Visualization of 3D Maps

A second application which makes use of the capabilities of RTVR is a visualization and analysis system for 3D dynamical systems (discrete non-invertible maps [38]) [4]. The application is used to analyze and visualize structures and events (bifurcations) within the phase-space of the systems.

Objects of interest are attractors (often complex or even chaotic), their basins of attraction (i.e., the set of all system states which are attracted by them in the limit) [20] and critical surfaces (surfaces which separate regions of phase space with different properties). Up to now, the most commonly used visualization technique for investigating this type of structures within phase space are simple planar cross-sections. Although they well depict the boundaries of a basin of attraction, for example, it is difficult to convey the 3D shape of the basin and even more the shape of a complex attractor by just viewing a set of sections (figure 7.2). The ability to view the objects in 3D with the option to interactively change viewing parameters and to manipulate objects, for example, to obtain cross sections, is of great help during investigation of the data. As attractors are contained within their basins of attraction, and frequently also basins are nested within each other, efficient tools to deal with occlusion are required, like the ability of object-wise tuning of opacity, render mode and lighting model which are provided by RTVR. The feature of mixing MIP with other compositing methods has proven to be especially useful for visualizing chaotic attractors. Their complex internal structure is well captured using MIP while producing little occlusion. At the same time, the attractor's basin of attraction can be rendered as a shaded surface.

Figure 7.2: Attractor, basin, and cross-sections through them: Deriving the 3D shape of the object just from sections is difficult.

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