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?