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More of the Stretched Model

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More of the Stretched Model

This continues my first post about a stretched model of the hyperbolic plane: plus.google.com/+RoiceNelson/posts/ESxR6WRvS9z

I've added pictures of various hyperbolic tilings (like the {4,∞} below) to the album: goo.gl/photos/QZvK9FyJQp1jGTbd9

I've also been able to reason out some properties of this model.

- Geodesics are half-hyperbolas with a center at the origin.
- Lines through the origin are degenerate hyperbolas and are geodesic as well.
- Hyperbolic circles with finite radius appear as circles if origin-centered, or ellipses otherwise.
- Horocycles are parabolas.

But perhaps the most useful feature of this model is that equidistant curves are also equidistant in the euclidean sense! For example, take a geodesic and simply offset it to or away from the origin and you have an equidistant curve of the hyperbolic plane. This also explains my observation in the first post that tile widths appeared to remain constant even though stretched in the radial…

Stretched Model of the Hyperbolic Plane

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Stretched Model of the Hyperbolic Plane

David Eppstein shared some very cool fisheye disk models of euclidean geometry recently.
plus.google.com/u/0/100003628603413742554/posts/5yz81rwZ4Q3

I generated some images of euclidean tilings in one of the models, as well as images for spherical tilings in a similar disk model.
goo.gl/photos/8bkvfQNLGtHP4p6n9

Then I realized we could apply exactly the opposite transformation to the Poincaré disk model of the hyperbolic plane to get a model covering the entire complex plane. That's what these images show for the {7,3} hyperbolic tiling (the dual {3,7} tiling is overlaid as well). Tiles stretch out dramatically in the radial direction, but seem to retain constant tangential width.

This non-conformal model may not be terribly useful, but there are some interesting patterns in the pictures. Clear fractal structure shows up in radial slices, and zooming the entire image by certain ratios makes this especially evident. The Fibonacci numbers appeared w…