Additional extensions

**Figure 4.13:** (a) Stream arrows shifted out of the stream surface plus anisotropic spot-noise. (b) Stream arrows outlines represented as 3D tubes and color coding of velocity magnitude.
$\framebox[\textwidth]{ \begin{tabular}{.93\linewidth}{@{}@{\extracolsep{\fill}... ...=62mm]{pics/tubearrows.ps} \\ {\small{}(a)} & {\small{}(b)} \end{tabular} }$

**Figure 4.14:** Misleading shifted stream arrows (bright: integrated stream arrow).
$\framebox[\textwidth]{ \includegraphics[scale=1]{figs/shifted-vs-real2.eps} }$

After the geometric separation of stream arrows has been performed two possibilities of extending stream arrows into 3D have been realized. One possibility is to shift the separated stream arrows slightly in a direction perpendicular to the remaining stream surface portions. To avoid any confusion - one could interpret the stream arrows as local solutions of the dynamical system, which is certainly not the case in most situations - the shifted stream arrows are connected with the remaining parts of the stream surface by semi-transparent patches. Generating 3D arrows using this technique improves the perception of spatial location and orientation of the stream surface (see Fig. 4.13(a)). In this image stream arrows are shown with anisotropic spot noise [87] illustrating stream lines and time lines.

Although this method of shifting stream arrows slightly out of the stream surface often generates useful results, it has its disadvantages in other cases. Especially if nearby stream surfaces are far from being coplanar to the enhanced stream surface - a stream surface as a stable system solution can be imagined as such a case (see Fig. 4.14) - this technique might give a wrong impression.

Therefore, another extension of the underlying dynamical system allows to build up the shifted stream arrow as a local solution. This is achieved by starting a new stream surface at the shifted tail of the stream arrow and cutting the arrow out of this stream surface. This technique will allow to represent the dynamical system not just at the stream surface, but also in its vicinity. Other shapes of 3D stream arrows, e.g., a tent-like shape or variations of the 2D base shape, also have been investigated.

Another 3D extension is the representation of the separating outline as a 3D tube (see Fig. 4.13(b)). Using this approach stream arrows can be realized without removing any part of the stream surface. Almost all of the advantages of the stream arrows technique are preserved as, e.g., indication of flow direction and local velocity.

Further extensions to the stream arrows technique might be 3D stream arrows which can be seen as glyphs with a set of free parameters. Scalar data such as velocity and helicity as well as vector data, for example, vorticity, may be mapped to a stream arrows parameter. Not only 3D shape attributes of stream arrows could be interpreted as glyph parameters, but also size, 2D shape, opacity, and color.

Related to the hierarchical stream arrows texture we worked on other extensions concerning the placement of stream arrows. One idea is to randomly position stream arrows on stream surfaces. Another idea is to use local stream surface attributes for the location of stream arrows. Surface curvature could be used for this purpose. This has already been shown to be useful [34]. The opacity of stream arrows could modulated according to the curvature of the stream surface such that rather opaque arrows are place in regions of high surface curvature. Finally we think of including viewing parameters to determine optimal placements of stream arrows. This idea was again inspired by the book of Abraham and Shaw [1], where these parameters are included in the hand-drawn illustrations as well. Location as well as length and width of arrows could be chosen in a way that occlusion of details behind is diminuished.