A Poincaré section is represented as a circular patch that is rendered semi-transparently. Therefore visualization icons before as well as behind are visible. The implementation of this technique is based on the existence of a base cycle within the 3D flow. Cycle defines the center of Poincaré section . The cycle is rendered as an opaque tube through 3D phase space together with a sphere at where the base cycle intersects the Poincaré section. See Fig. 5.4 or 5.6 for an image where , , and the intersection can be easily detected.
The module which generates this Poincaré map visualization takes care that initially a useful view point is chosen. is viewed under an angle which is little bit less than so that both the Poincaré map and the intersecting base cycle are easily recognizable (see Fig. 5.4). We also found it very useful to provide a relative placement capability such that the user can move the Poincaré section easily around the base cycle . The actual position of the map on the cycle is specified by a value between 0 and 1.
Additionally we suggest some more elaborated 3D extensions to Poincaré map visualization. Module TRAJECTORY, for example, generates either the entire trajectory which constructs the orbit , only short parts of this trajectory in the vicinity of , or just the spheres representing the consecutive intersections.
Drawing the entire trajectory that generates orbit , for example, helps to relate consecutive points on mentally (see Figs. 5.1 and 5.7). Long trajectories, however, may clutter the image.
Using only short parts of the trajectory near avoids the problem of visual clutter (see Fig. 5.12). Assume a periodic 3D system which exhibits some high frequency oscillation parallel to the rotational axis of the flow (see Fig. 5.12(a)). If the frequency (measured in oscillations per one entire revolution of the carrying periodic system around the rotational axis of the flow) is an integer number, the resulting Poincaré map is not affected by this oscillation at all. Results are the same as for a system without the modulated frequency. Compare Fig. 5.12(a) and 5.12(b). The difference between both systems which is not apparent in the Poincaré maps is visible through the embedding of the Poincaré sections in the 3D flow.
Both images in Fig. 5.12 are generated with techniques
already discussed previously in this chapter. Spot noise is used to
represent the entirety of
p, whereas white and red spheres are
used to visualize three
sequences
.
The
sequence starting near the origin of phase space is enhanced by
short parts of the constructing flow trajectory. This enhancement
is necessary to visualize the differences between both systems.
Another technique for the investigation of Poincaré maps
is realized as module SEEDLINE. By parameters r,
,
and
a line segment
of
length
is specified within
that is
perpendicular to the vector connecting
and the mid-point of
.
The length of this vector is specified by parameter
.
Parameters
and r are the polar coordinates of
one end-point of this line segment with respect to its
mid-point. SEEDLINE can be used to generate a
stream surface [33] or a rag of stream lines
alternatively (see figure on page ). In addition to the
stream surface, the
flow trajectory, which constructs a certain sequence of consecutive
applications of
p, was visualized by the use of a green tube.
Other techniques discussed previously in this chapter have been used
for visualizing Poincaré section
.