Differential Geom: Showing Hyperbolic Circle from Euclidean Circle

[dori1123]

Given \{(u,v)\inR^2:u^2+v^2<1\} with metric E = G =\frac{4}{(1-u^2-v^2)^2} and F = 0. How can I show that a Euclidean circle centered at the origin is a hyperbolic circle?

 

[Doodle Bob]

Note: rotations about the origin preserve this metric.

 

[dori1123]

Given \{(u,v)\inR^2:u^2+v^2<1\} with metric E = G =\frac{4}{(1-u^2-v^2)^2} and F = 0.
With a Euclidean circle centered at the origin with radius r, how can I find the hyperbolic radius by integrating \sqrt(E(u')^2+2Fu'v'+G(v')^2), what parametrized curve should I use?