Visualizing Poincaré Maps together with the Underlying Flow

We present a set of advanced techniques for the visualization of 2D Poincare maps. Since 2D Poincare maps are a mathematical abstraction of periodic or quasi-periodic 3D flows, we propose to embed the 2D visualization with standard 3D techniques to improve the understanding of the Poincare maps. Methods to enhance the representation of the relation x ↔ P(x), e.g., the use of spot noise, are presented as well as techniques to visualize the repeated application of P, e.g., the approximation of P as a warp function. It is shown that animation can be very useful to further improve the visualization. For example, the animation of the construction of Poincare map P is a very intuitive visualization. During the paper we present a set of examples which demonstrate the usefulness of our techniques.