Conclusions

A clever person solves a problem. A wise person avoids it.
Albert Einstein (1879-1955)

Those are my principles. If you don't like them I have others.
Groucho Marx (1890-1977)

After this project on the visualization of dynamical systems, i.e., after years of research in this field, a few conclusions can be drawn. Maybe the most important insight was the following: it is useful to intuitively communicate dynamics by the use of direct visualization. It is also useful to first derive topology data and then, afterwards, do the visualization. It seems, however, to be most useful to combine both approaches: first, see whether characteristic elements can be identified by the use of dynamical system analysis. Then, visualize these topological structures and add direct visualization to include some intuitive hints for reading the abstract representation.

Another experience learned from this work is an important implication of the rather obvious fact `the bandwidth of the visual channel available for communication via visualization is limited to a certain extent': visualization research is not only working on the question `How to visualize certain information'; equally important is the question `What information should be visualized, or what part of, i.e., what shall be omitted, etc.'.

Usually it is not sufficient to provide high-quality software to users. Often the visualization expert necessarily is included within the process of visualization to generate useful results. The knowledge of how to map data to visual information is in many cases not intuitive, and has to be learned also. Either users have to be trained to use visualization, or visualization experts are to be included within the visualization process.

An interesting point for doing visualization of three-dimensional data, is that 3D is more than just an extension of 2D. New challenges wait in 3D, for instance, characteristic structures of dynamical systems that simply do not exist in 2D like saddle cycles or tori. The rendering step involves a projection from 3D into 2D. This confronts the visualization expert with problems that are significantly more complex than 2D to 3D extensions would produce. Occlusion, for example, is a problem of a kind, which simply does not exist in 2D.

A conclusion can be drawn which is not limited to scientific visualization, computer graphics, or even technical science: the optimal visualization technique obviously depends on what the user wants to see! Especially in the case of visualization, the questions of users often vary in a significant way. New ideas in this field often do not replace other already existing methods, but, rather enrich the assortment of possible views onto the user's data.

mailto:helwig@cg.tuwien.ac.at.