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.