We study parallel redrawing graphs: graphs embedded on moving point sets in such a way that edges maintain their slopes all throughout the motion. The configuration space of such a graph is of an oriented-projective nature, and its combinatorial structure relates to rigidity theoretic parameters of the graph. A special type of kinetic structure emerges, whose events can be analyzed combinatorially. Of particular interest are those planar graph\uds which maintain non-crossing edges throughout the motion. Our main result is that they are (essentially) pseudo-triangulation mechanisms. These kinetic graph structures have potential applications in morphing of more complex shapes than just simple polygons.\ud
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