Phase Portraits of Hyperbolic Geometry

Scott B. Lindstrom; Paul Vrbik

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arXiv · 1903.01309

Phase Portraits of Hyperbolic Geometry

Scott B. Lindstrom·Paul Vrbik

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Abstract

Phase plotting is a useful way of visualising functions on complex space. We reinvent the method in the context of hyperbolic geometry, and we use it to plot functions on various representative surfaces for hyperbolic space, illustrating with direct motions in particular. The reinvention is nontrivial, and we discuss the essential features for ensuring that such visualisations convey useful information. Our approach is to exploit conformal maps between representative surfaces, in order to uniquely represent the preimages of geodesics. Our considerations and methods are prototypical of what one might consider for adapting similar methods of visualisation in other contexts.

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Scott B. Lindstrom, Paul Vrbik. 2019-03-04. Phase Portraits of Hyperbolic Geometry. https://arxiv.org/abs/1903.01309

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