differential geometry

dori1123

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