Animation aspects

Animation is a powerful approach to increase the dimensionality of visualization results. We found the following parameters of the modules suitable to be animated:

TRAJECTORY parameters  $\mathit{no}$ and  $\mathit{len}$ - Module TRAJECTORY generates a sequence  $\mathcal{O}$ of consecutive intersections of trajectory  $\mathcal{T}$ and Poincaré section  $\mathcal{S}$, i.e.,  $\mathcal{O}\!=\!\mathcal{T}$$\cap$ $\mathcal{S}$. Parameter  $\mathit{no}$ specifies how many intersections should be calculated. $\mathit{len}$ can be used to control $\left\vert\mathcal{O}\right\vert$ via the spatial length of  $\mathcal{T}$.

Animating one of these parameters, the construction of Poincaré map  p can be visualized. Furthermore the asymptotic behavior of  pⁿ(x), n$\to$$\infty$, can be investigated. This specific application of animation is capable of representing the inherent nature of Poincaré map  p.

TRAJECTORY parameters r and $\phi$ - Another pair of TRAJECTORY parameters, which is very well suited for animation, is ($\phi$, r). It encodes the initial condition  x of solution  $\mathcal{T}(\mathbf{x})$ in polar coordinates with respect to some arbitrary local coordinate system on  $\mathcal{S}$. In other words ($\phi$, r) specifies the starting point  x of sequence  $\left\{\mathbf{p}^j(\mathbf{x})\ \vert\ j\!\ge\!0\right\}$.

Animating these two parameters allows to investigate the development of arbitrary curves  $\mathcal{Q}$ within  $\mathcal{S}$. Such a curve  $\mathcal{Q}=(\mathcal{Q}_\phi(s), \mathcal{Q}_r(s))$ should be given as a parameterized subset of $\mathcal{S}$ with s as the parameter. Given such a curve  $\mathcal{Q}$, parameter s can be animated: The module TRAJECTORY takes $\mathbf{x}(s)=(\mathcal{Q}_\phi(s), \mathcal{Q}_r(s))$ as an initial condition for the generation of  $\left\{\mathbf{p}^j(\mathbf{x}(s))\ \vert\ j\!\ge\!0\right\}$. Initial condition  x(s) moves along curve  $\mathcal{Q}$, and simultaneously its long-term behavior  $\left\{\mathbf{p}^j(\mathbf{x}(s))\ \vert\ j\!\ge\!0\right\}$ is visualized.

WARP parameter n - The animation of WARP parameter n (number of applications) improves the expressiveness of the image warping approach. A sequence of images with consecutive applications of the warping function  w shows the overall behavior of  pⁿ, n$\to$$\infty$. Refer to Fig. 5.10 for three images resulting from consecutive applications of the warping function  w.

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