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Abstract

We present a set of advanced techniques for the visualization of 2D Poincare maps. Since 2D Poincare maps are a mathematical abstraction of periodic or quasiperiodic 3D flows, we propose to embed the 2D visualization with standard 3D techniques to improve the understanding of the Poincare maps. Methods to enhance the representation of the relation $x\leftrightarrow{}P(x)$, e.g., the use of spot noise, are presented as well as techniques to visualize the repeated application of $P$, e.g., the approximation of $P$ as a warp function. It is shown that animation can be very useful to further improve the visualization. For example, the animation of the construction of Poincare map $P$ is inherently a proper visualization. During the paper we present a set of examples which demonstrate the usefulness of our techniques.

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BibTeX

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@techreport{Loeffelmann-1997-VPM,
  title =      "Visualizing Poincare Maps together with the
		underlying
               flow",
  author =     "Helwig L{\"o}ffelmann and Thomas Kucera and Meister Eduard
               Gr{\"o}ller",
  year =       "1997",
  abstract =   "We present a set of advanced techniques for the
               visualization of 2D Poincare maps.  Since 2D Poincare maps
               are a mathematical abstraction of periodic or quasiperiodic
               3D flows, we propose to embed the 2D visualization with
               standard 3D techniques to improve the understanding of the
               Poincare maps.  Methods to enhance the representation of the
               relation $x\leftrightarrow{}P(x)$, e.g., the use of spot
               noise, are presented as well as techniques to visualize the
               repeated application of $P$, e.g., the approximation of $P$
               as a warp function.  It is shown that animation can be very
               useful to further improve the visualization.  For example,
               the animation of the construction of Poincare map $P$ is
               inherently a proper visualization.  During the paper we
               present a set of         examples which demonstrate the
               usefulness of our techniques.",
  address =    "Favoritenstrasse 9-11/186, A-1040 Vienna, Austria",
  institution = "Institute of Computer Graphics and Algorithms, Vienna
               University of Technology",
  note =       "human contact: technical-report@cg.tuwien.ac.at",
  month =      mar,
  number =     "TR-186-2-97-06",
  keywords =   "Poincare maps, dynamical systems, visualization",
  URL =        "http://www.cg.tuwien.ac.at/research/publications/1997/Loeffelmann-1997-VPM/",
}