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Differential geometry

by dori1123
Tags: differential, geometry
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dori1123
#1
Dec14-08, 03:43 PM
P: 11
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?
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Doodle Bob
#2
Dec15-08, 04:32 AM
P: 255
Note: rotations about the origin preserve this metric.
dori1123
#3
Dec16-08, 11:28 PM
P: 11
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?


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