Visualizing characteristic structures

Besides critical points, characteristic trajectories belong to the most important elements of flow topology. Similar to the visualization of critical points, it is useful to combine the visualization of abstract topology with direct visualization also in the case of characteristic curves.

In Chapter 7 the `thread of streamlets' technique is proposed to selectively place streamlets in the vicinity of characteristic trajectories. A certain probability function is used to randomly choose seed points for a numerous set of streamlets, i.e., a thread of streamlets. Using constant flow as a reference model a relation between number of streamlets, integration length and seed point distribution is derived. The actual spatial arrangement, shape, and length of the streamlets communicates the flow velocity near the base trajectory and local information about convergence/divergence, and rotation.

See Figs. 7.3(b) and 7.4(b) for two examples where characteristic trajectories belonging to the critical points of a linear dynamical systems are visualized using this technique. Compared to the visualization of topological information as shown in Fig. 7.3(a) advantages and disadvantages can be found:

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the dynamics related to the topological structure is more intuitively displayed using threads of streamlets. Having just topological information, it is usually impossible (or quite difficult, at least) to imagine, how a trajectory would evolve starting from a specific state near the critical point.

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the set of visual cues necessary for visualization is less dense using threads of streamlets than displaying topological information only. This reduces the remaining bandwidth of the visual channel used for visualization. The combination of this technique with other visualization results might cause problems due to visual overload.

An important topic when using 1D elements for visualization in 3D. As 1D elements in 3D have an infinite number of normals (opposed to surfaces with one pair of normals), more elaborate techniques must be used. A physically correct solution would require to evaluate an integral over all directions within the normal plane. Simple but useful approximations are available that allow efficient rendering of 1D elements in 3D.

Besides line shading the problem of line shadowing should be addressed. Again, a correct solution would require costly computations similar to volume rendering. Instead, depth cueing can be used as a rough approximation of line shadowing. Combining line shading and line shadowing, or, at least, some approximations of these aspects, allows to render useful results with the `threads of streamlets' technique.

Threads of streamlets are extended in several directions. Color coding is used for communicating local properties. Arrow heads can be attached to the streamlets, also providing parameters for visualization (see Figs. 7.4(a) and 7.5(a)).

The entire project on advanced visualization techniques concerning dynamical systems, including the sub-projects stream arrows, visualization based on Poincaré maps, visualizing critical points, and the visualization of characteristic structures, shares a common implementation platform, called DynSys3D. The system itself is based on AVS, which is a general purpose visualization system, featuring a data-flow model together with a scheme to build modules. A few design features characterize DynSys3D:

Principal components -

Each visualization module developed within the scope of DynSys3D consists of, at least, three principal components: DYNAMICAL SYSTEM, NUMERICAL INTEGRATOR, and VISUALIZATION TECHNIQUE. Rather strict interface specifications between the principal components allow to freely combine different components with each other. For example, it is very easy to reuse a new, integration technique with other existing visualization modules.

Separating Visualization and User Interface -

Each instance of the component VISUALIZATION TECHNIQUE is itself composed of a core providing the visualization, and a shell adding the user interface which is visible through the AVS environment. This separation allows to re-use already implemented visualization techniques like stream line integration within other, maybe more visualization methods like processing a rake of streamlets.

Focus on Visualization -

Instead of coming up with general solutions, in related fields that do not directly contribute to visualization research, certain problems like the identification of critical points or the evaluation of the Jacobian matrix where decided to be solved individually instead of coming up with a general solver for all cases. A dynamical system, for example, is required to know about its critical points instead of using some general piece of code which is able to find critical points for any dynamical system.

Interactivity -

One major aim of DynSys3D was to be a platform for rapid development of new ideas in the field of visualizing dynamical systems. To examine visualization results in a reasonable time it is necessary to prevent the rendering pipeline of being overloaded with too many geometry elements. The representation of the geometry being generated by DynSys3D modules is parameterized so that coarse approximations of the calculated geometry can be used for investigation. Decoupling numerical simulation accuracy and output geometry complexity allows to compute accurate solutions, but use only a small number of triangles for investigation.

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