2.3. Advanced SphereTufts

This technique is the most recent of the five fixed points visualization techniques developed in this project. Here the SphereTufts have been enhanced by some new features, which are the following:

• The Sphere is approximated by a n-times subdivided Icosaeder, the number of subdivisions can be chosen by the user. Each Vertex of this subdivided Icosaeder is taken as a startpoint for the integration, so the distribution around the fixed point is no more stochastic, but rather evenly distributed.
• A classification process for each vertex is applied, which determines, if the vertex is in an interesting region on the "sphere" around the fixed points. For details have a look at some some lines further down.
• All triangles containing at least one vertex which was classified as interesting, are adaptively subdivided, and then every vertex of all newly created triangles are integrated and classified again. This adaptive subdivision process is n-times repeated, again given by the user.
• This leads to the fact, that in the more interesting regions, the subdivision is finer, and more streamlets are renderd in these regions, so that the interesting directions come more clearly visible, while not being occluded by too many not so important streamlets in the other directions.

What is this classifications of the vertices for, and how does it work?

What we wanted to do with this last approach (the Advanced Spheretufts approach), was to put more weight on the visualization of the interesting directions of the "flow" around the fixed point. A fixed point, that has for example three eigenvectors, with the eigenvalues 1, 10, and 100 is a repellor, that repells in the three directions of the eigenvectors with different speed, according to the eignvalues.

But when integrating from evenly distributed points over the sphere around a fixed point, there is quite often one of these directions dominating the others so much, that you can't see, them, but only the dominant direction. This approach tries to overcome this problem by subdividing only in the directions of the eigenvectors (the interesting directions) more accurately, so that first of all these directions are more clearly visible, because they are not occluded by so many not so important streamlets. Secondly, by refining the interesting directions (eigenvector directions), you get with every subdivision step more and more accurate results for seeing these directions (check the diffrences in the pictures below between the SphereTufts and the Advanced SphereTufts approach). Especially the not so dominant directions are found more accurately, as they could be shown by simply starting one single streamlet at the one startpoint on the surface of the sphere, where the eigenvector should pass through (compare this with the CharDirs approach described in the next section).

Attention: The implementation of this technique has not been completed, and therefore the results are no final results!