lutterkort, Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon. Comput. Aided Geom. Des. 16, 613–631 (1999)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    G. Ramachandran, T. Schlick, Solvent effects on supercoiled DNA dynamics explored by Langevin dynamics simulations. Phys. Rev. E 51(6), 6188–6203 (1995)CrossRefGoogle Scholar
  17. 17.
    D. Russel, L. Guibas, Exploring protein folding conformations using spanners, in Pacific Symposium on Biocomputing, Hawaii, 2005, pp. 40–51Google Scholar
  18. 18.
    T. Schlick, Dynamic simulations of biomolecules,
  19. 19.
    T. Schlick, Modeling superhelical DNA: recent analytical and dynamic approaches. Curr. Opin. Struct. Biol. 5, 245–262 (1995)CrossRefGoogle Scholar
  20. 20.
    T. Schlick, Molecular Modeling and Simulation: An Interdisciplinary Guide (Springer, New York, 2002)CrossRefGoogle Scholar
  21. 21.
    T. Schlick, W.K. Olson, Trefoil knotting revealed by molecular dynamics simulations of supercoiled DNA. Science 21, 1110–1115 (1992)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Hugh P. Cassidy
    • 1
  • Thomas J. Peters
    • 1
  • Horea Ilies
    • 1
  • Kirk E. Jordan
    • 2
  1. 1.University of ConnecticutStorrsUSA
  2. 2.T.J. Watson Research Center, IBMCambridgeUSA

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