Stream arrows

The direct extension to stream lines in 2D are stream surfaces in 3D. A one-dimensional set of initial conditions is developed through phase space by the use of local integration of the flow data - a stream surface is constructed. Since three-dimensional space is much more difficult to investigate than 2D, some disadvantages are given for the use of stream surfaces. The projection of 3D data into image space and the use of large-scale surface structures (stream surfaces), often cause occlusion to become a severe problem - important parts of the visualization cues may be hidden from the viewer.

Stream arrows, i.e., the use of a special texture featuring semi-transparent arrows within the stream surface, enhances the visualization of three-dimensional flow by several means:

• The problem of occlusion is diminished, as parts of the stream surface are modeled semi-transparently and the viewer is able to perceive visualization cues placed behind. See Fig. 4.3 for a typical image featuring stream arrows.
• With stream arrows orientation, direction and velocity of the flow is shown additional to the spatial extent of the integral surface. Even convergence/divergence becomes visible by the use of stream arrows.

Extensions to stream arrows are:

• Anisotropic spot noise is combined with stream arrows to depict stream lines as well as time lines simultaneously on the stream surface (see Fig. 4.11).
• Stream arrows can be extended into 3D. For instance, the vicinity of a stream surface is visualized together with the stream surface. In the case of stable (stream) surface-sets the behavior near such a surface often is very important to be understood. The integrated stream arrow in Fig. 4.14 shows the attracting character of the stream surface.

The procedure for generating stream arrows is rather straight forward. First, a stream surface is calculated for a specific set of initial conditions. Texture coordinates are assigned to each vertex using the following principle: A 1D parameterization of the set of initial conditions, for instance, the arc-length parameterization, is assumed. All surface points lying on one stream line inherit the 1D parameter of the associated initial condition as u-coordinate. The v-coordinate of a surface point is the integration time, assuming the initial condition to have time zero (see Fig. 4.5).

Next, the stream arrows texture is defined. Vertical lines in texture space are correlated to stream lines, horizontal lines correspond to time lines within the stream surface. In the implementation the texture is defined on the basis of a base tile representing one arrow and a tiling mechanism generating implicitly as many stream arrows as necessary. Instead of one single stream arrows texture, an entire stack of textures can be used (hierarchical stream arrows). This eliminates some problems with the standard stream arrows technique in cases of great local divergence/convergence. See Fig. 4.8 for an illustration of this extension.

There are (at least) two possibilities to realize stream arrows within stream surfaces. One is to specify an alpha-texture for the surface element, and let the renderer care about semi-transparency. Another technique is to geometrically segment the stream surface into three separate triangle sets, one for the opaque parts, the border elements (1D), and the semi-transparent parts. An efficient segmentation algorithm is described in Sect. 4.3.

Anisotropic spot noise is generated using the same texture coordinates specification as used for the stream arrows and again exploiting the correlation between u-/v-lines and stream/time lines. A cyclic texture is generated in texture space. Constant flow along u-lines is assumed and an anisotropic spot is used to emphasis stream and time lines, simultaneously. See Fig. 4.10 for an (enlarged) image of the spot and the resulting texture. Mapping such an anisotropic spot noise texture to the stream surface illustrates stream and time lines.

Another approach to diminish the problem of occlusion is to use selective cuts through the visualization model. Parts in front of others, also important parts of the visualization are rendered almost transparently to allow insight within the model. Animation is used to move cut planes and help the viewer to understand the cutting operation performed. See Fig. 4.12 for two images out of an animation where the cut plane was moved through the model. The intersection of cut plane and stream surface was enhanced by white tubes.

In addition to selective cuts there are several other points in the stream arrows technique, where animation easily and useful can be hooked in. Arrows can be move over the stream surface into the direction of flow, the initial conditions may be altered within an animation. Viewpoint animation also improves the perceptibility of visualization models generated using the stream arrows technique.

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