Skip to main content

Visualizing the Topology of Symmetric, Second-Order, Time-Varying Two-Dimensional Tensor Fields

  • Chapter
Visualization and Processing of Tensor Fields

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

We introduce the underlying theory behind degenerate points in 2D tensor fields to study the local field properties in the vicinity of linear and nonlinear singularities. The structural stability of these features and their corresponding separatrices are also analyzed. From here, we highlight the main techniques for visualizing and simplifying the topology of both static and time-varying 2D tensor fields.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Andronov, E. Leontovich, I. Gordon, and A. Maier. Qualitative Theory of Second-Order Dynamic Systems. Israel Program for Scientific Translations, 1973.

    Google Scholar 

  2. B. Cabral and L. Leedom. Imaging vector fields using line integral convolution. Computer Graphics (SIGGRAPH Proceedings), 27(4):263–272, 1993.

    Google Scholar 

  3. T. Delmarcelle. The Visualization of Second Order Tensor Fields. PhD Thesis. Stanford University, 1994.

    Google Scholar 

  4. T. Delmarcelle and L. Hesselink. The topology of symmetric, second-order tensor fields. In IEEE Visualization, pp. 140–147, 1994.

    Google Scholar 

  5. R. Dickinson. A unified approach to the design of visualization software for the analysis of field problems. Three-Dimensional Visualization and Display Techniques, 1083:173–180, 1989.

    Google Scholar 

  6. B. Gray. Homotopy Theory. An Introduction to Algebraic Topology. Pure and Applied Mathematics. Academic Press, 1975.

    Google Scholar 

  7. J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems and Linear Algebra. Springer, 1983.

    Google Scholar 

  8. J. Helman and L. Hesselink. Representation and display of vector field topology in fluid flow flow data sets. IEEE Computer, 22(8):144–152, 1989.

    Google Scholar 

  9. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery. Numerical Recipes in C, Second Edition. Cambridge University Press, 1992.

    Google Scholar 

  10. G. Scheuermann, B. Hamann, K. Joy, and W. Kollmann. Visualizing local topology. Journal of Electronic Imaging, 9(4):356–367, 2000.

    Article  Google Scholar 

  11. X. Tricoche, G. Scheuermann, and H. Hagen. Tensor topology tracking: A visualization method for time-dependent 2D symmetric tensor fields, 2001.

    Google Scholar 

  12. X. Tricoche, G. Scheuermann, and H. Hagen. Scaling the topology of symmetric second order tensor fields. pp. 171–184, 2003.

    Google Scholar 

  13. X. Tricoche, G. Scheuermannn, and H. Hagen. Topology simplification of symmetric, second order 2D tensor fields. pp. 275–292, 2003.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Tricoche, X., Zheng, X., Pang, A. (2006). Visualizing the Topology of Symmetric, Second-Order, Time-Varying Two-Dimensional Tensor Fields. In: Weickert, J., Hagen, H. (eds) Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31272-2_13

Download citation

Publish with us

Policies and ethics