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The Klein model of the hyperbolic plane is a line-preserving map from $H^2$ to the disk.

Is there a model of the hyperbolic plane which is a line-preserving map from $H^2$ to $[0,1]^2$?

By line-preserving, I mean that geodesics in $H^2$ are mapped to line segments in the square.

This would be helpful in data visualization. The Poincaré model of the hyperbolic plane is often used as a way to display very large, complex graphs - because the hyperbolic plane has "more space" than the Euclidean plane, intuitively.

However, most computer screens are rectangular, not circular, and line segments are easier to draw than arcs, so this would be quite useful.

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    $\begingroup$ There is no such model. Every two elements of the boundary of the plane ie the line at infinity, must be connected by a unique geodesic. You can't draw a line from $(0,0)$ to $(1/2, 0)$ and have all of the points of the line be inside the interior of the square. $\endgroup$ – user24142 Jul 4 '15 at 2:01
  • $\begingroup$ @MikeBattaglia If you want to preserve angles instead of lines its possible though (the purple curves repersent straight lines in the hyperbolic plane). $\endgroup$ – PyRulez Jan 18 '18 at 5:21
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As user23142 mentioned there is no such model (in every model there should be unique lines between any two points on the boundary , and with a square that is not possible.

Maybe an alternative:

There is the Gans model ( https://en.wikipedia.org/wiki/Hyperbolic_geometry#The_Gans_Model ) which is more or less the hyperboloid model but then without the third coordinate, it uses the full euclidean plane, unfortunedly details needs a bit working on (still not sure what the function of the euclidean hyperbole is when you know two points where it goes trough. Also ideal points are outside the plane.

hope this helps

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